{"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := 3. G#M := 3. H#I := F#J / G#M. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := 3. G#M := 3. H#I := F#J / G#M.", "query": "H#I", "answer": 1, "cot": "F#J = 3 -> F#J = 3\nG#M = 3 -> G#M = 3\nH#I = F#J / G#M\n = 3 / 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => H#I = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#J": 1, "G#M": 1, "H#I": 2}, "id": "mod7_arith_d2_0000"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#K := E#N - J#O. J#O := 4. E#N := 4. G#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#K := E#N - J#O. J#O := 4. E#N := 4.", "query": "G#K", "answer": 0, "cot": "E#N = 4 -> E#N = 4\nJ#O = 4 -> J#O = 4\nG#K = E#N - J#O\n = 4 - 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => G#K = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#N": 1, "J#O": 1, "G#K": 2}, "id": "mod7_arith_d2_0001"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := 4. L#L := 4. J#N := 3. L#K := 4. I#P := 5 * L#K - J#N. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := 4. L#L := 4. J#N := 3. L#K := 4. I#P := 5 * L#K - J#N.", "query": "I#P", "answer": 3, "cot": "J#N = 3 -> J#N = 3\nL#K = 4 -> L#K = 4\nI#P = 5 * L#K - J#N\n = 5 * 4 - 3\n 5 * 4 = (5 × 4) mod 7 = 6\n 6 - 3 = (6 - 3) mod 7 = 3\n => I#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#P": 1, "J#N": 1, "L#K": 1, "L#L": 1, "I#P": 2}, "id": "mod7_arith_d2_0002"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#J := 3. E#L := J#J. H#P := 5. K#M := 6. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#J := 3. E#L := J#J. H#P := 5. K#M := 6.", "query": "E#L", "answer": 3, "cot": "J#J = 3 -> J#J = 3\nE#L = J#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#M": 1, "H#P": 1, "J#J": 1, "E#L": 2}, "id": "mod7_arith_d2_0003"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 2. K#I := 1. H#L := 1. H#I := F#K. F#K := 3. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 2. K#I := 1. H#L := 1. H#I := F#K. F#K := 3.", "query": "H#I", "answer": 3, "cot": "F#K = 3 -> F#K = 3\nH#I = F#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#K": 1, "H#L": 1, "K#I": 1, "J#N": 1, "H#I": 2}, "id": "mod7_arith_d2_0004"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := G#K. E#O := 5. G#K := 5. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := G#K. E#O := 5. G#K := 5.", "query": "E#L", "answer": 5, "cot": "G#K = 5 -> G#K = 5\nE#L = G#K = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#K": 1, "E#O": 1, "E#L": 2}, "id": "mod7_arith_d2_0005"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#M := 3. I#O := 5. L#I := 3. G#P := 2. F#N := F#M. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#M := 3. I#O := 5. L#I := 3. G#P := 2. F#N := F#M.", "query": "F#N", "answer": 3, "cot": "F#M = 3 -> F#M = 3\nF#N = F#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#M": 1, "L#I": 1, "I#O": 1, "G#P": 1, "F#N": 2}, "id": "mod7_arith_d2_0006"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := 6. G#I := K#I. J#M := 3. G#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := 6. G#I := K#I. J#M := 3.", "query": "G#I", "answer": 6, "cot": "K#I = 6 -> K#I = 6\nG#I = K#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#M": 1, "K#I": 1, "G#I": 2}, "id": "mod7_arith_d2_0007"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := 6. H#P := H#I * F#P. H#I := 2. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := 6. H#P := H#I * F#P. H#I := 2.", "query": "H#P", "answer": 5, "cot": "F#P = 6 -> F#P = 6\nH#I = 2 -> H#I = 2\nH#P = H#I * F#P\n = 2 * 6\n 2 * 6 = (2 × 6) mod 7 = 5\n => H#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#P": 1, "H#I": 1, "H#P": 2}, "id": "mod7_arith_d2_0008"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := 1. F#J := 2. K#O := 2. H#J := E#K. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := 1. F#J := 2. K#O := 2. H#J := E#K.", "query": "H#J", "answer": 1, "cot": "E#K = 1 -> E#K = 1\nH#J = E#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#K": 1, "K#O": 1, "F#J": 1, "H#J": 2}, "id": "mod7_arith_d2_0009"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := 4. G#J := H#O * E#I. E#I := 4. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := 4. G#J := H#O * E#I. E#I := 4.", "query": "G#J", "answer": 2, "cot": "H#O = 4 -> H#O = 4\nE#I = 4 -> E#I = 4\nG#J = H#O * E#I\n = 4 * 4\n 4 * 4 = (4 × 4) mod 7 = 2\n => G#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#O": 1, "E#I": 1, "G#J": 2}, "id": "mod7_arith_d2_0010"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#M := 2. E#O := 1. K#P := E#O * F#M. E#P := 1. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#M := 2. E#O := 1. K#P := E#O * F#M. E#P := 1.", "query": "K#P", "answer": 2, "cot": "E#O = 1 -> E#O = 1\nF#M = 2 -> F#M = 2\nK#P = E#O * F#M\n = 1 * 2\n 1 * 2 = (1 × 2) mod 7 = 2\n => K#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#O": 1, "F#M": 1, "E#P": 1, "K#P": 2}, "id": "mod7_arith_d2_0011"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := 6. L#L := 1. L#K := L#L / 4 - J#O. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := 6. L#L := 1. L#K := L#L / 4 - J#O.", "query": "L#K", "answer": 3, "cot": "L#L = 1 -> L#L = 1\nJ#O = 6 -> J#O = 6\nL#K = L#L / 4 - J#O\n = 1 / 4 - 6\n 1 / 4 = (1 × 4⁻¹) mod 7 = 2\n 2 - 6 = (2 - 6) mod 7 = 3\n => L#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#L": 1, "J#O": 1, "L#K": 2}, "id": "mod7_arith_d2_0012"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := 6. L#K := L#M / L#I. L#M := 3. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := 6. L#K := L#M / L#I. L#M := 3.", "query": "L#K", "answer": 4, "cot": "L#I = 6 -> L#I = 6\nL#M = 3 -> L#M = 3\nL#K = L#M / L#I\n = 3 / 6\n 3 / 6 = (3 × 6⁻¹) mod 7 = 4\n => L#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#I": 1, "L#M": 1, "L#K": 2}, "id": "mod7_arith_d2_0013"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#K := 2 - F#J. H#K := 4. I#O := 6. F#J := 1. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#K := 2 - F#J. H#K := 4. I#O := 6. F#J := 1.", "query": "J#K", "answer": 1, "cot": "F#J = 1 -> F#J = 1\nJ#K = 2 - F#J\n = 2 - 1\n 2 - 1 = (2 - 1) mod 7 = 1\n => J#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#K": 1, "I#O": 1, "F#J": 1, "J#K": 2}, "id": "mod7_arith_d2_0014"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#P := 2. G#P := 6 / I#J + K#J. I#J := 2. K#J := 3. H#M := 2. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#P := 2. G#P := 6 / I#J + K#J. I#J := 2. K#J := 3. H#M := 2.", "query": "G#P", "answer": 6, "cot": "I#J = 2 -> I#J = 2\nK#J = 3 -> K#J = 3\nG#P = 6 / I#J + K#J\n = 6 / 2 + 3\n 6 / 2 = (6 × 2⁻¹) mod 7 = 3\n 3 + 3 = (3 + 3) mod 7 = 6\n => G#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#P": 1, "I#J": 1, "K#J": 1, "H#M": 1, "G#P": 2}, "id": "mod7_arith_d2_0015"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := J#J. H#L := 4. K#P := 3. K#I := 3. J#J := 6. L#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := J#J. H#L := 4. K#P := 3. K#I := 3. J#J := 6.", "query": "L#P", "answer": 6, "cot": "J#J = 6 -> J#J = 6\nL#P = J#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#P": 1, "J#J": 1, "H#L": 1, "K#I": 1, "L#P": 2}, "id": "mod7_arith_d2_0016"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#P := 5. H#M := 6. G#I := 5. K#P := H#P + 2. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#P := 5. H#M := 6. G#I := 5. K#P := H#P + 2.", "query": "K#P", "answer": 0, "cot": "H#P = 5 -> H#P = 5\nK#P = H#P + 2\n = 5 + 2\n 5 + 2 = (5 + 2) mod 7 = 0\n => K#P = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#M": 1, "H#P": 1, "G#I": 1, "K#P": 2}, "id": "mod7_arith_d2_0017"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#I := I#P + K#M * F#N. I#P := 1. F#N := 2. K#M := 6. L#N := 4. G#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#I := I#P + K#M * F#N. I#P := 1. F#N := 2. K#M := 6. L#N := 4.", "query": "G#I", "answer": 0, "cot": "I#P = 1 -> I#P = 1\nF#N = 2 -> F#N = 2\nK#M = 6 -> K#M = 6\nG#I = I#P + K#M * F#N\n = 1 + 6 * 2\n 1 + 6 = (1 + 6) mod 7 = 0\n 0 * 2 = (0 × 2) mod 7 = 0\n => G#I = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#N": 1, "I#P": 1, "F#N": 1, "K#M": 1, "G#I": 2}, "id": "mod7_arith_d2_0018"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := 6. I#P := 6. I#J := G#P / I#P. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := 6. I#P := 6. I#J := G#P / I#P.", "query": "I#J", "answer": 1, "cot": "I#P = 6 -> I#P = 6\nG#P = 6 -> G#P = 6\nI#J = G#P / I#P\n = 6 / 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => I#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#P": 1, "G#P": 1, "I#J": 2}, "id": "mod7_arith_d2_0019"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := 5. F#J := H#M / H#O. H#M := 1. G#J := 3. E#L := 4. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := 5. F#J := H#M / H#O. H#M := 1. G#J := 3. E#L := 4.", "query": "F#J", "answer": 3, "cot": "H#O = 5 -> H#O = 5\nH#M = 1 -> H#M = 1\nF#J = H#M / H#O\n = 1 / 5\n 1 / 5 = (1 × 5⁻¹) mod 7 = 3\n => F#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#O": 1, "G#J": 1, "E#L": 1, "H#M": 1, "F#J": 2}, "id": "mod7_arith_d2_0020"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#K := 6. L#L := 3. H#P := L#L - L#K. J#K := 1. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#K := 6. L#L := 3. H#P := L#L - L#K. J#K := 1.", "query": "H#P", "answer": 4, "cot": "L#K = 6 -> L#K = 6\nL#L = 3 -> L#L = 3\nH#P = L#L - L#K\n = 3 - 6\n 3 - 6 = (3 - 6) mod 7 = 4\n => H#P = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#K": 1, "L#L": 1, "J#K": 1, "H#P": 2}, "id": "mod7_arith_d2_0021"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := 3. J#O := 6. I#O := J#O / 4. J#M := 6. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := 3. J#O := 6. I#O := J#O / 4. J#M := 6.", "query": "I#O", "answer": 5, "cot": "J#O = 6 -> J#O = 6\nI#O = J#O / 4\n = 6 / 4\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n => I#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#M": 1, "J#O": 1, "I#J": 1, "I#O": 2}, "id": "mod7_arith_d2_0022"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#O := 1. J#L := 4 / F#O + J#K. G#I := 1. J#K := 3. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#O := 1. J#L := 4 / F#O + J#K. G#I := 1. J#K := 3.", "query": "J#L", "answer": 0, "cot": "J#K = 3 -> J#K = 3\nF#O = 1 -> F#O = 1\nJ#L = 4 / F#O + J#K\n = 4 / 1 + 3\n 4 / 1 = (4 × 1⁻¹) mod 7 = 4\n 4 + 3 = (4 + 3) mod 7 = 0\n => J#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#I": 1, "J#K": 1, "F#O": 1, "J#L": 2}, "id": "mod7_arith_d2_0023"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#L := 6. G#O := 3. I#J := 3. H#I := 3 / I#J. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#L := 6. G#O := 3. I#J := 3. H#I := 3 / I#J.", "query": "H#I", "answer": 1, "cot": "I#J = 3 -> I#J = 3\nH#I = 3 / I#J\n = 3 / 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => H#I = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#O": 1, "I#J": 1, "H#L": 1, "H#I": 2}, "id": "mod7_arith_d2_0024"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := 1. F#N := 6. K#P := 4. J#J := K#P + K#L + 5. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := 1. F#N := 6. K#P := 4. J#J := K#P + K#L + 5.", "query": "J#J", "answer": 3, "cot": "K#L = 1 -> K#L = 1\nK#P = 4 -> K#P = 4\nJ#J = K#P + K#L + 5\n = 4 + 1 + 5\n 4 + 1 = (4 + 1) mod 7 = 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => J#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#N": 1, "K#L": 1, "K#P": 1, "J#J": 2}, "id": "mod7_arith_d2_0025"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#M := I#P - G#J. G#J := 4. I#P := 6. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#M := I#P - G#J. G#J := 4. I#P := 6.", "query": "E#M", "answer": 2, "cot": "G#J = 4 -> G#J = 4\nI#P = 6 -> I#P = 6\nE#M = I#P - G#J\n = 6 - 4\n 6 - 4 = (6 - 4) mod 7 = 2\n => E#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#J": 1, "I#P": 1, "E#M": 2}, "id": "mod7_arith_d2_0026"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#I := 2. E#I := L#I. L#I := 4. H#J := 4. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#I := 2. E#I := L#I. L#I := 4. H#J := 4.", "query": "E#I", "answer": 4, "cot": "L#I = 4 -> L#I = 4\nE#I = L#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"L#I": 1, "H#J": 1, "G#I": 1, "E#I": 2}, "id": "mod7_arith_d2_0027"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := 5. L#L := 1. H#O := E#K - F#L / 3. F#L := 1. H#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := 5. L#L := 1. H#O := E#K - F#L / 3. F#L := 1.", "query": "H#O", "answer": 6, "cot": "E#K = 5 -> E#K = 5\nF#L = 1 -> F#L = 1\nH#O = E#K - F#L / 3\n = 5 - 1 / 3\n 5 - 1 = (5 - 1) mod 7 = 4\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => H#O = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#K": 1, "F#L": 1, "L#L": 1, "H#O": 2}, "id": "mod7_arith_d2_0028"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := 4. I#M := K#N + F#J - 6. G#O := 2. K#N := 3. I#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := 4. I#M := K#N + F#J - 6. G#O := 2. K#N := 3.", "query": "I#M", "answer": 1, "cot": "K#N = 3 -> K#N = 3\nF#J = 4 -> F#J = 4\nI#M = K#N + F#J - 6\n = 3 + 4 - 6\n 3 + 4 = (3 + 4) mod 7 = 0\n 0 - 6 = (0 - 6) mod 7 = 1\n => I#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#O": 1, "K#N": 1, "F#J": 1, "I#M": 2}, "id": "mod7_arith_d2_0029"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#I := E#M. E#M := 6. L#I := 6. H#K := 5. E#K := 6. J#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#I := E#M. E#M := 6. L#I := 6. H#K := 5. E#K := 6.", "query": "J#I", "answer": 6, "cot": "E#M = 6 -> E#M = 6\nJ#I = E#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#K": 1, "E#M": 1, "L#I": 1, "H#K": 1, "J#I": 2}, "id": "mod7_arith_d2_0030"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := 4. K#I := G#N * I#J. G#N := 6. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := 4. K#I := G#N * I#J. G#N := 6.", "query": "K#I", "answer": 3, "cot": "G#N = 6 -> G#N = 6\nI#J = 4 -> I#J = 4\nK#I = G#N * I#J\n = 6 * 4\n 6 * 4 = (6 × 4) mod 7 = 3\n => K#I = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#N": 1, "I#J": 1, "K#I": 2}, "id": "mod7_arith_d2_0031"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := 4. L#N := J#M / K#L * G#K. J#M := 5. G#K := 4. L#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := 4. L#N := J#M / K#L * G#K. J#M := 5. G#K := 4.", "query": "L#N", "answer": 5, "cot": "J#M = 5 -> J#M = 5\nK#L = 4 -> K#L = 4\nG#K = 4 -> G#K = 4\nL#N = J#M / K#L * G#K\n = 5 / 4 * 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n 3 * 4 = (3 × 4) mod 7 = 5\n => L#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#M": 1, "K#L": 1, "G#K": 1, "L#N": 2}, "id": "mod7_arith_d2_0032"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#L := 5. G#M := 2. E#L := G#M. I#J := 1. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#L := 5. G#M := 2. E#L := G#M. I#J := 1.", "query": "E#L", "answer": 2, "cot": "G#M = 2 -> G#M = 2\nE#L = G#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"L#L": 1, "G#M": 1, "I#J": 1, "E#L": 2}, "id": "mod7_arith_d2_0033"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#M := 5. F#O := 4. G#O := G#L. G#L := 2. G#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#M := 5. F#O := 4. G#O := G#L. G#L := 2.", "query": "G#O", "answer": 2, "cot": "G#L = 2 -> G#L = 2\nG#O = G#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#O": 1, "E#M": 1, "G#L": 1, "G#O": 2}, "id": "mod7_arith_d2_0034"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := 5. L#L := 2. I#L := 2. K#L := L#L - J#O. K#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := 5. L#L := 2. I#L := 2. K#L := L#L - J#O.", "query": "K#L", "answer": 4, "cot": "L#L = 2 -> L#L = 2\nJ#O = 5 -> J#O = 5\nK#L = L#L - J#O\n = 2 - 5\n 2 - 5 = (2 - 5) mod 7 = 4\n => K#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#L": 1, "I#L": 1, "J#O": 1, "K#L": 2}, "id": "mod7_arith_d2_0035"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#I := K#M * H#L / K#O. K#M := 2. H#L := 1. K#O := 4. F#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#I := K#M * H#L / K#O. K#M := 2. H#L := 1. K#O := 4.", "query": "F#I", "answer": 4, "cot": "K#O = 4 -> K#O = 4\nK#M = 2 -> K#M = 2\nH#L = 1 -> H#L = 1\nF#I = K#M * H#L / K#O\n = 2 * 1 / 4\n 2 * 1 = (2 × 1) mod 7 = 2\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n => F#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#O": 1, "K#M": 1, "H#L": 1, "F#I": 2}, "id": "mod7_arith_d2_0036"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := G#K. H#M := 1. G#K := 2. J#L := 5. E#P := 1. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := G#K. H#M := 1. G#K := 2. J#L := 5. E#P := 1.", "query": "E#K", "answer": 2, "cot": "G#K = 2 -> G#K = 2\nE#K = G#K = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#M": 1, "E#P": 1, "G#K": 1, "J#L": 1, "E#K": 2}, "id": "mod7_arith_d2_0037"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := I#M / E#N. I#M := 4. E#N := 3. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := I#M / E#N. I#M := 4. E#N := 3.", "query": "F#N", "answer": 6, "cot": "I#M = 4 -> I#M = 4\nE#N = 3 -> E#N = 3\nF#N = I#M / E#N\n = 4 / 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => F#N = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#M": 1, "E#N": 1, "F#N": 2}, "id": "mod7_arith_d2_0038"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := 4. E#J := 6. K#O := 3 + E#J. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := 4. E#J := 6. K#O := 3 + E#J.", "query": "K#O", "answer": 2, "cot": "E#J = 6 -> E#J = 6\nK#O = 3 + E#J\n = 3 + 6\n 3 + 6 = (3 + 6) mod 7 = 2\n => K#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#N": 1, "E#J": 1, "K#O": 2}, "id": "mod7_arith_d2_0039"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := 1. J#J := 5. G#M := 1. E#L := 5. F#L := G#M * E#L. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := 1. J#J := 5. G#M := 1. E#L := 5. F#L := G#M * E#L.", "query": "F#L", "answer": 5, "cot": "G#M = 1 -> G#M = 1\nE#L = 5 -> E#L = 5\nF#L = G#M * E#L\n = 1 * 5\n 1 * 5 = (1 × 5) mod 7 = 5\n => F#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#M": 1, "E#L": 1, "H#O": 1, "J#J": 1, "F#L": 2}, "id": "mod7_arith_d2_0040"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#K := 4. K#P := 3 - G#K / G#P. G#P := 6. J#I := 5. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#K := 4. K#P := 3 - G#K / G#P. G#P := 6. J#I := 5.", "query": "K#P", "answer": 1, "cot": "G#K = 4 -> G#K = 4\nG#P = 6 -> G#P = 6\nK#P = 3 - G#K / G#P\n = 3 - 4 / 6\n 3 - 4 = (3 - 4) mod 7 = 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => K#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#K": 1, "G#P": 1, "J#I": 1, "K#P": 2}, "id": "mod7_arith_d2_0041"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := E#O * I#L / H#O. H#O := 4. E#O := 5. I#L := 3. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := E#O * I#L / H#O. H#O := 4. E#O := 5. I#L := 3.", "query": "I#I", "answer": 2, "cot": "H#O = 4 -> H#O = 4\nI#L = 3 -> I#L = 3\nE#O = 5 -> E#O = 5\nI#I = E#O * I#L / H#O\n = 5 * 3 / 4\n 5 * 3 = (5 × 3) mod 7 = 1\n 1 / 4 = (1 × 4⁻¹) mod 7 = 2\n => I#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#O": 1, "I#L": 1, "E#O": 1, "I#I": 2}, "id": "mod7_arith_d2_0042"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := I#N - K#J - 4. I#N := 4. K#J := 4. F#L := 6. L#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := I#N - K#J - 4. I#N := 4. K#J := 4. F#L := 6.", "query": "L#P", "answer": 3, "cot": "I#N = 4 -> I#N = 4\nK#J = 4 -> K#J = 4\nL#P = I#N - K#J - 4\n = 4 - 4 - 4\n 4 - 4 = (4 - 4) mod 7 = 0\n 0 - 4 = (0 - 4) mod 7 = 3\n => L#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#N": 1, "F#L": 1, "K#J": 1, "L#P": 2}, "id": "mod7_arith_d2_0043"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := 3. G#O := 3. I#N := J#O. L#P := 6. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := 3. G#O := 3. I#N := J#O. L#P := 6.", "query": "I#N", "answer": 3, "cot": "J#O = 3 -> J#O = 3\nI#N = J#O = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"L#P": 1, "G#O": 1, "J#O": 1, "I#N": 2}, "id": "mod7_arith_d2_0044"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#I := 3. J#N := 6. K#K := J#N. K#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#I := 3. J#N := 6. K#K := J#N.", "query": "K#K", "answer": 6, "cot": "J#N = 6 -> J#N = 6\nK#K = J#N = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#I": 1, "J#N": 1, "K#K": 2}, "id": "mod7_arith_d2_0045"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := H#P - I#P - E#N. E#N := 1. H#P := 1. I#P := 2. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := H#P - I#P - E#N. E#N := 1. H#P := 1. I#P := 2.", "query": "E#I", "answer": 5, "cot": "H#P = 1 -> H#P = 1\nE#N = 1 -> E#N = 1\nI#P = 2 -> I#P = 2\nE#I = H#P - I#P - E#N\n = 1 - 2 - 1\n 1 - 2 = (1 - 2) mod 7 = 6\n 6 - 1 = (6 - 1) mod 7 = 5\n => E#I = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#P": 1, "E#N": 1, "I#P": 1, "E#I": 2}, "id": "mod7_arith_d2_0046"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#N := 2. K#P := 4. I#J := 5. J#N := K#P - I#J. J#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#N := 2. K#P := 4. I#J := 5. J#N := K#P - I#J.", "query": "J#N", "answer": 6, "cot": "K#P = 4 -> K#P = 4\nI#J = 5 -> I#J = 5\nJ#N = K#P - I#J\n = 4 - 5\n 4 - 5 = (4 - 5) mod 7 = 6\n => J#N = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#P": 1, "I#J": 1, "H#N": 1, "J#N": 2}, "id": "mod7_arith_d2_0047"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#N := 6. G#I := 4. E#K := G#I - L#N. G#P := 5. G#L := 6. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#N := 6. G#I := 4. E#K := G#I - L#N. G#P := 5. G#L := 6.", "query": "E#K", "answer": 5, "cot": "G#I = 4 -> G#I = 4\nL#N = 6 -> L#N = 6\nE#K = G#I - L#N\n = 4 - 6\n 4 - 6 = (4 - 6) mod 7 = 5\n => E#K = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#I": 1, "L#N": 1, "G#L": 1, "G#P": 1, "E#K": 2}, "id": "mod7_arith_d2_0048"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := J#J + E#N. J#J := 3. E#N := 4. K#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := J#J + E#N. J#J := 3. E#N := 4.", "query": "K#J", "answer": 0, "cot": "J#J = 3 -> J#J = 3\nE#N = 4 -> E#N = 4\nK#J = J#J + E#N\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => K#J = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#J": 1, "E#N": 1, "K#J": 2}, "id": "mod7_arith_d2_0049"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := 1. K#K := 2. F#M := 3. I#P := F#M. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := 1. K#K := 2. F#M := 3. I#P := F#M.", "query": "I#P", "answer": 3, "cot": "F#M = 3 -> F#M = 3\nI#P = F#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#M": 1, "E#L": 1, "K#K": 1, "I#P": 2}, "id": "mod7_arith_d2_0050"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#I := F#L. J#M := 6. F#L := 2. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#I := F#L. J#M := 6. F#L := 2.", "query": "H#I", "answer": 2, "cot": "F#L = 2 -> F#L = 2\nH#I = F#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#M": 1, "F#L": 1, "H#I": 2}, "id": "mod7_arith_d2_0051"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := 3. G#O := 6. H#M := 5. E#I := 2 / G#O. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := 3. G#O := 6. H#M := 5. E#I := 2 / G#O.", "query": "E#I", "answer": 5, "cot": "G#O = 6 -> G#O = 6\nE#I = 2 / G#O\n = 2 / 6\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => E#I = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#K": 1, "G#O": 1, "H#M": 1, "E#I": 2}, "id": "mod7_arith_d2_0052"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := 3. G#J := 1. G#M := 3 * F#K. I#M := 1. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := 3. G#J := 1. G#M := 3 * F#K. I#M := 1.", "query": "G#M", "answer": 2, "cot": "F#K = 3 -> F#K = 3\nG#M = 3 * F#K\n = 3 * 3\n 3 * 3 = (3 × 3) mod 7 = 2\n => G#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#M": 1, "G#J": 1, "F#K": 1, "G#M": 2}, "id": "mod7_arith_d2_0053"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#K := K#I. K#P := 6. K#I := 3. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#K := K#I. K#P := 6. K#I := 3.", "query": "J#K", "answer": 3, "cot": "K#I = 3 -> K#I = 3\nJ#K = K#I = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#I": 1, "K#P": 1, "J#K": 2}, "id": "mod7_arith_d2_0054"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := 6. I#J := 1. G#L := I#J - K#L. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := 6. I#J := 1. G#L := I#J - K#L.", "query": "G#L", "answer": 2, "cot": "K#L = 6 -> K#L = 6\nI#J = 1 -> I#J = 1\nG#L = I#J - K#L\n = 1 - 6\n 1 - 6 = (1 - 6) mod 7 = 2\n => G#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#L": 1, "I#J": 1, "G#L": 2}, "id": "mod7_arith_d2_0055"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#N := J#J - J#L. J#J := 4. K#P := 1. E#J := 6. J#L := 1. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#N := J#J - J#L. J#J := 4. K#P := 1. E#J := 6. J#L := 1.", "query": "I#N", "answer": 3, "cot": "J#L = 1 -> J#L = 1\nJ#J = 4 -> J#J = 4\nI#N = J#J - J#L\n = 4 - 1\n 4 - 1 = (4 - 1) mod 7 = 3\n => I#N = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#P": 1, "J#L": 1, "J#J": 1, "E#J": 1, "I#N": 2}, "id": "mod7_arith_d2_0056"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#O := 4. E#L := 4. E#J := 4. G#M := E#L / 4. K#N := 2. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#O := 4. E#L := 4. E#J := 4. G#M := E#L / 4. K#N := 2.", "query": "G#M", "answer": 1, "cot": "E#L = 4 -> E#L = 4\nG#M = E#L / 4\n = 4 / 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => G#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#O": 1, "E#J": 1, "E#L": 1, "K#N": 1, "G#M": 2}, "id": "mod7_arith_d2_0057"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#O := 5. E#M := L#O - 6 - G#O. G#O := 3. J#J := 5. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#O := 5. E#M := L#O - 6 - G#O. G#O := 3. J#J := 5.", "query": "E#M", "answer": 3, "cot": "G#O = 3 -> G#O = 3\nL#O = 5 -> L#O = 5\nE#M = L#O - 6 - G#O\n = 5 - 6 - 3\n 5 - 6 = (5 - 6) mod 7 = 6\n 6 - 3 = (6 - 3) mod 7 = 3\n => E#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#O": 1, "L#O": 1, "J#J": 1, "E#M": 2}, "id": "mod7_arith_d2_0058"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := 2. F#M := H#O - 2. K#K := 5. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := 2. F#M := H#O - 2. K#K := 5.", "query": "F#M", "answer": 0, "cot": "H#O = 2 -> H#O = 2\nF#M = H#O - 2\n = 2 - 2\n 2 - 2 = (2 - 2) mod 7 = 0\n => F#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#K": 1, "H#O": 1, "F#M": 2}, "id": "mod7_arith_d2_0059"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := 2. H#M := K#L + I#L. I#L := 2. E#O := 5. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := 2. H#M := K#L + I#L. I#L := 2. E#O := 5.", "query": "H#M", "answer": 4, "cot": "I#L = 2 -> I#L = 2\nK#L = 2 -> K#L = 2\nH#M = K#L + I#L\n = 2 + 2\n 2 + 2 = (2 + 2) mod 7 = 4\n => H#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#O": 1, "I#L": 1, "K#L": 1, "H#M": 2}, "id": "mod7_arith_d2_0060"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#P := 3. L#M := G#P. G#P := 1. H#N := 3. J#M := 6. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#P := 3. L#M := G#P. G#P := 1. H#N := 3. J#M := 6.", "query": "L#M", "answer": 1, "cot": "G#P = 1 -> G#P = 1\nL#M = G#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#M": 1, "E#P": 1, "G#P": 1, "H#N": 1, "L#M": 2}, "id": "mod7_arith_d2_0061"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := 1. K#P := 2. G#I := K#M. G#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := 1. K#P := 2. G#I := K#M.", "query": "G#I", "answer": 1, "cot": "K#M = 1 -> K#M = 1\nG#I = K#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#P": 1, "K#M": 1, "G#I": 2}, "id": "mod7_arith_d2_0062"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#O := 6. J#J := E#N. E#N := 2. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#O := 6. J#J := E#N. E#N := 2.", "query": "J#J", "answer": 2, "cot": "E#N = 2 -> E#N = 2\nJ#J = E#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#N": 1, "L#O": 1, "J#J": 2}, "id": "mod7_arith_d2_0063"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := 4. G#O := 5 * E#K. E#I := 3. I#M := 1. E#K := 1. G#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := 4. G#O := 5 * E#K. E#I := 3. I#M := 1. E#K := 1.", "query": "G#O", "answer": 5, "cot": "E#K = 1 -> E#K = 1\nG#O = 5 * E#K\n = 5 * 1\n 5 * 1 = (5 × 1) mod 7 = 5\n => G#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#K": 1, "I#M": 1, "E#I": 1, "I#J": 1, "G#O": 2}, "id": "mod7_arith_d2_0064"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#I := 2. H#L := 6. F#L := G#I - H#L. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#I := 2. H#L := 6. F#L := G#I - H#L.", "query": "F#L", "answer": 3, "cot": "H#L = 6 -> H#L = 6\nG#I = 2 -> G#I = 2\nF#L = G#I - H#L\n = 2 - 6\n 2 - 6 = (2 - 6) mod 7 = 3\n => F#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#L": 1, "G#I": 1, "F#L": 2}, "id": "mod7_arith_d2_0065"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := 3. E#J := 2. L#K := 2. H#O := 3. I#N := H#O - 2. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := 3. E#J := 2. L#K := 2. H#O := 3. I#N := H#O - 2.", "query": "I#N", "answer": 1, "cot": "H#O = 3 -> H#O = 3\nI#N = H#O - 2\n = 3 - 2\n 3 - 2 = (3 - 2) mod 7 = 1\n => I#N = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#J": 1, "L#K": 1, "H#O": 1, "E#J": 1, "I#N": 2}, "id": "mod7_arith_d2_0066"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := 4. G#J := 4. E#K := G#L / F#K. K#J := 5. G#L := 5. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := 4. G#J := 4. E#K := G#L / F#K. K#J := 5. G#L := 5.", "query": "E#K", "answer": 3, "cot": "G#L = 5 -> G#L = 5\nF#K = 4 -> F#K = 4\nE#K = G#L / F#K\n = 5 / 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => E#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#J": 1, "G#L": 1, "G#J": 1, "F#K": 1, "E#K": 2}, "id": "mod7_arith_d2_0067"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := I#P. E#I := 1. K#K := 2. I#P := 3. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := I#P. E#I := 1. K#K := 2. I#P := 3.", "query": "F#L", "answer": 3, "cot": "I#P = 3 -> I#P = 3\nF#L = I#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#K": 1, "I#P": 1, "E#I": 1, "F#L": 2}, "id": "mod7_arith_d2_0068"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := K#J - E#I. K#J := 4. E#I := 1. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := K#J - E#I. K#J := 4. E#I := 1.", "query": "K#I", "answer": 3, "cot": "E#I = 1 -> E#I = 1\nK#J = 4 -> K#J = 4\nK#I = K#J - E#I\n = 4 - 1\n 4 - 1 = (4 - 1) mod 7 = 3\n => K#I = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#I": 1, "K#J": 1, "K#I": 2}, "id": "mod7_arith_d2_0069"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := E#I. I#J := 2. E#I := 6. F#K := 6. I#L := 6. F#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := E#I. I#J := 2. E#I := 6. F#K := 6. I#L := 6.", "query": "F#P", "answer": 6, "cot": "E#I = 6 -> E#I = 6\nF#P = E#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#J": 1, "I#L": 1, "F#K": 1, "E#I": 1, "F#P": 2}, "id": "mod7_arith_d2_0070"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := H#J * I#J. I#J := 2. H#J := 4. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := H#J * I#J. I#J := 2. H#J := 4.", "query": "I#P", "answer": 1, "cot": "I#J = 2 -> I#J = 2\nH#J = 4 -> H#J = 4\nI#P = H#J * I#J\n = 4 * 2\n 4 * 2 = (4 × 2) mod 7 = 1\n => I#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#J": 1, "H#J": 1, "I#P": 2}, "id": "mod7_arith_d2_0071"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#P := 4 * 6 * L#J. H#L := 4. K#N := 6. L#J := 4. E#P := 1. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#P := 4 * 6 * L#J. H#L := 4. K#N := 6. L#J := 4. E#P := 1.", "query": "K#P", "answer": 5, "cot": "L#J = 4 -> L#J = 4\nK#P = 4 * 6 * L#J\n = 4 * 6 * 4\n 4 * 6 = (4 × 6) mod 7 = 3\n 3 * 4 = (3 × 4) mod 7 = 5\n => K#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#L": 1, "L#J": 1, "K#N": 1, "E#P": 1, "K#P": 2}, "id": "mod7_arith_d2_0072"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := 6. H#I := K#I - 2. I#K := 4. G#I := 1. H#J := 1. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := 6. H#I := K#I - 2. I#K := 4. G#I := 1. H#J := 1.", "query": "H#I", "answer": 4, "cot": "K#I = 6 -> K#I = 6\nH#I = K#I - 2\n = 6 - 2\n 6 - 2 = (6 - 2) mod 7 = 4\n => H#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#J": 1, "I#K": 1, "K#I": 1, "G#I": 1, "H#I": 2}, "id": "mod7_arith_d2_0073"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := H#L. H#L := 6. H#M := 1. L#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := H#L. H#L := 6. H#M := 1.", "query": "L#P", "answer": 6, "cot": "H#L = 6 -> H#L = 6\nL#P = H#L = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#M": 1, "H#L": 1, "L#P": 2}, "id": "mod7_arith_d2_0074"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#K := 3. H#O := 1. L#P := 5. G#J := H#O * G#K. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#K := 3. H#O := 1. L#P := 5. G#J := H#O * G#K.", "query": "G#J", "answer": 3, "cot": "H#O = 1 -> H#O = 1\nG#K = 3 -> G#K = 3\nG#J = H#O * G#K\n = 1 * 3\n 1 * 3 = (1 × 3) mod 7 = 3\n => G#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#O": 1, "L#P": 1, "G#K": 1, "G#J": 2}, "id": "mod7_arith_d2_0075"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 1. H#K := 5. H#I := H#K / J#N. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 1. H#K := 5. H#I := H#K / J#N.", "query": "H#I", "answer": 5, "cot": "H#K = 5 -> H#K = 5\nJ#N = 1 -> J#N = 1\nH#I = H#K / J#N\n = 5 / 1\n 5 / 1 = (5 × 1⁻¹) mod 7 = 5\n => H#I = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#K": 1, "J#N": 1, "H#I": 2}, "id": "mod7_arith_d2_0076"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#O := 1. H#J := 3. J#K := 5. I#P := I#N + J#K. I#N := 4. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#O := 1. H#J := 3. J#K := 5. I#P := I#N + J#K. I#N := 4.", "query": "I#P", "answer": 2, "cot": "J#K = 5 -> J#K = 5\nI#N = 4 -> I#N = 4\nI#P = I#N + J#K\n = 4 + 5\n 4 + 5 = (4 + 5) mod 7 = 2\n => I#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#O": 1, "J#K": 1, "H#J": 1, "I#N": 1, "I#P": 2}, "id": "mod7_arith_d2_0077"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := 5. I#O := 1. J#M := G#N. I#I := 1. G#N := 2. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := 5. I#O := 1. J#M := G#N. I#I := 1. G#N := 2.", "query": "J#M", "answer": 2, "cot": "G#N = 2 -> G#N = 2\nJ#M = G#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#O": 1, "I#I": 1, "G#N": 1, "H#J": 1, "J#M": 2}, "id": "mod7_arith_d2_0078"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := I#M. J#P := 4. I#M := 5. I#P := 5. K#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := I#M. J#P := 4. I#M := 5. I#P := 5.", "query": "K#J", "answer": 5, "cot": "I#M = 5 -> I#M = 5\nK#J = I#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#P": 1, "I#M": 1, "J#P": 1, "K#J": 2}, "id": "mod7_arith_d2_0079"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := E#P + F#O. L#M := 4. F#O := 4. E#P := 5. J#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := E#P + F#O. L#M := 4. F#O := 4. E#P := 5.", "query": "J#O", "answer": 2, "cot": "F#O = 4 -> F#O = 4\nE#P = 5 -> E#P = 5\nJ#O = E#P + F#O\n = 5 + 4\n 5 + 4 = (5 + 4) mod 7 = 2\n => J#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#O": 1, "L#M": 1, "E#P": 1, "J#O": 2}, "id": "mod7_arith_d2_0080"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#M := K#M - K#L. K#L := 6. K#M := 5. K#O := 3. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#M := K#M - K#L. K#L := 6. K#M := 5. K#O := 3.", "query": "E#M", "answer": 6, "cot": "K#L = 6 -> K#L = 6\nK#M = 5 -> K#M = 5\nE#M = K#M - K#L\n = 5 - 6\n 5 - 6 = (5 - 6) mod 7 = 6\n => E#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#L": 1, "K#M": 1, "K#O": 1, "E#M": 2}, "id": "mod7_arith_d2_0081"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#I := 1. J#O := 5. G#M := J#O / G#I / 4. F#L := 4. F#O := 3. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#I := 1. J#O := 5. G#M := J#O / G#I / 4. F#L := 4. F#O := 3.", "query": "G#M", "answer": 3, "cot": "J#O = 5 -> J#O = 5\nG#I = 1 -> G#I = 1\nG#M = J#O / G#I / 4\n = 5 / 1 / 4\n 5 / 1 = (5 × 1⁻¹) mod 7 = 5\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => G#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#O": 1, "G#I": 1, "F#L": 1, "F#O": 1, "G#M": 2}, "id": "mod7_arith_d2_0082"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := 2. L#O := 1. I#M := L#M + L#L. L#L := 6. I#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := 2. L#O := 1. I#M := L#M + L#L. L#L := 6.", "query": "I#M", "answer": 1, "cot": "L#L = 6 -> L#L = 6\nL#M = 2 -> L#M = 2\nI#M = L#M + L#L\n = 2 + 6\n 2 + 6 = (2 + 6) mod 7 = 1\n => I#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#L": 1, "L#M": 1, "L#O": 1, "I#M": 2}, "id": "mod7_arith_d2_0083"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := J#M - H#L. H#L := 6. J#M := 5. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := J#M - H#L. H#L := 6. J#M := 5.", "query": "F#J", "answer": 6, "cot": "J#M = 5 -> J#M = 5\nH#L = 6 -> H#L = 6\nF#J = J#M - H#L\n = 5 - 6\n 5 - 6 = (5 - 6) mod 7 = 6\n => F#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#M": 1, "H#L": 1, "F#J": 2}, "id": "mod7_arith_d2_0084"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := 4. K#K := 2. E#O := 5. J#M := 2. J#O := L#P - E#O * J#M. J#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := 4. K#K := 2. E#O := 5. J#M := 2. J#O := L#P - E#O * J#M.", "query": "J#O", "answer": 5, "cot": "L#P = 4 -> L#P = 4\nJ#M = 2 -> J#M = 2\nE#O = 5 -> E#O = 5\nJ#O = L#P - E#O * J#M\n = 4 - 5 * 2\n 4 - 5 = (4 - 5) mod 7 = 6\n 6 * 2 = (6 × 2) mod 7 = 5\n => J#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#P": 1, "J#M": 1, "E#O": 1, "K#K": 1, "J#O": 2}, "id": "mod7_arith_d2_0085"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := 1. G#M := 5. H#M := 6. L#I := H#J - H#M. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := 1. G#M := 5. H#M := 6. L#I := H#J - H#M.", "query": "L#I", "answer": 2, "cot": "H#M = 6 -> H#M = 6\nH#J = 1 -> H#J = 1\nL#I = H#J - H#M\n = 1 - 6\n 1 - 6 = (1 - 6) mod 7 = 2\n => L#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#M": 1, "G#M": 1, "H#J": 1, "L#I": 2}, "id": "mod7_arith_d2_0086"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := E#P + K#K. K#K := 2. E#P := 1. J#K := 5. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := E#P + K#K. K#K := 2. E#P := 1. J#K := 5.", "query": "H#J", "answer": 3, "cot": "E#P = 1 -> E#P = 1\nK#K = 2 -> K#K = 2\nH#J = E#P + K#K\n = 1 + 2\n 1 + 2 = (1 + 2) mod 7 = 3\n => H#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#K": 1, "E#P": 1, "K#K": 1, "H#J": 2}, "id": "mod7_arith_d2_0087"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := 4. H#P := F#L. F#L := 6. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := 4. H#P := F#L. F#L := 6.", "query": "H#P", "answer": 6, "cot": "F#L = 6 -> F#L = 6\nH#P = F#L = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#L": 1, "K#L": 1, "H#P": 2}, "id": "mod7_arith_d2_0088"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#L := 3. G#P := 6. I#J := G#P - H#L. L#N := 3. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#L := 3. G#P := 6. I#J := G#P - H#L. L#N := 3.", "query": "I#J", "answer": 3, "cot": "H#L = 3 -> H#L = 3\nG#P = 6 -> G#P = 6\nI#J = G#P - H#L\n = 6 - 3\n 6 - 3 = (6 - 3) mod 7 = 3\n => I#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#L": 1, "L#N": 1, "G#P": 1, "I#J": 2}, "id": "mod7_arith_d2_0089"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#M := 6. L#K := 3. E#O := 5. J#J := G#M + E#O + L#K. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#M := 6. L#K := 3. E#O := 5. J#J := G#M + E#O + L#K.", "query": "J#J", "answer": 0, "cot": "G#M = 6 -> G#M = 6\nE#O = 5 -> E#O = 5\nL#K = 3 -> L#K = 3\nJ#J = G#M + E#O + L#K\n = 6 + 5 + 3\n 6 + 5 = (6 + 5) mod 7 = 4\n 4 + 3 = (4 + 3) mod 7 = 0\n => J#J = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#M": 1, "E#O": 1, "L#K": 1, "J#J": 2}, "id": "mod7_arith_d2_0090"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#K := L#O / J#O. J#O := 3. J#M := 5. L#O := 4. E#K := 2. G#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#K := L#O / J#O. J#O := 3. J#M := 5. L#O := 4. E#K := 2.", "query": "G#K", "answer": 6, "cot": "J#O = 3 -> J#O = 3\nL#O = 4 -> L#O = 4\nG#K = L#O / J#O\n = 4 / 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => G#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#K": 1, "J#O": 1, "J#M": 1, "L#O": 1, "G#K": 2}, "id": "mod7_arith_d2_0091"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := 4. K#K := 2 - J#O. J#O := 2. K#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := 4. K#K := 2 - J#O. J#O := 2.", "query": "K#K", "answer": 0, "cot": "J#O = 2 -> J#O = 2\nK#K = 2 - J#O\n = 2 - 2\n 2 - 2 = (2 - 2) mod 7 = 0\n => K#K = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#O": 1, "E#I": 1, "K#K": 2}, "id": "mod7_arith_d2_0092"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#L := H#J - E#K + E#L. E#M := 2. E#L := 3. H#J := 3. E#K := 3. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#L := H#J - E#K + E#L. E#M := 2. E#L := 3. H#J := 3. E#K := 3.", "query": "G#L", "answer": 3, "cot": "H#J = 3 -> H#J = 3\nE#L = 3 -> E#L = 3\nE#K = 3 -> E#K = 3\nG#L = H#J - E#K + E#L\n = 3 - 3 + 3\n 3 - 3 = (3 - 3) mod 7 = 0\n 0 + 3 = (0 + 3) mod 7 = 3\n => G#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#J": 1, "E#L": 1, "E#K": 1, "E#M": 1, "G#L": 2}, "id": "mod7_arith_d2_0093"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := 4. J#P := 1. H#K := F#L * H#P - 4. H#P := 1. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := 4. J#P := 1. H#K := F#L * H#P - 4. H#P := 1.", "query": "H#K", "answer": 0, "cot": "F#L = 4 -> F#L = 4\nH#P = 1 -> H#P = 1\nH#K = F#L * H#P - 4\n = 4 * 1 - 4\n 4 * 1 = (4 × 1) mod 7 = 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => H#K = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#L": 1, "J#P": 1, "H#P": 1, "H#K": 2}, "id": "mod7_arith_d2_0094"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := F#J + K#J. F#J := 1. E#M := 1. K#J := 3. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := F#J + K#J. F#J := 1. E#M := 1. K#J := 3.", "query": "H#K", "answer": 4, "cot": "F#J = 1 -> F#J = 1\nK#J = 3 -> K#J = 3\nH#K = F#J + K#J\n = 1 + 3\n 1 + 3 = (1 + 3) mod 7 = 4\n => H#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#J": 1, "K#J": 1, "E#M": 1, "H#K": 2}, "id": "mod7_arith_d2_0095"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#M := I#K * 6 * E#O. E#O := 3. I#K := 6. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#M := I#K * 6 * E#O. E#O := 3. I#K := 6.", "query": "F#M", "answer": 3, "cot": "I#K = 6 -> I#K = 6\nE#O = 3 -> E#O = 3\nF#M = I#K * 6 * E#O\n = 6 * 6 * 3\n 6 * 6 = (6 × 6) mod 7 = 1\n 1 * 3 = (1 × 3) mod 7 = 3\n => F#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#K": 1, "E#O": 1, "F#M": 2}, "id": "mod7_arith_d2_0096"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#M := 3. E#K := J#P + H#J. J#P := 5. H#J := 1. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#M := 3. E#K := J#P + H#J. J#P := 5. H#J := 1.", "query": "E#K", "answer": 6, "cot": "H#J = 1 -> H#J = 1\nJ#P = 5 -> J#P = 5\nE#K = J#P + H#J\n = 5 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n => E#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#M": 1, "H#J": 1, "J#P": 1, "E#K": 2}, "id": "mod7_arith_d2_0097"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := K#O / I#K. I#K := 6. K#O := 6. G#M := 3. E#O := 2. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := K#O / I#K. I#K := 6. K#O := 6. G#M := 3. E#O := 2.", "query": "F#N", "answer": 1, "cot": "I#K = 6 -> I#K = 6\nK#O = 6 -> K#O = 6\nF#N = K#O / I#K\n = 6 / 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => F#N = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#M": 1, "I#K": 1, "E#O": 1, "K#O": 1, "F#N": 2}, "id": "mod7_arith_d2_0098"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#O := 1. L#I := G#J / I#O. G#J := 4. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#O := 1. L#I := G#J / I#O. G#J := 4.", "query": "L#I", "answer": 4, "cot": "G#J = 4 -> G#J = 4\nI#O = 1 -> I#O = 1\nL#I = G#J / I#O\n = 4 / 1\n 4 / 1 = (4 × 1⁻¹) mod 7 = 4\n => L#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#J": 1, "I#O": 1, "L#I": 2}, "id": "mod7_arith_d2_0099"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#K := 2. H#J := 6. L#L := 3. L#I := I#K * 6. L#J := 2. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#K := 2. H#J := 6. L#L := 3. L#I := I#K * 6. L#J := 2.", "query": "L#I", "answer": 5, "cot": "I#K = 2 -> I#K = 2\nL#I = I#K * 6\n = 2 * 6\n 2 * 6 = (2 × 6) mod 7 = 5\n => L#I = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#J": 1, "I#K": 1, "L#L": 1, "L#J": 1, "L#I": 2}, "id": "mod7_arith_d2_0100"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#L := 3. E#I := 3. H#P := J#L - K#I / E#I. K#I := 5. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#L := 3. E#I := 3. H#P := J#L - K#I / E#I. K#I := 5.", "query": "H#P", "answer": 4, "cot": "J#L = 3 -> J#L = 3\nE#I = 3 -> E#I = 3\nK#I = 5 -> K#I = 5\nH#P = J#L - K#I / E#I\n = 3 - 5 / 3\n 3 - 5 = (3 - 5) mod 7 = 5\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n => H#P = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#L": 1, "E#I": 1, "K#I": 1, "H#P": 2}, "id": "mod7_arith_d2_0101"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := 6. K#M := 1. G#M := G#J / I#N. J#M := 2. I#N := 5. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := 6. K#M := 1. G#M := G#J / I#N. J#M := 2. I#N := 5.", "query": "G#M", "answer": 4, "cot": "I#N = 5 -> I#N = 5\nG#J = 6 -> G#J = 6\nG#M = G#J / I#N\n = 6 / 5\n 6 / 5 = (6 × 5⁻¹) mod 7 = 4\n => G#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#M": 1, "K#M": 1, "I#N": 1, "G#J": 1, "G#M": 2}, "id": "mod7_arith_d2_0102"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#P := K#L / K#I. K#I := 1. F#J := 5. K#L := 3. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#P := K#L / K#I. K#I := 1. F#J := 5. K#L := 3.", "query": "H#P", "answer": 3, "cot": "K#I = 1 -> K#I = 1\nK#L = 3 -> K#L = 3\nH#P = K#L / K#I\n = 3 / 1\n 3 / 1 = (3 × 1⁻¹) mod 7 = 3\n => H#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#J": 1, "K#I": 1, "K#L": 1, "H#P": 2}, "id": "mod7_arith_d2_0103"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := F#O * L#L. F#O := 5. L#L := 6. G#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := F#O * L#L. F#O := 5. L#L := 6.", "query": "G#O", "answer": 2, "cot": "L#L = 6 -> L#L = 6\nF#O = 5 -> F#O = 5\nG#O = F#O * L#L\n = 5 * 6\n 5 * 6 = (5 × 6) mod 7 = 2\n => G#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#L": 1, "F#O": 1, "G#O": 2}, "id": "mod7_arith_d2_0104"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := I#I / I#M. I#M := 1. I#I := 3. F#L := 5. E#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := I#I / I#M. I#M := 1. I#I := 3. F#L := 5.", "query": "E#N", "answer": 3, "cot": "I#I = 3 -> I#I = 3\nI#M = 1 -> I#M = 1\nE#N = I#I / I#M\n = 3 / 1\n 3 / 1 = (3 × 1⁻¹) mod 7 = 3\n => E#N = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#I": 1, "F#L": 1, "I#M": 1, "E#N": 2}, "id": "mod7_arith_d2_0105"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := 4. L#P := 2. E#J := 1. H#O := 4. J#M := H#O + L#P / K#M. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := 4. L#P := 2. E#J := 1. H#O := 4. J#M := H#O + L#P / K#M.", "query": "J#M", "answer": 5, "cot": "L#P = 2 -> L#P = 2\nH#O = 4 -> H#O = 4\nK#M = 4 -> K#M = 4\nJ#M = H#O + L#P / K#M\n = 4 + 2 / 4\n 4 + 2 = (4 + 2) mod 7 = 6\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n => J#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#P": 1, "H#O": 1, "E#J": 1, "K#M": 1, "J#M": 2}, "id": "mod7_arith_d2_0106"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := 1. L#O := 3. J#O := 5. K#P := L#I + 5. L#I := 2. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := 1. L#O := 3. J#O := 5. K#P := L#I + 5. L#I := 2.", "query": "K#P", "answer": 0, "cot": "L#I = 2 -> L#I = 2\nK#P = L#I + 5\n = 2 + 5\n 2 + 5 = (2 + 5) mod 7 = 0\n => K#P = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#O": 1, "L#O": 1, "E#L": 1, "L#I": 1, "K#P": 2}, "id": "mod7_arith_d2_0107"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#J := 2. J#K := 4. G#P := F#N * E#J - J#K. F#N := 5. H#N := 5. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#J := 2. J#K := 4. G#P := F#N * E#J - J#K. F#N := 5. H#N := 5.", "query": "G#P", "answer": 6, "cot": "J#K = 4 -> J#K = 4\nF#N = 5 -> F#N = 5\nE#J = 2 -> E#J = 2\nG#P = F#N * E#J - J#K\n = 5 * 2 - 4\n 5 * 2 = (5 × 2) mod 7 = 3\n 3 - 4 = (3 - 4) mod 7 = 6\n => G#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#K": 1, "H#N": 1, "F#N": 1, "E#J": 1, "G#P": 2}, "id": "mod7_arith_d2_0108"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#M := E#N * 5. E#N := 1. K#M := 5. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#M := E#N * 5. E#N := 1. K#M := 5.", "query": "J#M", "answer": 5, "cot": "E#N = 1 -> E#N = 1\nJ#M = E#N * 5\n = 1 * 5\n 1 * 5 = (1 × 5) mod 7 = 5\n => J#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#M": 1, "E#N": 1, "J#M": 2}, "id": "mod7_arith_d2_0109"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#J := 4. I#P := 6. K#K := J#J + I#P. K#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#J := 4. I#P := 6. K#K := J#J + I#P.", "query": "K#K", "answer": 3, "cot": "I#P = 6 -> I#P = 6\nJ#J = 4 -> J#J = 4\nK#K = J#J + I#P\n = 4 + 6\n 4 + 6 = (4 + 6) mod 7 = 3\n => K#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#P": 1, "J#J": 1, "K#K": 2}, "id": "mod7_arith_d2_0110"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := 1. L#L := J#J - F#K. E#I := 2. J#J := 1. G#I := 3. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := 1. L#L := J#J - F#K. E#I := 2. J#J := 1. G#I := 3.", "query": "L#L", "answer": 0, "cot": "F#K = 1 -> F#K = 1\nJ#J = 1 -> J#J = 1\nL#L = J#J - F#K\n = 1 - 1\n 1 - 1 = (1 - 1) mod 7 = 0\n => L#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#K": 1, "E#I": 1, "G#I": 1, "J#J": 1, "L#L": 2}, "id": "mod7_arith_d2_0111"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := 4. I#K := 4. K#L := G#O. K#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := 4. I#K := 4. K#L := G#O.", "query": "K#L", "answer": 4, "cot": "G#O = 4 -> G#O = 4\nK#L = G#O = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#O": 1, "I#K": 1, "K#L": 2}, "id": "mod7_arith_d2_0112"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := F#I - H#P. H#K := 3. H#P := 2. F#I := 1. E#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := F#I - H#P. H#K := 3. H#P := 2. F#I := 1.", "query": "E#N", "answer": 6, "cot": "F#I = 1 -> F#I = 1\nH#P = 2 -> H#P = 2\nE#N = F#I - H#P\n = 1 - 2\n 1 - 2 = (1 - 2) mod 7 = 6\n => E#N = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#I": 1, "H#P": 1, "H#K": 1, "E#N": 2}, "id": "mod7_arith_d2_0113"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := 3. L#K := 6. G#J := E#L - K#K * L#K. E#L := 4. K#K := 1. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := 3. L#K := 6. G#J := E#L - K#K * L#K. E#L := 4. K#K := 1.", "query": "G#J", "answer": 4, "cot": "L#K = 6 -> L#K = 6\nK#K = 1 -> K#K = 1\nE#L = 4 -> E#L = 4\nG#J = E#L - K#K * L#K\n = 4 - 1 * 6\n 4 - 1 = (4 - 1) mod 7 = 3\n 3 * 6 = (3 × 6) mod 7 = 4\n => G#J = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#K": 1, "K#K": 1, "E#L": 1, "L#J": 1, "G#J": 2}, "id": "mod7_arith_d2_0114"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#M := 4. I#K := F#M + H#L. H#L := 4. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#M := 4. I#K := F#M + H#L. H#L := 4.", "query": "I#K", "answer": 1, "cot": "F#M = 4 -> F#M = 4\nH#L = 4 -> H#L = 4\nI#K = F#M + H#L\n = 4 + 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => I#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#M": 1, "H#L": 1, "I#K": 2}, "id": "mod7_arith_d2_0115"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := I#M * E#M - I#I. L#I := 5. E#M := 3. I#I := 4. I#M := 3. K#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := I#M * E#M - I#I. L#I := 5. E#M := 3. I#I := 4. I#M := 3.", "query": "K#L", "answer": 5, "cot": "I#I = 4 -> I#I = 4\nI#M = 3 -> I#M = 3\nE#M = 3 -> E#M = 3\nK#L = I#M * E#M - I#I\n = 3 * 3 - 4\n 3 * 3 = (3 × 3) mod 7 = 2\n 2 - 4 = (2 - 4) mod 7 = 5\n => K#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#I": 1, "I#M": 1, "E#M": 1, "L#I": 1, "K#L": 2}, "id": "mod7_arith_d2_0116"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#J := 6. G#K := 2. K#N := G#K. K#P := 5. K#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#J := 6. G#K := 2. K#N := G#K. K#P := 5.", "query": "K#N", "answer": 2, "cot": "G#K = 2 -> G#K = 2\nK#N = G#K = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#K": 1, "K#P": 1, "J#J": 1, "K#N": 2}, "id": "mod7_arith_d2_0117"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#I := 3. I#L := L#I. L#L := 3. L#I := 3. J#O := 6. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#I := 3. I#L := L#I. L#L := 3. L#I := 3. J#O := 6.", "query": "I#L", "answer": 3, "cot": "L#I = 3 -> L#I = 3\nI#L = L#I = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#O": 1, "L#L": 1, "L#I": 1, "H#I": 1, "I#L": 2}, "id": "mod7_arith_d2_0118"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := 2. H#P := 3. J#L := F#L * 6. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := 2. H#P := 3. J#L := F#L * 6.", "query": "J#L", "answer": 5, "cot": "F#L = 2 -> F#L = 2\nJ#L = F#L * 6\n = 2 * 6\n 2 * 6 = (2 × 6) mod 7 = 5\n => J#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#L": 1, "H#P": 1, "J#L": 2}, "id": "mod7_arith_d2_0119"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#L := J#L * 6 * H#M. J#L := 6. H#M := 4. L#K := 5. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#L := J#L * 6 * H#M. J#L := 6. H#M := 4. L#K := 5.", "query": "I#L", "answer": 4, "cot": "H#M = 4 -> H#M = 4\nJ#L = 6 -> J#L = 6\nI#L = J#L * 6 * H#M\n = 6 * 6 * 4\n 6 * 6 = (6 × 6) mod 7 = 1\n 1 * 4 = (1 × 4) mod 7 = 4\n => I#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#K": 1, "H#M": 1, "J#L": 1, "I#L": 2}, "id": "mod7_arith_d2_0120"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := H#I - K#I - 2. E#M := 2. H#I := 4. K#I := 5. G#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := H#I - K#I - 2. E#M := 2. H#I := 4. K#I := 5.", "query": "G#N", "answer": 4, "cot": "K#I = 5 -> K#I = 5\nH#I = 4 -> H#I = 4\nG#N = H#I - K#I - 2\n = 4 - 5 - 2\n 4 - 5 = (4 - 5) mod 7 = 6\n 6 - 2 = (6 - 2) mod 7 = 4\n => G#N = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#M": 1, "K#I": 1, "H#I": 1, "G#N": 2}, "id": "mod7_arith_d2_0121"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := 5. H#N := E#K. L#I := 6. E#K := 1. H#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := 5. H#N := E#K. L#I := 6. E#K := 1.", "query": "H#N", "answer": 1, "cot": "E#K = 1 -> E#K = 1\nH#N = E#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#K": 1, "E#K": 1, "L#I": 1, "H#N": 2}, "id": "mod7_arith_d2_0122"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := 6. G#M := H#K / G#N + 4. G#N := 4. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := 6. G#M := H#K / G#N + 4. G#N := 4.", "query": "G#M", "answer": 2, "cot": "G#N = 4 -> G#N = 4\nH#K = 6 -> H#K = 6\nG#M = H#K / G#N + 4\n = 6 / 4 + 4\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n 5 + 4 = (5 + 4) mod 7 = 2\n => G#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#N": 1, "H#K": 1, "G#M": 2}, "id": "mod7_arith_d2_0123"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := 4. L#K := 1. L#O := 3 + 4 - J#O. L#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := 4. L#K := 1. L#O := 3 + 4 - J#O.", "query": "L#O", "answer": 3, "cot": "J#O = 4 -> J#O = 4\nL#O = 3 + 4 - J#O\n = 3 + 4 - 4\n 3 + 4 = (3 + 4) mod 7 = 0\n 0 - 4 = (0 - 4) mod 7 = 3\n => L#O = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#O": 1, "L#K": 1, "L#O": 2}, "id": "mod7_arith_d2_0124"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := 2. F#P := H#P / K#L. H#P := 5. J#N := 4. F#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := 2. F#P := H#P / K#L. H#P := 5. J#N := 4.", "query": "F#P", "answer": 6, "cot": "H#P = 5 -> H#P = 5\nK#L = 2 -> K#L = 2\nF#P = H#P / K#L\n = 5 / 2\n 5 / 2 = (5 × 2⁻¹) mod 7 = 6\n => F#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#P": 1, "J#N": 1, "K#L": 1, "F#P": 2}, "id": "mod7_arith_d2_0125"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#L := 2. H#M := L#L - J#L. L#L := 4. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#L := 2. H#M := L#L - J#L. L#L := 4.", "query": "H#M", "answer": 2, "cot": "L#L = 4 -> L#L = 4\nJ#L = 2 -> J#L = 2\nH#M = L#L - J#L\n = 4 - 2\n 4 - 2 = (4 - 2) mod 7 = 2\n => H#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#L": 1, "J#L": 1, "H#M": 2}, "id": "mod7_arith_d2_0126"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#P := 3. G#P := L#N - I#I. J#L := 3. I#I := 5. L#N := 3. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#P := 3. G#P := L#N - I#I. J#L := 3. I#I := 5. L#N := 3.", "query": "G#P", "answer": 5, "cot": "I#I = 5 -> I#I = 5\nL#N = 3 -> L#N = 3\nG#P = L#N - I#I\n = 3 - 5\n 3 - 5 = (3 - 5) mod 7 = 5\n => G#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#I": 1, "K#P": 1, "J#L": 1, "L#N": 1, "G#P": 2}, "id": "mod7_arith_d2_0127"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 4. J#M := 2. F#K := J#M / J#N. F#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 4. J#M := 2. F#K := J#M / J#N.", "query": "F#K", "answer": 4, "cot": "J#N = 4 -> J#N = 4\nJ#M = 2 -> J#M = 2\nF#K = J#M / J#N\n = 2 / 4\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n => F#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#N": 1, "J#M": 1, "F#K": 2}, "id": "mod7_arith_d2_0128"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#M := F#J. H#J := 5. J#N := 5. F#J := 2. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#M := F#J. H#J := 5. J#N := 5. F#J := 2.", "query": "J#M", "answer": 2, "cot": "F#J = 2 -> F#J = 2\nJ#M = F#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#J": 1, "J#N": 1, "F#J": 1, "J#M": 2}, "id": "mod7_arith_d2_0129"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := 1. F#P := 2. L#J := 3. I#K := F#K / L#J / F#P. F#K := 1. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := 1. F#P := 2. L#J := 3. I#K := F#K / L#J / F#P. F#K := 1.", "query": "I#K", "answer": 6, "cot": "F#K = 1 -> F#K = 1\nF#P = 2 -> F#P = 2\nL#J = 3 -> L#J = 3\nI#K = F#K / L#J / F#P\n = 1 / 3 / 2\n 1 / 3 = (1 × 3⁻¹) mod 7 = 5\n 5 / 2 = (5 × 2⁻¹) mod 7 = 6\n => I#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#K": 1, "F#P": 1, "L#J": 1, "L#M": 1, "I#K": 2}, "id": "mod7_arith_d2_0130"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#I := 1. J#K := 4. I#K := 5. J#L := J#K * 6. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#I := 1. J#K := 4. I#K := 5. J#L := J#K * 6.", "query": "J#L", "answer": 3, "cot": "J#K = 4 -> J#K = 4\nJ#L = J#K * 6\n = 4 * 6\n 4 * 6 = (4 × 6) mod 7 = 3\n => J#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#I": 1, "I#K": 1, "J#K": 1, "J#L": 2}, "id": "mod7_arith_d2_0131"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := 2. L#O := 4 * F#M + K#I. E#M := 6. F#M := 6. L#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := 2. L#O := 4 * F#M + K#I. E#M := 6. F#M := 6.", "query": "L#O", "answer": 5, "cot": "K#I = 2 -> K#I = 2\nF#M = 6 -> F#M = 6\nL#O = 4 * F#M + K#I\n = 4 * 6 + 2\n 4 * 6 = (4 × 6) mod 7 = 3\n 3 + 2 = (3 + 2) mod 7 = 5\n => L#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#I": 1, "F#M": 1, "E#M": 1, "L#O": 2}, "id": "mod7_arith_d2_0132"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := G#I + J#P. J#P := 1. I#J := 2. G#I := 2. I#K := 6. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := G#I + J#P. J#P := 1. I#J := 2. G#I := 2. I#K := 6.", "query": "F#L", "answer": 3, "cot": "J#P = 1 -> J#P = 1\nG#I = 2 -> G#I = 2\nF#L = G#I + J#P\n = 2 + 1\n 2 + 1 = (2 + 1) mod 7 = 3\n => F#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#P": 1, "I#J": 1, "I#K": 1, "G#I": 1, "F#L": 2}, "id": "mod7_arith_d2_0133"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#P := 2. L#M := 4. F#K := 1. L#J := F#K + L#M + 4. L#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#P := 2. L#M := 4. F#K := 1. L#J := F#K + L#M + 4.", "query": "L#J", "answer": 2, "cot": "F#K = 1 -> F#K = 1\nL#M = 4 -> L#M = 4\nL#J = F#K + L#M + 4\n = 1 + 4 + 4\n 1 + 4 = (1 + 4) mod 7 = 5\n 5 + 4 = (5 + 4) mod 7 = 2\n => L#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#K": 1, "E#P": 1, "L#M": 1, "L#J": 2}, "id": "mod7_arith_d2_0134"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := H#I - L#O + 6. H#I := 1. L#I := 3. L#O := 1. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := H#I - L#O + 6. H#I := 1. L#I := 3. L#O := 1.", "query": "L#M", "answer": 6, "cot": "L#O = 1 -> L#O = 1\nH#I = 1 -> H#I = 1\nL#M = H#I - L#O + 6\n = 1 - 1 + 6\n 1 - 1 = (1 - 1) mod 7 = 0\n 0 + 6 = (0 + 6) mod 7 = 6\n => L#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#O": 1, "H#I": 1, "L#I": 1, "L#M": 2}, "id": "mod7_arith_d2_0135"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := 1. I#J := 5. H#O := I#J + I#I. H#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := 1. I#J := 5. H#O := I#J + I#I.", "query": "H#O", "answer": 6, "cot": "I#I = 1 -> I#I = 1\nI#J = 5 -> I#J = 5\nH#O = I#J + I#I\n = 5 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n => H#O = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#I": 1, "I#J": 1, "H#O": 2}, "id": "mod7_arith_d2_0136"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#I := 1. I#K := 2. F#L := 3. F#M := 6. G#P := I#K / G#I. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#I := 1. I#K := 2. F#L := 3. F#M := 6. G#P := I#K / G#I.", "query": "G#P", "answer": 2, "cot": "I#K = 2 -> I#K = 2\nG#I = 1 -> G#I = 1\nG#P = I#K / G#I\n = 2 / 1\n 2 / 1 = (2 × 1⁻¹) mod 7 = 2\n => G#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#L": 1, "F#M": 1, "I#K": 1, "G#I": 1, "G#P": 2}, "id": "mod7_arith_d2_0137"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 3. F#O := 6. F#K := 3. F#N := J#N + F#K * F#O. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 3. F#O := 6. F#K := 3. F#N := J#N + F#K * F#O.", "query": "F#N", "answer": 1, "cot": "J#N = 3 -> J#N = 3\nF#O = 6 -> F#O = 6\nF#K = 3 -> F#K = 3\nF#N = J#N + F#K * F#O\n = 3 + 3 * 6\n 3 + 3 = (3 + 3) mod 7 = 6\n 6 * 6 = (6 × 6) mod 7 = 1\n => F#N = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#N": 1, "F#O": 1, "F#K": 1, "F#N": 2}, "id": "mod7_arith_d2_0138"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#M := 1. H#P := 5. H#K := F#M - H#P. K#J := 5. K#L := 6. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#M := 1. H#P := 5. H#K := F#M - H#P. K#J := 5. K#L := 6.", "query": "H#K", "answer": 3, "cot": "F#M = 1 -> F#M = 1\nH#P = 5 -> H#P = 5\nH#K = F#M - H#P\n = 1 - 5\n 1 - 5 = (1 - 5) mod 7 = 3\n => H#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#J": 1, "K#L": 1, "F#M": 1, "H#P": 1, "H#K": 2}, "id": "mod7_arith_d2_0139"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#O := E#O / K#O - J#N. J#N := 3. E#O := 5. K#O := 3. F#M := 5. L#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#O := E#O / K#O - J#N. J#N := 3. E#O := 5. K#O := 3. F#M := 5.", "query": "L#O", "answer": 1, "cot": "E#O = 5 -> E#O = 5\nK#O = 3 -> K#O = 3\nJ#N = 3 -> J#N = 3\nL#O = E#O / K#O - J#N\n = 5 / 3 - 3\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n 4 - 3 = (4 - 3) mod 7 = 1\n => L#O = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#O": 1, "F#M": 1, "K#O": 1, "J#N": 1, "L#O": 2}, "id": "mod7_arith_d2_0140"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := G#P. E#J := 2. G#P := 1. G#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := G#P. E#J := 2. G#P := 1.", "query": "G#O", "answer": 1, "cot": "G#P = 1 -> G#P = 1\nG#O = G#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#J": 1, "G#P": 1, "G#O": 2}, "id": "mod7_arith_d2_0141"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := 3. E#L := 4. K#J := 1. H#P := K#J. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := 3. E#L := 4. K#J := 1. H#P := K#J.", "query": "H#P", "answer": 1, "cot": "K#J = 1 -> K#J = 1\nH#P = K#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"L#J": 1, "E#L": 1, "K#J": 1, "H#P": 2}, "id": "mod7_arith_d2_0142"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := F#K + I#M. F#K := 1. I#M := 6. G#J := 4. H#I := 3. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := F#K + I#M. F#K := 1. I#M := 6. G#J := 4. H#I := 3.", "query": "L#M", "answer": 0, "cot": "I#M = 6 -> I#M = 6\nF#K = 1 -> F#K = 1\nL#M = F#K + I#M\n = 1 + 6\n 1 + 6 = (1 + 6) mod 7 = 0\n => L#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#M": 1, "F#K": 1, "G#J": 1, "H#I": 1, "L#M": 2}, "id": "mod7_arith_d2_0143"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#M := 2. I#I := I#P + G#M. I#P := 3. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#M := 2. I#I := I#P + G#M. I#P := 3.", "query": "I#I", "answer": 5, "cot": "I#P = 3 -> I#P = 3\nG#M = 2 -> G#M = 2\nI#I = I#P + G#M\n = 3 + 2\n 3 + 2 = (3 + 2) mod 7 = 5\n => I#I = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#P": 1, "G#M": 1, "I#I": 2}, "id": "mod7_arith_d2_0144"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#L := 1. F#M := I#L - 5. K#O := 4. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#L := 1. F#M := I#L - 5. K#O := 4.", "query": "F#M", "answer": 3, "cot": "I#L = 1 -> I#L = 1\nF#M = I#L - 5\n = 1 - 5\n 1 - 5 = (1 - 5) mod 7 = 3\n => F#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#O": 1, "I#L": 1, "F#M": 2}, "id": "mod7_arith_d2_0145"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := 5. K#P := E#L. K#N := 6. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := 5. K#P := E#L. K#N := 6.", "query": "K#P", "answer": 5, "cot": "E#L = 5 -> E#L = 5\nK#P = E#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#N": 1, "E#L": 1, "K#P": 2}, "id": "mod7_arith_d2_0146"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 5. J#P := 2. K#P := 4. G#N := 5. J#K := J#N. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 5. J#P := 2. K#P := 4. G#N := 5. J#K := J#N.", "query": "J#K", "answer": 5, "cot": "J#N = 5 -> J#N = 5\nJ#K = J#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#N": 1, "J#P": 1, "K#P": 1, "J#N": 1, "J#K": 2}, "id": "mod7_arith_d2_0147"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#O := 6. F#L := E#I - I#O. E#I := 6. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#O := 6. F#L := E#I - I#O. E#I := 6.", "query": "F#L", "answer": 0, "cot": "E#I = 6 -> E#I = 6\nI#O = 6 -> I#O = 6\nF#L = E#I - I#O\n = 6 - 6\n 6 - 6 = (6 - 6) mod 7 = 0\n => F#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#I": 1, "I#O": 1, "F#L": 2}, "id": "mod7_arith_d2_0148"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := 5. H#P := 1. G#I := H#P * L#P. L#P := 2. G#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := 5. H#P := 1. G#I := H#P * L#P. L#P := 2.", "query": "G#I", "answer": 2, "cot": "H#P = 1 -> H#P = 1\nL#P = 2 -> L#P = 2\nG#I = H#P * L#P\n = 1 * 2\n 1 * 2 = (1 × 2) mod 7 = 2\n => G#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#I": 1, "H#P": 1, "L#P": 1, "G#I": 2}, "id": "mod7_arith_d2_0149"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#M := 3 - G#N. J#O := 1. G#N := 5. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#M := 3 - G#N. J#O := 1. G#N := 5.", "query": "J#M", "answer": 5, "cot": "G#N = 5 -> G#N = 5\nJ#M = 3 - G#N\n = 3 - 5\n 3 - 5 = (3 - 5) mod 7 = 5\n => J#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#N": 1, "J#O": 1, "J#M": 2}, "id": "mod7_arith_d2_0150"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := 2. H#L := 3. F#K := 2. E#O := F#K / H#L. E#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := 2. H#L := 3. F#K := 2. E#O := F#K / H#L.", "query": "E#O", "answer": 3, "cot": "F#K = 2 -> F#K = 2\nH#L = 3 -> H#L = 3\nE#O = F#K / H#L\n = 2 / 3\n 2 / 3 = (2 × 3⁻¹) mod 7 = 3\n => E#O = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#J": 1, "F#K": 1, "H#L": 1, "E#O": 2}, "id": "mod7_arith_d2_0151"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := 5. E#K := F#P * L#N. L#N := 4. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := 5. E#K := F#P * L#N. L#N := 4.", "query": "E#K", "answer": 6, "cot": "L#N = 4 -> L#N = 4\nF#P = 5 -> F#P = 5\nE#K = F#P * L#N\n = 5 * 4\n 5 * 4 = (5 × 4) mod 7 = 6\n => E#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#N": 1, "F#P": 1, "E#K": 2}, "id": "mod7_arith_d2_0152"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#L := G#O * I#O. I#J := 3. I#O := 4. G#O := 4. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#L := G#O * I#O. I#J := 3. I#O := 4. G#O := 4.", "query": "L#L", "answer": 2, "cot": "I#O = 4 -> I#O = 4\nG#O = 4 -> G#O = 4\nL#L = G#O * I#O\n = 4 * 4\n 4 * 4 = (4 × 4) mod 7 = 2\n => L#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#O": 1, "I#J": 1, "G#O": 1, "L#L": 2}, "id": "mod7_arith_d2_0153"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#M := 6. J#N := 5. J#J := 2 - J#N. L#L := 2. G#L := 5. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#M := 6. J#N := 5. J#J := 2 - J#N. L#L := 2. G#L := 5.", "query": "J#J", "answer": 4, "cot": "J#N = 5 -> J#N = 5\nJ#J = 2 - J#N\n = 2 - 5\n 2 - 5 = (2 - 5) mod 7 = 4\n => J#J = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"L#L": 1, "G#L": 1, "I#M": 1, "J#N": 1, "J#J": 2}, "id": "mod7_arith_d2_0154"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := E#N - 6 * J#L. J#L := 3. E#N := 1. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := E#N - 6 * J#L. J#L := 3. E#N := 1.", "query": "H#K", "answer": 6, "cot": "J#L = 3 -> J#L = 3\nE#N = 1 -> E#N = 1\nH#K = E#N - 6 * J#L\n = 1 - 6 * 3\n 1 - 6 = (1 - 6) mod 7 = 2\n 2 * 3 = (2 × 3) mod 7 = 6\n => H#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#L": 1, "E#N": 1, "H#K": 2}, "id": "mod7_arith_d2_0155"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#L := E#J. E#J := 6. E#O := 1. E#K := 6. J#O := 2. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#L := E#J. E#J := 6. E#O := 1. E#K := 6. J#O := 2.", "query": "G#L", "answer": 6, "cot": "E#J = 6 -> E#J = 6\nG#L = E#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#O": 1, "E#J": 1, "E#K": 1, "J#O": 1, "G#L": 2}, "id": "mod7_arith_d2_0156"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := 6. E#J := 2. I#N := F#L / E#J / G#N. F#L := 5. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := 6. E#J := 2. I#N := F#L / E#J / G#N. F#L := 5.", "query": "I#N", "answer": 1, "cot": "G#N = 6 -> G#N = 6\nE#J = 2 -> E#J = 2\nF#L = 5 -> F#L = 5\nI#N = F#L / E#J / G#N\n = 5 / 2 / 6\n 5 / 2 = (5 × 2⁻¹) mod 7 = 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => I#N = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#N": 1, "E#J": 1, "F#L": 1, "I#N": 2}, "id": "mod7_arith_d2_0157"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#M := 2 + J#L. F#M := 3. G#M := 6. G#K := 2. J#L := 4. I#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#M := 2 + J#L. F#M := 3. G#M := 6. G#K := 2. J#L := 4.", "query": "I#M", "answer": 6, "cot": "J#L = 4 -> J#L = 4\nI#M = 2 + J#L\n = 2 + 4\n 2 + 4 = (2 + 4) mod 7 = 6\n => I#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#K": 1, "G#M": 1, "F#M": 1, "J#L": 1, "I#M": 2}, "id": "mod7_arith_d2_0158"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#M := 2. H#M := 5. F#I := 1. E#P := F#I / L#J - E#M. L#J := 5. E#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#M := 2. H#M := 5. F#I := 1. E#P := F#I / L#J - E#M. L#J := 5.", "query": "E#P", "answer": 1, "cot": "L#J = 5 -> L#J = 5\nE#M = 2 -> E#M = 2\nF#I = 1 -> F#I = 1\nE#P = F#I / L#J - E#M\n = 1 / 5 - 2\n 1 / 5 = (1 × 5⁻¹) mod 7 = 3\n 3 - 2 = (3 - 2) mod 7 = 1\n => E#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#J": 1, "H#M": 1, "E#M": 1, "F#I": 1, "E#P": 2}, "id": "mod7_arith_d2_0159"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#M := 4. J#P := 6. F#P := 4. H#M := 4. H#P := I#M. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#M := 4. J#P := 6. F#P := 4. H#M := 4. H#P := I#M.", "query": "H#P", "answer": 4, "cot": "I#M = 4 -> I#M = 4\nH#P = I#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#M": 1, "J#P": 1, "I#M": 1, "F#P": 1, "H#P": 2}, "id": "mod7_arith_d2_0160"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := 1. J#I := 1. G#O := 6. F#M := H#J / L#I. L#I := 4. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := 1. J#I := 1. G#O := 6. F#M := H#J / L#I. L#I := 4.", "query": "F#M", "answer": 2, "cot": "H#J = 1 -> H#J = 1\nL#I = 4 -> L#I = 4\nF#M = H#J / L#I\n = 1 / 4\n 1 / 4 = (1 × 4⁻¹) mod 7 = 2\n => F#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#I": 1, "H#J": 1, "L#I": 1, "G#O": 1, "F#M": 2}, "id": "mod7_arith_d2_0161"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := 4. H#L := 3. L#I := 4. H#P := K#I. H#I := 4. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := 4. H#L := 3. L#I := 4. H#P := K#I. H#I := 4.", "query": "H#P", "answer": 4, "cot": "K#I = 4 -> K#I = 4\nH#P = K#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#I": 1, "H#L": 1, "K#I": 1, "L#I": 1, "H#P": 2}, "id": "mod7_arith_d2_0162"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := 6. I#L := F#N + 2. J#K := 4. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := 6. I#L := F#N + 2. J#K := 4.", "query": "I#L", "answer": 1, "cot": "F#N = 6 -> F#N = 6\nI#L = F#N + 2\n = 6 + 2\n 6 + 2 = (6 + 2) mod 7 = 1\n => I#L = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#N": 1, "J#K": 1, "I#L": 2}, "id": "mod7_arith_d2_0163"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#J := L#M * H#N. F#M := 1. H#N := 2. L#M := 6. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#J := L#M * H#N. F#M := 1. H#N := 2. L#M := 6.", "query": "J#J", "answer": 5, "cot": "L#M = 6 -> L#M = 6\nH#N = 2 -> H#N = 2\nJ#J = L#M * H#N\n = 6 * 2\n 6 * 2 = (6 × 2) mod 7 = 5\n => J#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#M": 1, "F#M": 1, "H#N": 1, "J#J": 2}, "id": "mod7_arith_d2_0164"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := 4. J#M := E#K. J#K := 1. G#I := 2. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := 4. J#M := E#K. J#K := 1. G#I := 2.", "query": "J#M", "answer": 4, "cot": "E#K = 4 -> E#K = 4\nJ#M = E#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#I": 1, "J#K": 1, "E#K": 1, "J#M": 2}, "id": "mod7_arith_d2_0165"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := 5. L#K := 2. H#P := L#K - K#L. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := 5. L#K := 2. H#P := L#K - K#L.", "query": "H#P", "answer": 4, "cot": "K#L = 5 -> K#L = 5\nL#K = 2 -> L#K = 2\nH#P = L#K - K#L\n = 2 - 5\n 2 - 5 = (2 - 5) mod 7 = 4\n => H#P = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#L": 1, "L#K": 1, "H#P": 2}, "id": "mod7_arith_d2_0166"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := 6. K#K := 3. L#P := 6. E#I := I#J. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := 6. K#K := 3. L#P := 6. E#I := I#J.", "query": "E#I", "answer": 6, "cot": "I#J = 6 -> I#J = 6\nE#I = I#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#J": 1, "L#P": 1, "K#K": 1, "E#I": 2}, "id": "mod7_arith_d2_0167"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := 3 / K#M. G#N := 3. K#M := 1. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := 3 / K#M. G#N := 3. K#M := 1.", "query": "E#I", "answer": 3, "cot": "K#M = 1 -> K#M = 1\nE#I = 3 / K#M\n = 3 / 1\n 3 / 1 = (3 × 1⁻¹) mod 7 = 3\n => E#I = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#M": 1, "G#N": 1, "E#I": 2}, "id": "mod7_arith_d2_0168"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#M := 4. F#K := 2. J#J := F#K - 6. G#I := 6. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#M := 4. F#K := 2. J#J := F#K - 6. G#I := 6.", "query": "J#J", "answer": 3, "cot": "F#K = 2 -> F#K = 2\nJ#J = F#K - 6\n = 2 - 6\n 2 - 6 = (2 - 6) mod 7 = 3\n => J#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#K": 1, "I#M": 1, "G#I": 1, "J#J": 2}, "id": "mod7_arith_d2_0169"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := 6. L#N := 6. L#O := L#P. J#I := 5. L#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := 6. L#N := 6. L#O := L#P. J#I := 5.", "query": "L#O", "answer": 6, "cot": "L#P = 6 -> L#P = 6\nL#O = L#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#I": 1, "L#P": 1, "L#N": 1, "L#O": 2}, "id": "mod7_arith_d2_0170"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := 3. F#M := E#P. E#P := 1. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := 3. F#M := E#P. E#P := 1.", "query": "F#M", "answer": 1, "cot": "E#P = 1 -> E#P = 1\nF#M = E#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#N": 1, "E#P": 1, "F#M": 2}, "id": "mod7_arith_d2_0171"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := E#J + 4. E#J := 3. E#L := 6. G#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := E#J + 4. E#J := 3. E#L := 6.", "query": "G#N", "answer": 0, "cot": "E#J = 3 -> E#J = 3\nG#N = E#J + 4\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => G#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#J": 1, "E#L": 1, "G#N": 2}, "id": "mod7_arith_d2_0172"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := 6. E#N := 5. E#K := G#N + L#I. L#I := 6. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := 6. E#N := 5. E#K := G#N + L#I. L#I := 6.", "query": "E#K", "answer": 5, "cot": "G#N = 6 -> G#N = 6\nL#I = 6 -> L#I = 6\nE#K = G#N + L#I\n = 6 + 6\n 6 + 6 = (6 + 6) mod 7 = 5\n => E#K = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#N": 1, "E#N": 1, "L#I": 1, "E#K": 2}, "id": "mod7_arith_d2_0173"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#K := I#O / 4. F#P := 4. I#L := 6. L#L := 3. I#O := 2. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#K := I#O / 4. F#P := 4. I#L := 6. L#L := 3. I#O := 2.", "query": "L#K", "answer": 4, "cot": "I#O = 2 -> I#O = 2\nL#K = I#O / 4\n = 2 / 4\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n => L#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#L": 1, "L#L": 1, "F#P": 1, "I#O": 1, "L#K": 2}, "id": "mod7_arith_d2_0174"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := H#J * E#K * 3. H#J := 6. K#P := 4. E#K := 6. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := H#J * E#K * 3. H#J := 6. K#P := 4. E#K := 6.", "query": "F#J", "answer": 3, "cot": "H#J = 6 -> H#J = 6\nE#K = 6 -> E#K = 6\nF#J = H#J * E#K * 3\n = 6 * 6 * 3\n 6 * 6 = (6 × 6) mod 7 = 1\n 1 * 3 = (1 × 3) mod 7 = 3\n => F#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#J": 1, "K#P": 1, "E#K": 1, "F#J": 2}, "id": "mod7_arith_d2_0175"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#L := L#N. L#N := 4. K#J := 3. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#L := L#N. L#N := 4. K#J := 3.", "query": "L#L", "answer": 4, "cot": "L#N = 4 -> L#N = 4\nL#L = L#N = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"L#N": 1, "K#J": 1, "L#L": 2}, "id": "mod7_arith_d2_0176"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := 3. E#J := 5. G#N := F#N - E#J. L#J := 4. G#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := 3. E#J := 5. G#N := F#N - E#J. L#J := 4.", "query": "G#N", "answer": 5, "cot": "E#J = 5 -> E#J = 5\nF#N = 3 -> F#N = 3\nG#N = F#N - E#J\n = 3 - 5\n 3 - 5 = (3 - 5) mod 7 = 5\n => G#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#J": 1, "E#J": 1, "F#N": 1, "G#N": 2}, "id": "mod7_arith_d2_0177"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#M := 6. L#J := 1. J#M := 2. G#P := J#M / L#J - E#M. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#M := 6. L#J := 1. J#M := 2. G#P := J#M / L#J - E#M.", "query": "G#P", "answer": 3, "cot": "J#M = 2 -> J#M = 2\nL#J = 1 -> L#J = 1\nE#M = 6 -> E#M = 6\nG#P = J#M / L#J - E#M\n = 2 / 1 - 6\n 2 / 1 = (2 × 1⁻¹) mod 7 = 2\n 2 - 6 = (2 - 6) mod 7 = 3\n => G#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#M": 1, "L#J": 1, "E#M": 1, "G#P": 2}, "id": "mod7_arith_d2_0178"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#K := K#K * J#L. J#L := 5. K#K := 6. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#K := K#K * J#L. J#L := 5. K#K := 6.", "query": "L#K", "answer": 2, "cot": "J#L = 5 -> J#L = 5\nK#K = 6 -> K#K = 6\nL#K = K#K * J#L\n = 6 * 5\n 6 * 5 = (6 × 5) mod 7 = 2\n => L#K = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#L": 1, "K#K": 1, "L#K": 2}, "id": "mod7_arith_d2_0179"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#K := 2. L#L := 1. F#M := G#N - 5 * 4. G#N := 3. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#K := 2. L#L := 1. F#M := G#N - 5 * 4. G#N := 3.", "query": "F#M", "answer": 6, "cot": "G#N = 3 -> G#N = 3\nF#M = G#N - 5 * 4\n = 3 - 5 * 4\n 3 - 5 = (3 - 5) mod 7 = 5\n 5 * 4 = (5 × 4) mod 7 = 6\n => F#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#K": 1, "G#N": 1, "L#L": 1, "F#M": 2}, "id": "mod7_arith_d2_0180"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := F#J / F#O / 6. K#J := 6. F#O := 3. F#J := 6. J#J := 5. J#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := F#J / F#O / 6. K#J := 6. F#O := 3. F#J := 6. J#J := 5.", "query": "J#O", "answer": 5, "cot": "F#O = 3 -> F#O = 3\nF#J = 6 -> F#J = 6\nJ#O = F#J / F#O / 6\n = 6 / 3 / 6\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => J#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#J": 1, "J#J": 1, "F#O": 1, "F#J": 1, "J#O": 2}, "id": "mod7_arith_d2_0181"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#M := 1. J#P := 5. J#N := 6. G#P := 4 + J#P. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#M := 1. J#P := 5. J#N := 6. G#P := 4 + J#P.", "query": "G#P", "answer": 2, "cot": "J#P = 5 -> J#P = 5\nG#P = 4 + J#P\n = 4 + 5\n 4 + 5 = (4 + 5) mod 7 = 2\n => G#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#N": 1, "J#P": 1, "H#M": 1, "G#P": 2}, "id": "mod7_arith_d2_0182"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#M := 3. I#I := H#O + 5 * E#M. H#O := 1. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#M := 3. I#I := H#O + 5 * E#M. H#O := 1.", "query": "I#I", "answer": 4, "cot": "H#O = 1 -> H#O = 1\nE#M = 3 -> E#M = 3\nI#I = H#O + 5 * E#M\n = 1 + 5 * 3\n 1 + 5 = (1 + 5) mod 7 = 6\n 6 * 3 = (6 × 3) mod 7 = 4\n => I#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#O": 1, "E#M": 1, "I#I": 2}, "id": "mod7_arith_d2_0183"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := F#K / J#O. F#K := 2. J#O := 6. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := F#K / J#O. F#K := 2. J#O := 6.", "query": "H#K", "answer": 5, "cot": "F#K = 2 -> F#K = 2\nJ#O = 6 -> J#O = 6\nH#K = F#K / J#O\n = 2 / 6\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => H#K = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#K": 1, "J#O": 1, "H#K": 2}, "id": "mod7_arith_d2_0184"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#L := F#M * H#L. H#L := 6. F#M := 3. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#L := F#M * H#L. H#L := 6. F#M := 3.", "query": "I#L", "answer": 4, "cot": "F#M = 3 -> F#M = 3\nH#L = 6 -> H#L = 6\nI#L = F#M * H#L\n = 3 * 6\n 3 * 6 = (3 × 6) mod 7 = 4\n => I#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#M": 1, "H#L": 1, "I#L": 2}, "id": "mod7_arith_d2_0185"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := 6. H#P := 5. J#J := I#J - E#K. H#K := 6. E#K := 1. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := 6. H#P := 5. J#J := I#J - E#K. H#K := 6. E#K := 1.", "query": "J#J", "answer": 5, "cot": "I#J = 6 -> I#J = 6\nE#K = 1 -> E#K = 1\nJ#J = I#J - E#K\n = 6 - 1\n 6 - 1 = (6 - 1) mod 7 = 5\n => J#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#J": 1, "H#K": 1, "H#P": 1, "E#K": 1, "J#J": 2}, "id": "mod7_arith_d2_0186"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := 2. I#N := 2. F#I := 3. I#I := 6. L#P := I#I / K#I. L#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := 2. I#N := 2. F#I := 3. I#I := 6. L#P := I#I / K#I.", "query": "L#P", "answer": 3, "cot": "K#I = 2 -> K#I = 2\nI#I = 6 -> I#I = 6\nL#P = I#I / K#I\n = 6 / 2\n 6 / 2 = (6 × 2⁻¹) mod 7 = 3\n => L#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#N": 1, "K#I": 1, "F#I": 1, "I#I": 1, "L#P": 2}, "id": "mod7_arith_d2_0187"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#K := J#I. G#L := 4. J#I := 2. G#M := 5. E#P := 4. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#K := J#I. G#L := 4. J#I := 2. G#M := 5. E#P := 4.", "query": "J#K", "answer": 2, "cot": "J#I = 2 -> J#I = 2\nJ#K = J#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#I": 1, "E#P": 1, "G#M": 1, "G#L": 1, "J#K": 2}, "id": "mod7_arith_d2_0188"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#O := 4. F#M := I#L. I#L := 5. L#K := 1. H#I := 4. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#O := 4. F#M := I#L. I#L := 5. L#K := 1. H#I := 4.", "query": "F#M", "answer": 5, "cot": "I#L = 5 -> I#L = 5\nF#M = I#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#L": 1, "H#I": 1, "K#O": 1, "L#K": 1, "F#M": 2}, "id": "mod7_arith_d2_0189"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := G#O + F#L - 3. L#M := 3. F#L := 2. G#O := 3. F#I := 2. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := G#O + F#L - 3. L#M := 3. F#L := 2. G#O := 3. F#I := 2.", "query": "G#J", "answer": 2, "cot": "F#L = 2 -> F#L = 2\nG#O = 3 -> G#O = 3\nG#J = G#O + F#L - 3\n = 3 + 2 - 3\n 3 + 2 = (3 + 2) mod 7 = 5\n 5 - 3 = (5 - 3) mod 7 = 2\n => G#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#M": 1, "F#L": 1, "F#I": 1, "G#O": 1, "G#J": 2}, "id": "mod7_arith_d2_0190"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#O := 3. K#M := I#M * I#O. I#M := 2. K#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#O := 3. K#M := I#M * I#O. I#M := 2.", "query": "K#M", "answer": 6, "cot": "I#O = 3 -> I#O = 3\nI#M = 2 -> I#M = 2\nK#M = I#M * I#O\n = 2 * 3\n 2 * 3 = (2 × 3) mod 7 = 6\n => K#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#O": 1, "I#M": 1, "K#M": 2}, "id": "mod7_arith_d2_0191"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#P := 2. L#O := E#P + H#M. H#M := 1. K#O := 4. E#L := 3. L#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#P := 2. L#O := E#P + H#M. H#M := 1. K#O := 4. E#L := 3.", "query": "L#O", "answer": 3, "cot": "E#P = 2 -> E#P = 2\nH#M = 1 -> H#M = 1\nL#O = E#P + H#M\n = 2 + 1\n 2 + 1 = (2 + 1) mod 7 = 3\n => L#O = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#P": 1, "E#L": 1, "K#O": 1, "H#M": 1, "L#O": 2}, "id": "mod7_arith_d2_0192"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#O := 1. G#N := K#J - K#O * 4. K#J := 1. G#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#O := 1. G#N := K#J - K#O * 4. K#J := 1.", "query": "G#N", "answer": 0, "cot": "K#O = 1 -> K#O = 1\nK#J = 1 -> K#J = 1\nG#N = K#J - K#O * 4\n = 1 - 1 * 4\n 1 - 1 = (1 - 1) mod 7 = 0\n 0 * 4 = (0 × 4) mod 7 = 0\n => G#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#O": 1, "K#J": 1, "G#N": 2}, "id": "mod7_arith_d2_0193"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := 3. J#O := 2. I#O := J#O. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := 3. J#O := 2. I#O := J#O.", "query": "I#O", "answer": 2, "cot": "J#O = 2 -> J#O = 2\nI#O = J#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#I": 1, "J#O": 1, "I#O": 2}, "id": "mod7_arith_d2_0194"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := F#M - I#O. I#L := 6. F#M := 3. H#I := 3. I#O := 6. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := F#M - I#O. I#L := 6. F#M := 3. H#I := 3. I#O := 6.", "query": "E#I", "answer": 4, "cot": "I#O = 6 -> I#O = 6\nF#M = 3 -> F#M = 3\nE#I = F#M - I#O\n = 3 - 6\n 3 - 6 = (3 - 6) mod 7 = 4\n => E#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#O": 1, "I#L": 1, "F#M": 1, "H#I": 1, "E#I": 2}, "id": "mod7_arith_d2_0195"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#M := 5 + K#I. K#I := 6. E#P := 1. H#I := 4. G#P := 2. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#M := 5 + K#I. K#I := 6. E#P := 1. H#I := 4. G#P := 2.", "query": "E#M", "answer": 4, "cot": "K#I = 6 -> K#I = 6\nE#M = 5 + K#I\n = 5 + 6\n 5 + 6 = (5 + 6) mod 7 = 4\n => E#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#P": 1, "G#P": 1, "K#I": 1, "H#I": 1, "E#M": 2}, "id": "mod7_arith_d2_0196"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := E#P - F#M. E#P := 2. F#M := 4. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := E#P - F#M. E#P := 2. F#M := 4.", "query": "G#J", "answer": 5, "cot": "F#M = 4 -> F#M = 4\nE#P = 2 -> E#P = 2\nG#J = E#P - F#M\n = 2 - 4\n 2 - 4 = (2 - 4) mod 7 = 5\n => G#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#M": 1, "E#P": 1, "G#J": 2}, "id": "mod7_arith_d2_0197"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := F#K - 3. J#O := 5. E#L := 1. E#M := 6. F#K := 2. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := F#K - 3. J#O := 5. E#L := 1. E#M := 6. F#K := 2.", "query": "L#I", "answer": 6, "cot": "F#K = 2 -> F#K = 2\nL#I = F#K - 3\n = 2 - 3\n 2 - 3 = (2 - 3) mod 7 = 6\n => L#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#L": 1, "E#M": 1, "J#O": 1, "F#K": 1, "L#I": 2}, "id": "mod7_arith_d2_0198"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#O := 2. L#N := 3. E#I := L#O * L#N. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#O := 2. L#N := 3. E#I := L#O * L#N.", "query": "E#I", "answer": 6, "cot": "L#N = 3 -> L#N = 3\nL#O = 2 -> L#O = 2\nE#I = L#O * L#N\n = 2 * 3\n 2 * 3 = (2 × 3) mod 7 = 6\n => E#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#N": 1, "L#O": 1, "E#I": 2}, "id": "mod7_arith_d2_0199"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := 2 + J#N * G#P. J#K := 1. G#P := 3. J#N := 2. L#O := 3. K#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := 2 + J#N * G#P. J#K := 1. G#P := 3. J#N := 2. L#O := 3.", "query": "K#M", "answer": 5, "cot": "J#N = 2 -> J#N = 2\nG#P = 3 -> G#P = 3\nK#M = 2 + J#N * G#P\n = 2 + 2 * 3\n 2 + 2 = (2 + 2) mod 7 = 4\n 4 * 3 = (4 × 3) mod 7 = 5\n => K#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#O": 1, "J#N": 1, "G#P": 1, "J#K": 1, "K#M": 2}, "id": "mod7_arith_d2_0200"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#N := H#P - H#L. H#L := 6. H#P := 4. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#N := H#P - H#L. H#L := 6. H#P := 4.", "query": "I#N", "answer": 5, "cot": "H#L = 6 -> H#L = 6\nH#P = 4 -> H#P = 4\nI#N = H#P - H#L\n = 4 - 6\n 4 - 6 = (4 - 6) mod 7 = 5\n => I#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#L": 1, "H#P": 1, "I#N": 2}, "id": "mod7_arith_d2_0201"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := 3. L#J := 6. I#K := 4. L#L := I#K / F#P - L#J. F#P := 3. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := 3. L#J := 6. I#K := 4. L#L := I#K / F#P - L#J. F#P := 3.", "query": "L#L", "answer": 0, "cot": "I#K = 4 -> I#K = 4\nL#J = 6 -> L#J = 6\nF#P = 3 -> F#P = 3\nL#L = I#K / F#P - L#J\n = 4 / 3 - 6\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n 6 - 6 = (6 - 6) mod 7 = 0\n => L#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#J": 1, "I#K": 1, "L#J": 1, "F#P": 1, "L#L": 2}, "id": "mod7_arith_d2_0202"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := 6. L#I := 1. K#P := 1. J#P := K#P - 3. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := 6. L#I := 1. K#P := 1. J#P := K#P - 3.", "query": "J#P", "answer": 5, "cot": "K#P = 1 -> K#P = 1\nJ#P = K#P - 3\n = 1 - 3\n 1 - 3 = (1 - 3) mod 7 = 5\n => J#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#P": 1, "L#I": 1, "I#I": 1, "J#P": 2}, "id": "mod7_arith_d2_0203"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := 5. F#P := 4. J#J := 2. L#N := 5. I#K := F#L - J#J. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := 5. F#P := 4. J#J := 2. L#N := 5. I#K := F#L - J#J.", "query": "I#K", "answer": 3, "cot": "F#L = 5 -> F#L = 5\nJ#J = 2 -> J#J = 2\nI#K = F#L - J#J\n = 5 - 2\n 5 - 2 = (5 - 2) mod 7 = 3\n => I#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#L": 1, "F#P": 1, "L#N": 1, "J#J": 1, "I#K": 2}, "id": "mod7_arith_d2_0204"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#M := F#N. F#N := 3. L#J := 4. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#M := F#N. F#N := 3. L#J := 4.", "query": "H#M", "answer": 3, "cot": "F#N = 3 -> F#N = 3\nH#M = F#N = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"L#J": 1, "F#N": 1, "H#M": 2}, "id": "mod7_arith_d2_0205"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := H#L - I#M + 4. I#M := 6. H#L := 4. F#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := H#L - I#M + 4. I#M := 6. H#L := 4.", "query": "F#P", "answer": 2, "cot": "I#M = 6 -> I#M = 6\nH#L = 4 -> H#L = 4\nF#P = H#L - I#M + 4\n = 4 - 6 + 4\n 4 - 6 = (4 - 6) mod 7 = 5\n 5 + 4 = (5 + 4) mod 7 = 2\n => F#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#M": 1, "H#L": 1, "F#P": 2}, "id": "mod7_arith_d2_0206"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#L := 1. J#O := I#I * 4. I#I := 1. E#L := 6. J#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#L := 1. J#O := I#I * 4. I#I := 1. E#L := 6.", "query": "J#O", "answer": 4, "cot": "I#I = 1 -> I#I = 1\nJ#O = I#I * 4\n = 1 * 4\n 1 * 4 = (1 × 4) mod 7 = 4\n => J#O = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#L": 1, "L#L": 1, "I#I": 1, "J#O": 2}, "id": "mod7_arith_d2_0207"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#K := G#O. G#O := 4. J#J := 5. I#M := 2. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#K := G#O. G#O := 4. J#J := 5. I#M := 2.", "query": "I#K", "answer": 4, "cot": "G#O = 4 -> G#O = 4\nI#K = G#O = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#M": 1, "G#O": 1, "J#J": 1, "I#K": 2}, "id": "mod7_arith_d2_0208"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#P := 3. K#P := 5. I#P := E#P. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#P := 3. K#P := 5. I#P := E#P.", "query": "I#P", "answer": 3, "cot": "E#P = 3 -> E#P = 3\nI#P = E#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#P": 1, "E#P": 1, "I#P": 2}, "id": "mod7_arith_d2_0209"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := 1. J#K := E#M. E#M := 3. H#I := 1. E#O := 5. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := 1. J#K := E#M. E#M := 3. H#I := 1. E#O := 5.", "query": "J#K", "answer": 3, "cot": "E#M = 3 -> E#M = 3\nJ#K = E#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#O": 1, "G#O": 1, "H#I": 1, "E#M": 1, "J#K": 2}, "id": "mod7_arith_d2_0210"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := 1. L#M := 3. I#L := L#M - H#J. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := 1. L#M := 3. I#L := L#M - H#J.", "query": "I#L", "answer": 2, "cot": "H#J = 1 -> H#J = 1\nL#M = 3 -> L#M = 3\nI#L = L#M - H#J\n = 3 - 1\n 3 - 1 = (3 - 1) mod 7 = 2\n => I#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#J": 1, "L#M": 1, "I#L": 2}, "id": "mod7_arith_d2_0211"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := 2. I#O := G#M + J#N. J#N := 3. G#M := 3. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := 2. I#O := G#M + J#N. J#N := 3. G#M := 3.", "query": "I#O", "answer": 6, "cot": "G#M = 3 -> G#M = 3\nJ#N = 3 -> J#N = 3\nI#O = G#M + J#N\n = 3 + 3\n 3 + 3 = (3 + 3) mod 7 = 6\n => I#O = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#P": 1, "G#M": 1, "J#N": 1, "I#O": 2}, "id": "mod7_arith_d2_0212"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := 5. K#N := 5 + 3 - F#I. F#I := 6. K#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := 5. K#N := 5 + 3 - F#I. F#I := 6.", "query": "K#N", "answer": 2, "cot": "F#I = 6 -> F#I = 6\nK#N = 5 + 3 - F#I\n = 5 + 3 - 6\n 5 + 3 = (5 + 3) mod 7 = 1\n 1 - 6 = (1 - 6) mod 7 = 2\n => K#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"F#P": 1, "F#I": 1, "K#N": 2}, "id": "mod7_arith_d2_0213"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#M := 4. J#M := I#M. G#J := 4. E#M := 2. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#M := 4. J#M := I#M. G#J := 4. E#M := 2.", "query": "J#M", "answer": 4, "cot": "I#M = 4 -> I#M = 4\nJ#M = I#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#M": 1, "G#J": 1, "I#M": 1, "J#M": 2}, "id": "mod7_arith_d2_0214"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#N := 5. L#N := 1. L#O := K#I. I#J := 2. K#I := 4. L#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#N := 5. L#N := 1. L#O := K#I. I#J := 2. K#I := 4.", "query": "L#O", "answer": 4, "cot": "K#I = 4 -> K#I = 4\nL#O = K#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#N": 1, "I#J": 1, "L#N": 1, "K#I": 1, "L#O": 2}, "id": "mod7_arith_d2_0215"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := 5. F#L := L#M. L#M := 3. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := 5. F#L := L#M. L#M := 3.", "query": "F#L", "answer": 3, "cot": "L#M = 3 -> L#M = 3\nF#L = L#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#O": 1, "L#M": 1, "F#L": 2}, "id": "mod7_arith_d2_0216"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#O := 6. E#M := H#P. G#N := 4. H#P := 5. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#O := 6. E#M := H#P. G#N := 4. H#P := 5.", "query": "E#M", "answer": 5, "cot": "H#P = 5 -> H#P = 5\nE#M = H#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#P": 1, "G#N": 1, "F#O": 1, "E#M": 2}, "id": "mod7_arith_d2_0217"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := 4. I#P := L#I * K#I. F#P := 4. K#I := 6. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := 4. I#P := L#I * K#I. F#P := 4. K#I := 6.", "query": "I#P", "answer": 3, "cot": "K#I = 6 -> K#I = 6\nL#I = 4 -> L#I = 4\nI#P = L#I * K#I\n = 4 * 6\n 4 * 6 = (4 × 6) mod 7 = 3\n => I#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#I": 1, "F#P": 1, "L#I": 1, "I#P": 2}, "id": "mod7_arith_d2_0218"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#O := 5. F#J := E#O / G#M. G#M := 2. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#O := 5. F#J := E#O / G#M. G#M := 2.", "query": "F#J", "answer": 6, "cot": "G#M = 2 -> G#M = 2\nE#O = 5 -> E#O = 5\nF#J = E#O / G#M\n = 5 / 2\n 5 / 2 = (5 × 2⁻¹) mod 7 = 6\n => F#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#M": 1, "E#O": 1, "F#J": 2}, "id": "mod7_arith_d2_0219"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := 6. F#N := E#M + F#L. E#K := 2. E#M := 1. H#N := 6. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := 6. F#N := E#M + F#L. E#K := 2. E#M := 1. H#N := 6.", "query": "F#N", "answer": 0, "cot": "F#L = 6 -> F#L = 6\nE#M = 1 -> E#M = 1\nF#N = E#M + F#L\n = 1 + 6\n 1 + 6 = (1 + 6) mod 7 = 0\n => F#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#L": 1, "H#N": 1, "E#K": 1, "E#M": 1, "F#N": 2}, "id": "mod7_arith_d2_0220"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := 5. K#N := G#M. G#M := 5. K#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := 5. K#N := G#M. G#M := 5.", "query": "K#N", "answer": 5, "cot": "G#M = 5 -> G#M = 5\nK#N = G#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#M": 1, "K#M": 1, "K#N": 2}, "id": "mod7_arith_d2_0221"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := 2. I#N := 2. G#K := 5. L#P := G#P - G#K * I#N. G#P := 6. L#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := 2. I#N := 2. G#K := 5. L#P := G#P - G#K * I#N. G#P := 6.", "query": "L#P", "answer": 2, "cot": "I#N = 2 -> I#N = 2\nG#K = 5 -> G#K = 5\nG#P = 6 -> G#P = 6\nL#P = G#P - G#K * I#N\n = 6 - 5 * 2\n 6 - 5 = (6 - 5) mod 7 = 1\n 1 * 2 = (1 × 2) mod 7 = 2\n => L#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#N": 1, "G#K": 1, "E#N": 1, "G#P": 1, "L#P": 2}, "id": "mod7_arith_d2_0222"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := 6. I#N := 1. I#I := 2. E#M := E#I - I#I + I#N. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := 6. I#N := 1. I#I := 2. E#M := E#I - I#I + I#N.", "query": "E#M", "answer": 5, "cot": "I#I = 2 -> I#I = 2\nI#N = 1 -> I#N = 1\nE#I = 6 -> E#I = 6\nE#M = E#I - I#I + I#N\n = 6 - 2 + 1\n 6 - 2 = (6 - 2) mod 7 = 4\n 4 + 1 = (4 + 1) mod 7 = 5\n => E#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#I": 1, "I#N": 1, "E#I": 1, "E#M": 2}, "id": "mod7_arith_d2_0223"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := I#O / E#P. E#P := 2. I#O := 4. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := I#O / E#P. E#P := 2. I#O := 4.", "query": "E#K", "answer": 2, "cot": "E#P = 2 -> E#P = 2\nI#O = 4 -> I#O = 4\nE#K = I#O / E#P\n = 4 / 2\n 4 / 2 = (4 × 2⁻¹) mod 7 = 2\n => E#K = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#P": 1, "I#O": 1, "E#K": 2}, "id": "mod7_arith_d2_0224"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#K := E#N. H#K := 3. E#N := 1. F#N := 4. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#K := E#N. H#K := 3. E#N := 1. F#N := 4.", "query": "I#K", "answer": 1, "cot": "E#N = 1 -> E#N = 1\nI#K = E#N = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#N": 1, "H#K": 1, "F#N": 1, "I#K": 2}, "id": "mod7_arith_d2_0225"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#M := L#L. H#N := 5. L#L := 1. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#M := L#L. H#N := 5. L#L := 1.", "query": "E#M", "answer": 1, "cot": "L#L = 1 -> L#L = 1\nE#M = L#L = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"L#L": 1, "H#N": 1, "E#M": 2}, "id": "mod7_arith_d2_0226"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := 2. L#L := J#J / 6. J#J := 2. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := 2. L#L := J#J / 6. J#J := 2.", "query": "L#L", "answer": 5, "cot": "J#J = 2 -> J#J = 2\nL#L = J#J / 6\n = 2 / 6\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => L#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#J": 1, "J#J": 1, "L#L": 2}, "id": "mod7_arith_d2_0227"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#O := H#L. H#L := 4. K#N := 6. H#J := 4. J#K := 5. F#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#O := H#L. H#L := 4. K#N := 6. H#J := 4. J#K := 5.", "query": "F#O", "answer": 4, "cot": "H#L = 4 -> H#L = 4\nF#O = H#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#N": 1, "H#J": 1, "H#L": 1, "J#K": 1, "F#O": 2}, "id": "mod7_arith_d2_0228"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := 3. L#L := 3. L#M := H#M - L#L. H#M := 6. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := 3. L#L := 3. L#M := H#M - L#L. H#M := 6.", "query": "L#M", "answer": 3, "cot": "H#M = 6 -> H#M = 6\nL#L = 3 -> L#L = 3\nL#M = H#M - L#L\n = 6 - 3\n 6 - 3 = (6 - 3) mod 7 = 3\n => L#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#M": 1, "L#L": 1, "E#N": 1, "L#M": 2}, "id": "mod7_arith_d2_0229"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := 6. K#O := 2 / L#N. G#J := 1. K#L := 4. L#N := 1. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := 6. K#O := 2 / L#N. G#J := 1. K#L := 4. L#N := 1.", "query": "K#O", "answer": 2, "cot": "L#N = 1 -> L#N = 1\nK#O = 2 / L#N\n = 2 / 1\n 2 / 1 = (2 × 1⁻¹) mod 7 = 2\n => K#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"L#N": 1, "G#J": 1, "K#L": 1, "E#I": 1, "K#O": 2}, "id": "mod7_arith_d2_0230"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#I := 1. G#J := 1. K#O := 4. J#J := G#J / J#K * F#I. J#K := 4. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#I := 1. G#J := 1. K#O := 4. J#J := G#J / J#K * F#I. J#K := 4.", "query": "J#J", "answer": 2, "cot": "J#K = 4 -> J#K = 4\nG#J = 1 -> G#J = 1\nF#I = 1 -> F#I = 1\nJ#J = G#J / J#K * F#I\n = 1 / 4 * 1\n 1 / 4 = (1 × 4⁻¹) mod 7 = 2\n 2 * 1 = (2 × 1) mod 7 = 2\n => J#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#K": 1, "G#J": 1, "F#I": 1, "K#O": 1, "J#J": 2}, "id": "mod7_arith_d2_0231"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#M := 2. J#O := 4. F#J := J#O. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#M := 2. J#O := 4. F#J := J#O.", "query": "F#J", "answer": 4, "cot": "J#O = 4 -> J#O = 4\nF#J = J#O = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"J#O": 1, "J#M": 1, "F#J": 2}, "id": "mod7_arith_d2_0232"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#J := 5. K#K := 1. I#I := 2. E#K := I#I + K#K. I#L := 4. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#J := 5. K#K := 1. I#I := 2. E#K := I#I + K#K. I#L := 4.", "query": "E#K", "answer": 3, "cot": "I#I = 2 -> I#I = 2\nK#K = 1 -> K#K = 1\nE#K = I#I + K#K\n = 2 + 1\n 2 + 1 = (2 + 1) mod 7 = 3\n => E#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#L": 1, "I#I": 1, "K#K": 1, "J#J": 1, "E#K": 2}, "id": "mod7_arith_d2_0233"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#K := I#L. I#L := 5. I#N := 5. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#K := I#L. I#L := 5. I#N := 5.", "query": "I#K", "answer": 5, "cot": "I#L = 5 -> I#L = 5\nI#K = I#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"I#N": 1, "I#L": 1, "I#K": 2}, "id": "mod7_arith_d2_0234"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := 5. E#J := 1. I#K := E#J + 2. G#L := 3. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := 5. E#J := 1. I#K := E#J + 2. G#L := 3.", "query": "I#K", "answer": 3, "cot": "E#J = 1 -> E#J = 1\nI#K = E#J + 2\n = 1 + 2\n 1 + 2 = (1 + 2) mod 7 = 3\n => I#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"G#L": 1, "E#J": 1, "H#O": 1, "I#K": 2}, "id": "mod7_arith_d2_0235"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := 3. G#I := 3. E#M := 5. J#N := 3. J#I := E#M. J#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := 3. G#I := 3. E#M := 5. J#N := 3. J#I := E#M.", "query": "J#I", "answer": 5, "cot": "E#M = 5 -> E#M = 5\nJ#I = E#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"K#I": 1, "J#N": 1, "E#M": 1, "G#I": 1, "J#I": 2}, "id": "mod7_arith_d2_0236"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := 2. H#K := 5. I#J := 2. J#N := H#K + I#P. J#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := 2. H#K := 5. I#J := 2. J#N := H#K + I#P.", "query": "J#N", "answer": 0, "cot": "H#K = 5 -> H#K = 5\nI#P = 2 -> I#P = 2\nJ#N = H#K + I#P\n = 5 + 2\n 5 + 2 = (5 + 2) mod 7 = 0\n => J#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#K": 1, "I#J": 1, "I#P": 1, "J#N": 2}, "id": "mod7_arith_d2_0237"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := 4. E#P := 2. E#N := 3. I#I := E#N - F#J. F#J := 1. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := 4. E#P := 2. E#N := 3. I#I := E#N - F#J. F#J := 1.", "query": "I#I", "answer": 2, "cot": "E#N = 3 -> E#N = 3\nF#J = 1 -> F#J = 1\nI#I = E#N - F#J\n = 3 - 1\n 3 - 1 = (3 - 1) mod 7 = 2\n => I#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#N": 1, "F#J": 1, "K#I": 1, "E#P": 1, "I#I": 2}, "id": "mod7_arith_d2_0238"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#O := H#I. L#M := 2. E#I := 6. H#I := 4. H#O := 1. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#O := H#I. L#M := 2. E#I := 6. H#I := 4. H#O := 1.", "query": "K#O", "answer": 4, "cot": "H#I = 4 -> H#I = 4\nK#O = H#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#O": 1, "L#M": 1, "E#I": 1, "H#I": 1, "K#O": 2}, "id": "mod7_arith_d2_0239"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := 6. J#P := G#N. K#P := 2. H#M := 3. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := 6. J#P := G#N. K#P := 2. H#M := 3.", "query": "J#P", "answer": 6, "cot": "G#N = 6 -> G#N = 6\nJ#P = G#N = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#M": 1, "K#P": 1, "G#N": 1, "J#P": 2}, "id": "mod7_arith_d2_0240"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#M := E#I / E#K. K#P := 5. E#I := 2. E#K := 1. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#M := E#I / E#K. K#P := 5. E#I := 2. E#K := 1.", "query": "H#M", "answer": 2, "cot": "E#I = 2 -> E#I = 2\nE#K = 1 -> E#K = 1\nH#M = E#I / E#K\n = 2 / 1\n 2 / 1 = (2 × 1⁻¹) mod 7 = 2\n => H#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#I": 1, "K#P": 1, "E#K": 1, "H#M": 2}, "id": "mod7_arith_d2_0241"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#M := 4. G#P := G#I / G#M. G#I := 6. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#M := 4. G#P := G#I / G#M. G#I := 6.", "query": "G#P", "answer": 5, "cot": "G#M = 4 -> G#M = 4\nG#I = 6 -> G#I = 6\nG#P = G#I / G#M\n = 6 / 4\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n => G#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#M": 1, "G#I": 1, "G#P": 2}, "id": "mod7_arith_d2_0242"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := 1. J#M := 2. J#L := L#J + K#N. K#N := 4. L#J := 4. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := 1. J#M := 2. J#L := L#J + K#N. K#N := 4. L#J := 4.", "query": "J#L", "answer": 1, "cot": "L#J = 4 -> L#J = 4\nK#N = 4 -> K#N = 4\nJ#L = L#J + K#N\n = 4 + 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => J#L = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#M": 1, "L#J": 1, "K#N": 1, "J#M": 1, "J#L": 2}, "id": "mod7_arith_d2_0243"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#K := 1. E#P := 1. J#L := 6. G#O := J#L. G#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#K := 1. E#P := 1. J#L := 6. G#O := J#L.", "query": "G#O", "answer": 6, "cot": "J#L = 6 -> J#L = 6\nG#O = J#L = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#P": 1, "I#K": 1, "J#L": 1, "G#O": 2}, "id": "mod7_arith_d2_0244"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := 5. L#K := J#P + F#N. J#P := 4. I#M := 3. I#N := 2. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := 5. L#K := J#P + F#N. J#P := 4. I#M := 3. I#N := 2.", "query": "L#K", "answer": 2, "cot": "J#P = 4 -> J#P = 4\nF#N = 5 -> F#N = 5\nL#K = J#P + F#N\n = 4 + 5\n 4 + 5 = (4 + 5) mod 7 = 2\n => L#K = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#N": 1, "J#P": 1, "F#N": 1, "I#M": 1, "L#K": 2}, "id": "mod7_arith_d2_0245"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#M := 2. G#L := 3. I#K := I#M - 6 * G#L. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#M := 2. G#L := 3. I#K := I#M - 6 * G#L.", "query": "I#K", "answer": 2, "cot": "I#M = 2 -> I#M = 2\nG#L = 3 -> G#L = 3\nI#K = I#M - 6 * G#L\n = 2 - 6 * 3\n 2 - 6 = (2 - 6) mod 7 = 3\n 3 * 3 = (3 × 3) mod 7 = 2\n => I#K = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#M": 1, "G#L": 1, "I#K": 2}, "id": "mod7_arith_d2_0246"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := K#M - 4. H#L := 5. E#O := 1. K#M := 1. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := K#M - 4. H#L := 5. E#O := 1. K#M := 1.", "query": "I#P", "answer": 4, "cot": "K#M = 1 -> K#M = 1\nI#P = K#M - 4\n = 1 - 4\n 1 - 4 = (1 - 4) mod 7 = 4\n => I#P = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 4, "num_necessary": 2, "num_distractors": 0, "var_layer": {"E#O": 1, "H#L": 1, "K#M": 1, "I#P": 2}, "id": "mod7_arith_d2_0247"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#M := 5. H#J := 5 / G#M. H#N := 4. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#M := 5. H#J := 5 / G#M. H#N := 4.", "query": "H#J", "answer": 1, "cot": "G#M = 5 -> G#M = 5\nH#J = 5 / G#M\n = 5 / 5\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n => H#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 3, "num_necessary": 2, "num_distractors": 0, "var_layer": {"H#N": 1, "G#M": 1, "H#J": 2}, "id": "mod7_arith_d2_0248"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := 2. I#L := 4. K#M := 3. E#P := K#M / F#L. K#P := 4. E#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := 2. I#L := 4. K#M := 3. E#P := K#M / F#L. K#P := 4.", "query": "E#P", "answer": 5, "cot": "F#L = 2 -> F#L = 2\nK#M = 3 -> K#M = 3\nE#P = K#M / F#L\n = 3 / 2\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => E#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 2, "achieved_depth": 2, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#P": 1, "I#L": 1, "F#L": 1, "K#M": 1, "E#P": 2}, "id": "mod7_arith_d2_0249"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#L := I#J. F#L := 1. I#J := 5. I#O := 6 + 5 - F#L. J#K := I#O - L#L / F#N. F#N := 4. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#L := I#J. F#L := 1. I#J := 5. I#O := 6 + 5 - F#L. J#K := I#O - L#L / F#N. F#N := 4.", "query": "J#K", "answer": 3, "cot": "I#J = 5 -> I#J = 5\nF#N = 4 -> F#N = 4\nF#L = 1 -> F#L = 1\nI#O = 6 + 5 - F#L\n = 6 + 5 - 1\n 6 + 5 = (6 + 5) mod 7 = 4\n 4 - 1 = (4 - 1) mod 7 = 3\n => I#O = 3\nL#L = I#J = 5\nJ#K = I#O - L#L / F#N\n = 3 - 5 / 4\n 3 - 5 = (3 - 5) mod 7 = 5\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => J#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 6, "num_distractors": 0, "var_layer": {"I#J": 1, "F#N": 1, "F#L": 1, "I#O": 2, "L#L": 2, "J#K": 3}, "id": "mod7_arith_d3_0000"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := L#O + 2. G#N := 1. H#K := G#P / G#N. G#P := 2. L#M := L#J. L#O := 4. F#M := G#N * G#P - L#O. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := L#O + 2. G#N := 1. H#K := G#P / G#N. G#P := 2. L#M := L#J. L#O := 4. F#M := G#N * G#P - L#O.", "query": "L#M", "answer": 6, "cot": "L#O = 4 -> L#O = 4\nL#J = L#O + 2\n = 4 + 2\n 4 + 2 = (4 + 2) mod 7 = 6\n => L#J = 6\nL#M = L#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#N": 1, "L#O": 1, "G#P": 1, "H#K": 2, "L#J": 2, "F#M": 2, "L#M": 3}, "id": "mod7_arith_d3_0001"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#O := L#P + H#P. I#I := 3. H#P := L#P. H#J := 5. I#O := I#I * H#J. L#P := 1. L#I := H#J. L#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#O := L#P + H#P. I#I := 3. H#P := L#P. H#J := 5. I#O := I#I * H#J. L#P := 1. L#I := H#J.", "query": "L#O", "answer": 2, "cot": "L#P = 1 -> L#P = 1\nH#P = L#P = 1\nL#O = L#P + H#P\n = 1 + 1\n 1 + 1 = (1 + 1) mod 7 = 2\n => L#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#J": 1, "I#I": 1, "L#P": 1, "H#P": 2, "L#I": 2, "I#O": 2, "L#O": 3}, "id": "mod7_arith_d3_0002"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := 1. J#I := J#K. J#K := 5. J#M := F#M. H#P := E#K. K#K := H#P * E#K. F#M := 1. K#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := 1. J#I := J#K. J#K := 5. J#M := F#M. H#P := E#K. K#K := H#P * E#K. F#M := 1.", "query": "K#K", "answer": 1, "cot": "E#K = 1 -> E#K = 1\nH#P = E#K = 1\nK#K = H#P * E#K\n = 1 * 1\n 1 * 1 = (1 × 1) mod 7 = 1\n => K#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#M": 1, "J#K": 1, "E#K": 1, "H#P": 2, "J#I": 2, "J#M": 2, "K#K": 3}, "id": "mod7_arith_d3_0003"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#J := 3. L#J := 4 - E#J. I#L := 6 + K#O. K#O := H#J - 5. H#J := 6. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#J := 3. L#J := 4 - E#J. I#L := 6 + K#O. K#O := H#J - 5. H#J := 6.", "query": "I#L", "answer": 0, "cot": "H#J = 6 -> H#J = 6\nK#O = H#J - 5\n = 6 - 5\n 6 - 5 = (6 - 5) mod 7 = 1\n => K#O = 1\nI#L = 6 + K#O\n = 6 + 1\n 6 + 1 = (6 + 1) mod 7 = 0\n => I#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#J": 1, "E#J": 1, "K#O": 2, "L#J": 2, "I#L": 3}, "id": "mod7_arith_d3_0004"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := 2. F#I := K#N - L#K. K#L := L#I / K#N. L#K := 6. H#O := F#I. I#M := K#N. K#N := 5. H#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := 2. F#I := K#N - L#K. K#L := L#I / K#N. L#K := 6. H#O := F#I. I#M := K#N. K#N := 5.", "query": "H#O", "answer": 6, "cot": "K#N = 5 -> K#N = 5\nL#K = 6 -> L#K = 6\nF#I = K#N - L#K\n = 5 - 6\n 5 - 6 = (5 - 6) mod 7 = 6\n => F#I = 6\nH#O = F#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#I": 1, "K#N": 1, "L#K": 1, "I#M": 2, "K#L": 2, "F#I": 2, "H#O": 3}, "id": "mod7_arith_d3_0005"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#N := I#P - 5 * H#J. H#J := 5. E#J := 3. L#K := F#L / K#I. I#P := H#J. F#L := 1. K#I := 6. H#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#N := I#P - 5 * H#J. H#J := 5. E#J := 3. L#K := F#L / K#I. I#P := H#J. F#L := 1. K#I := 6.", "query": "H#N", "answer": 0, "cot": "H#J = 5 -> H#J = 5\nI#P = H#J = 5\nH#N = I#P - 5 * H#J\n = 5 - 5 * 5\n 5 - 5 = (5 - 5) mod 7 = 0\n 0 * 5 = (0 × 5) mod 7 = 0\n => H#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#L": 1, "E#J": 1, "H#J": 1, "K#I": 1, "I#P": 2, "L#K": 2, "H#N": 3}, "id": "mod7_arith_d3_0006"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := 6. K#K := 6. L#N := 1. E#K := 5. L#M := E#K - I#I * G#J. G#J := K#K * L#N. E#J := L#N - I#I. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := 6. K#K := 6. L#N := 1. E#K := 5. L#M := E#K - I#I * G#J. G#J := K#K * L#N. E#J := L#N - I#I.", "query": "L#M", "answer": 1, "cot": "L#N = 1 -> L#N = 1\nK#K = 6 -> K#K = 6\nE#K = 5 -> E#K = 5\nI#I = 6 -> I#I = 6\nG#J = K#K * L#N\n = 6 * 1\n 6 * 1 = (6 × 1) mod 7 = 6\n => G#J = 6\nL#M = E#K - I#I * G#J\n = 5 - 6 * 6\n 5 - 6 = (5 - 6) mod 7 = 6\n 6 * 6 = (6 × 6) mod 7 = 1\n => L#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"L#N": 1, "K#K": 1, "E#K": 1, "I#I": 1, "G#J": 2, "E#J": 2, "L#M": 3}, "id": "mod7_arith_d3_0007"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := L#M / G#P. L#M := 4. L#J := F#O * I#P * G#P. F#O := L#M / G#P + 4. G#P := 6. L#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := L#M / G#P. L#M := 4. L#J := F#O * I#P * G#P. F#O := L#M / G#P + 4. G#P := 6.", "query": "L#J", "answer": 0, "cot": "G#P = 6 -> G#P = 6\nL#M = 4 -> L#M = 4\nF#O = L#M / G#P + 4\n = 4 / 6 + 4\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n 3 + 4 = (3 + 4) mod 7 = 0\n => F#O = 0\nI#P = L#M / G#P\n = 4 / 6\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n => I#P = 3\nL#J = F#O * I#P * G#P\n = 0 * 3 * 6\n 0 * 3 = (0 × 3) mod 7 = 0\n 0 * 6 = (0 × 6) mod 7 = 0\n => L#J = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#P": 1, "L#M": 1, "F#O": 2, "I#P": 2, "L#J": 3}, "id": "mod7_arith_d3_0008"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := 6. K#M := 2. E#L := 1. G#K := E#J + 4. E#J := 3. I#N := E#L. I#K := 5 + G#K * E#J. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := 6. K#M := 2. E#L := 1. G#K := E#J + 4. E#J := 3. I#N := E#L. I#K := 5 + G#K * E#J.", "query": "I#K", "answer": 1, "cot": "E#J = 3 -> E#J = 3\nG#K = E#J + 4\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => G#K = 0\nI#K = 5 + G#K * E#J\n = 5 + 0 * 3\n 5 + 0 = (5 + 0) mod 7 = 5\n 5 * 3 = (5 × 3) mod 7 = 1\n => I#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#O": 1, "E#J": 1, "E#L": 1, "K#M": 1, "G#K": 2, "I#N": 2, "I#K": 3}, "id": "mod7_arith_d3_0009"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#I := G#L + F#J. L#O := 2. G#L := 2. I#K := 5. F#J := L#O / G#L - 4. I#O := L#O. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#I := G#L + F#J. L#O := 2. G#L := 2. I#K := 5. F#J := L#O / G#L - 4. I#O := L#O.", "query": "H#I", "answer": 6, "cot": "L#O = 2 -> L#O = 2\nG#L = 2 -> G#L = 2\nF#J = L#O / G#L - 4\n = 2 / 2 - 4\n 2 / 2 = (2 × 2⁻¹) mod 7 = 1\n 1 - 4 = (1 - 4) mod 7 = 4\n => F#J = 4\nH#I = G#L + F#J\n = 2 + 4\n 2 + 4 = (2 + 4) mod 7 = 6\n => H#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#O": 1, "I#K": 1, "G#L": 1, "I#O": 2, "F#J": 2, "H#I": 3}, "id": "mod7_arith_d3_0010"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#N := 5. K#P := K#O + 2. E#O := 3 - H#N. H#N := 5 - K#O. K#O := 5. E#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#N := 5. K#P := K#O + 2. E#O := 3 - H#N. H#N := 5 - K#O. K#O := 5.", "query": "E#O", "answer": 3, "cot": "K#O = 5 -> K#O = 5\nH#N = 5 - K#O\n = 5 - 5\n 5 - 5 = (5 - 5) mod 7 = 0\n => H#N = 0\nE#O = 3 - H#N\n = 3 - 0\n 3 - 0 = (3 - 0) mod 7 = 3\n => E#O = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#N": 1, "K#O": 1, "K#P": 2, "H#N": 2, "E#O": 3}, "id": "mod7_arith_d3_0011"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := 4 / L#N. I#I := 1. I#O := L#N / 6 / K#O. I#P := I#O * 4. L#N := 5. K#O := 2. J#P := I#I * 6 - 5. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := 4 / L#N. I#I := 1. I#O := L#N / 6 / K#O. I#P := I#O * 4. L#N := 5. K#O := 2. J#P := I#I * 6 - 5.", "query": "I#P", "answer": 4, "cot": "L#N = 5 -> L#N = 5\nK#O = 2 -> K#O = 2\nI#O = L#N / 6 / K#O\n = 5 / 6 / 2\n 5 / 6 = (5 × 6⁻¹) mod 7 = 2\n 2 / 2 = (2 × 2⁻¹) mod 7 = 1\n => I#O = 1\nI#P = I#O * 4\n = 1 * 4\n 1 * 4 = (1 × 4) mod 7 = 4\n => I#P = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#N": 1, "K#O": 1, "I#I": 1, "L#M": 2, "I#O": 2, "J#P": 2, "I#P": 3}, "id": "mod7_arith_d3_0012"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := 6 / K#K / L#N. J#L := K#K. K#K := 3. H#K := L#P / 5. L#N := 6. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := 6 / K#K / L#N. J#L := K#K. K#K := 3. H#K := L#P / 5. L#N := 6.", "query": "H#K", "answer": 1, "cot": "L#N = 6 -> L#N = 6\nK#K = 3 -> K#K = 3\nL#P = 6 / K#K / L#N\n = 6 / 3 / 6\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => L#P = 5\nH#K = L#P / 5\n = 5 / 5\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n => H#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#N": 1, "K#K": 1, "L#P": 2, "J#L": 2, "H#K": 3}, "id": "mod7_arith_d3_0013"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#P := 3. H#J := 3 - E#P. J#J := 6. L#P := K#O + G#N. K#O := E#P * G#N. G#N := 5. L#N := 4. L#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#P := 3. H#J := 3 - E#P. J#J := 6. L#P := K#O + G#N. K#O := E#P * G#N. G#N := 5. L#N := 4.", "query": "L#P", "answer": 6, "cot": "G#N = 5 -> G#N = 5\nE#P = 3 -> E#P = 3\nK#O = E#P * G#N\n = 3 * 5\n 3 * 5 = (3 × 5) mod 7 = 1\n => K#O = 1\nL#P = K#O + G#N\n = 1 + 5\n 1 + 5 = (1 + 5) mod 7 = 6\n => L#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#N": 1, "L#N": 1, "E#P": 1, "J#J": 1, "K#O": 2, "H#J": 2, "L#P": 3}, "id": "mod7_arith_d3_0014"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := H#M. H#M := 4. E#K := 5. J#P := I#I / H#M. F#L := H#M - E#K. I#I := E#K. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := H#M. H#M := 4. E#K := 5. J#P := I#I / H#M. F#L := H#M - E#K. I#I := E#K.", "query": "J#P", "answer": 3, "cot": "E#K = 5 -> E#K = 5\nH#M = 4 -> H#M = 4\nI#I = E#K = 5\nJ#P = I#I / H#M\n = 5 / 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => J#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#K": 1, "H#M": 1, "L#P": 2, "F#L": 2, "I#I": 2, "J#P": 3}, "id": "mod7_arith_d3_0015"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := 2. E#P := H#L + J#P + I#P. I#L := J#P. I#P := 5. H#L := 5 * E#J * 6. E#J := 2. J#P := 3. E#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := 2. E#P := H#L + J#P + I#P. I#L := J#P. I#P := 5. H#L := 5 * E#J * 6. E#J := 2. J#P := 3.", "query": "E#P", "answer": 5, "cot": "I#P = 5 -> I#P = 5\nE#J = 2 -> E#J = 2\nJ#P = 3 -> J#P = 3\nH#L = 5 * E#J * 6\n = 5 * 2 * 6\n 5 * 2 = (5 × 2) mod 7 = 3\n 3 * 6 = (3 × 6) mod 7 = 4\n => H#L = 4\nE#P = H#L + J#P + I#P\n = 4 + 3 + 5\n 4 + 3 = (4 + 3) mod 7 = 0\n 0 + 5 = (0 + 5) mod 7 = 5\n => E#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#P": 1, "E#J": 1, "J#P": 1, "F#N": 1, "I#L": 2, "H#L": 2, "E#P": 3}, "id": "mod7_arith_d3_0016"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := H#L + I#L. I#L := 3. H#L := 6. I#P := J#J. L#P := H#L + I#L. J#J := H#L / I#L. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := H#L + I#L. I#L := 3. H#L := 6. I#P := J#J. L#P := H#L + I#L. J#J := H#L / I#L.", "query": "I#P", "answer": 2, "cot": "I#L = 3 -> I#L = 3\nH#L = 6 -> H#L = 6\nJ#J = H#L / I#L\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => J#J = 2\nI#P = J#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#L": 1, "H#L": 1, "K#J": 2, "L#P": 2, "J#J": 2, "I#P": 3}, "id": "mod7_arith_d3_0017"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#K := 3. G#L := 6. E#O := 3. J#L := G#K + E#O. G#P := 5. G#K := K#K. H#K := G#L / E#O + K#K. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#K := 3. G#L := 6. E#O := 3. J#L := G#K + E#O. G#P := 5. G#K := K#K. H#K := G#L / E#O + K#K.", "query": "J#L", "answer": 6, "cot": "E#O = 3 -> E#O = 3\nK#K = 3 -> K#K = 3\nG#K = K#K = 3\nJ#L = G#K + E#O\n = 3 + 3\n 3 + 3 = (3 + 3) mod 7 = 6\n => J#L = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#O": 1, "K#K": 1, "G#P": 1, "G#L": 1, "G#K": 2, "H#K": 2, "J#L": 3}, "id": "mod7_arith_d3_0018"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := E#P. G#K := 1. H#M := G#K * F#M. G#J := E#L + H#M. F#M := 1. E#P := 6. J#M := 4. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := E#P. G#K := 1. H#M := G#K * F#M. G#J := E#L + H#M. F#M := 1. E#P := 6. J#M := 4.", "query": "G#J", "answer": 0, "cot": "F#M = 1 -> F#M = 1\nE#P = 6 -> E#P = 6\nG#K = 1 -> G#K = 1\nE#L = E#P = 6\nH#M = G#K * F#M\n = 1 * 1\n 1 * 1 = (1 × 1) mod 7 = 1\n => H#M = 1\nG#J = E#L + H#M\n = 6 + 1\n 6 + 1 = (6 + 1) mod 7 = 0\n => G#J = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#M": 1, "J#M": 1, "E#P": 1, "G#K": 1, "E#L": 2, "H#M": 2, "G#J": 3}, "id": "mod7_arith_d3_0019"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#K := K#O - F#I. K#K := K#O / 3. E#J := K#O + G#P * I#N. E#N := 3. K#O := 5. F#I := 5. I#N := E#N / 3. G#P := 3. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#K := K#O - F#I. K#K := K#O / 3. E#J := K#O + G#P * I#N. E#N := 3. K#O := 5. F#I := 5. I#N := E#N / 3. G#P := 3.", "query": "E#J", "answer": 1, "cot": "K#O = 5 -> K#O = 5\nE#N = 3 -> E#N = 3\nG#P = 3 -> G#P = 3\nI#N = E#N / 3\n = 3 / 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => I#N = 1\nE#J = K#O + G#P * I#N\n = 5 + 3 * 1\n 5 + 3 = (5 + 3) mod 7 = 1\n 1 * 1 = (1 × 1) mod 7 = 1\n => E#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#O": 1, "E#N": 1, "G#P": 1, "F#I": 1, "I#K": 2, "I#N": 2, "K#K": 2, "E#J": 3}, "id": "mod7_arith_d3_0020"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := F#O * J#P / 4. F#O := 2. J#P := 1. E#P := 6 / F#O. I#P := F#M * E#P - 6. L#N := F#M / F#O. F#M := 3. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := F#O * J#P / 4. F#O := 2. J#P := 1. E#P := 6 / F#O. I#P := F#M * E#P - 6. L#N := F#M / F#O. F#M := 3.", "query": "I#P", "answer": 3, "cot": "F#O = 2 -> F#O = 2\nF#M = 3 -> F#M = 3\nE#P = 6 / F#O\n = 6 / 2\n 6 / 2 = (6 × 2⁻¹) mod 7 = 3\n => E#P = 3\nI#P = F#M * E#P - 6\n = 3 * 3 - 6\n 3 * 3 = (3 × 3) mod 7 = 2\n 2 - 6 = (2 - 6) mod 7 = 3\n => I#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#O": 1, "F#M": 1, "J#P": 1, "E#P": 2, "G#J": 2, "L#N": 2, "I#P": 3}, "id": "mod7_arith_d3_0021"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := 5. K#I := I#L / 3. J#O := 5. J#P := 5. I#L := J#I / J#O / J#P. E#M := J#O + I#P + 5. J#I := 1. G#O := J#P / J#I. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := 5. K#I := I#L / 3. J#O := 5. J#P := 5. I#L := J#I / J#O / J#P. E#M := J#O + I#P + 5. J#I := 1. G#O := J#P / J#I.", "query": "K#I", "answer": 3, "cot": "J#P = 5 -> J#P = 5\nJ#I = 1 -> J#I = 1\nJ#O = 5 -> J#O = 5\nI#L = J#I / J#O / J#P\n = 1 / 5 / 5\n 1 / 5 = (1 × 5⁻¹) mod 7 = 3\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n => I#L = 2\nK#I = I#L / 3\n = 2 / 3\n 2 / 3 = (2 × 3⁻¹) mod 7 = 3\n => K#I = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#P": 1, "I#P": 1, "J#I": 1, "J#O": 1, "I#L": 2, "G#O": 2, "E#M": 2, "K#I": 3}, "id": "mod7_arith_d3_0022"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := H#N + 6 / 5. H#N := 6 - G#J * F#O. F#L := F#O - G#J + K#L. F#O := 4. K#L := 3. F#J := K#L - G#J. G#J := 5. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := H#N + 6 / 5. H#N := 6 - G#J * F#O. F#L := F#O - G#J + K#L. F#O := 4. K#L := 3. F#J := K#L - G#J. G#J := 5.", "query": "E#I", "answer": 2, "cot": "G#J = 5 -> G#J = 5\nF#O = 4 -> F#O = 4\nH#N = 6 - G#J * F#O\n = 6 - 5 * 4\n 6 - 5 = (6 - 5) mod 7 = 1\n 1 * 4 = (1 × 4) mod 7 = 4\n => H#N = 4\nE#I = H#N + 6 / 5\n = 4 + 6 / 5\n 4 + 6 = (4 + 6) mod 7 = 3\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n => E#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#J": 1, "K#L": 1, "F#O": 1, "F#L": 2, "F#J": 2, "H#N": 2, "E#I": 3}, "id": "mod7_arith_d3_0023"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#P := 1. J#K := 5. F#I := 4. F#J := 5. L#O := F#I. L#K := H#P + F#J. H#M := F#I / J#K / H#P. K#O := L#K - 6 / 3. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#P := 1. J#K := 5. F#I := 4. F#J := 5. L#O := F#I. L#K := H#P + F#J. H#M := F#I / J#K / H#P. K#O := L#K - 6 / 3.", "query": "K#O", "answer": 0, "cot": "F#J = 5 -> F#J = 5\nH#P = 1 -> H#P = 1\nL#K = H#P + F#J\n = 1 + 5\n 1 + 5 = (1 + 5) mod 7 = 6\n => L#K = 6\nK#O = L#K - 6 / 3\n = 6 - 6 / 3\n 6 - 6 = (6 - 6) mod 7 = 0\n 0 / 3 = (0 × 3⁻¹) mod 7 = 0\n => K#O = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#J": 1, "F#I": 1, "J#K": 1, "H#P": 1, "L#O": 2, "L#K": 2, "H#M": 2, "K#O": 3}, "id": "mod7_arith_d3_0024"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#K := 4. G#L := J#K / K#J. K#J := 4. G#O := K#J / J#K. I#K := G#O + G#L * E#O. E#O := K#J. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#K := 4. G#L := J#K / K#J. K#J := 4. G#O := K#J / J#K. I#K := G#O + G#L * E#O. E#O := K#J.", "query": "I#K", "answer": 1, "cot": "K#J = 4 -> K#J = 4\nJ#K = 4 -> J#K = 4\nE#O = K#J = 4\nG#L = J#K / K#J\n = 4 / 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => G#L = 1\nG#O = K#J / J#K\n = 4 / 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => G#O = 1\nI#K = G#O + G#L * E#O\n = 1 + 1 * 4\n 1 + 1 = (1 + 1) mod 7 = 2\n 2 * 4 = (2 × 4) mod 7 = 1\n => I#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#J": 1, "J#K": 1, "E#O": 2, "G#L": 2, "G#O": 2, "I#K": 3}, "id": "mod7_arith_d3_0025"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := 5. E#M := 4. F#O := H#K + J#P * I#I. H#J := 3. J#P := 2 - E#M. I#I := H#J * 2. F#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := 5. E#M := 4. F#O := H#K + J#P * I#I. H#J := 3. J#P := 2 - E#M. I#I := H#J * 2.", "query": "F#O", "answer": 4, "cot": "H#K = 5 -> H#K = 5\nE#M = 4 -> E#M = 4\nH#J = 3 -> H#J = 3\nI#I = H#J * 2\n = 3 * 2\n 3 * 2 = (3 × 2) mod 7 = 6\n => I#I = 6\nJ#P = 2 - E#M\n = 2 - 4\n 2 - 4 = (2 - 4) mod 7 = 5\n => J#P = 5\nF#O = H#K + J#P * I#I\n = 5 + 5 * 6\n 5 + 5 = (5 + 5) mod 7 = 3\n 3 * 6 = (3 × 6) mod 7 = 4\n => F#O = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 6, "num_distractors": 0, "var_layer": {"H#K": 1, "E#M": 1, "H#J": 1, "I#I": 2, "J#P": 2, "F#O": 3}, "id": "mod7_arith_d3_0026"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := H#M * L#I. G#M := 1. L#I := G#P - J#P. G#P := 5. J#P := 5. H#M := 2 * H#I / G#P. H#I := 2. L#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := H#M * L#I. G#M := 1. L#I := G#P - J#P. G#P := 5. J#P := 5. H#M := 2 * H#I / G#P. H#I := 2.", "query": "L#P", "answer": 0, "cot": "H#I = 2 -> H#I = 2\nJ#P = 5 -> J#P = 5\nG#P = 5 -> G#P = 5\nH#M = 2 * H#I / G#P\n = 2 * 2 / 5\n 2 * 2 = (2 × 2) mod 7 = 4\n 4 / 5 = (4 × 5⁻¹) mod 7 = 5\n => H#M = 5\nL#I = G#P - J#P\n = 5 - 5\n 5 - 5 = (5 - 5) mod 7 = 0\n => L#I = 0\nL#P = H#M * L#I\n = 5 * 0\n 5 * 0 = (5 × 0) mod 7 = 0\n => L#P = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#M": 1, "H#I": 1, "J#P": 1, "G#P": 1, "H#M": 2, "L#I": 2, "L#P": 3}, "id": "mod7_arith_d3_0027"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#I := 2. E#O := G#I. G#I := 4. J#P := H#I * G#I. E#I := J#P / H#I. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#I := 2. E#O := G#I. G#I := 4. J#P := H#I * G#I. E#I := J#P / H#I.", "query": "E#I", "answer": 4, "cot": "H#I = 2 -> H#I = 2\nG#I = 4 -> G#I = 4\nJ#P = H#I * G#I\n = 2 * 4\n 2 * 4 = (2 × 4) mod 7 = 1\n => J#P = 1\nE#I = J#P / H#I\n = 1 / 2\n 1 / 2 = (1 × 2⁻¹) mod 7 = 4\n => E#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#I": 1, "G#I": 1, "J#P": 2, "E#O": 2, "E#I": 3}, "id": "mod7_arith_d3_0028"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := 6. G#L := 2. K#P := 5 + E#O / H#K. K#N := E#O - G#L + E#J. E#O := 1. L#K := 1. E#J := L#K / G#L. J#O := L#K / 5. K#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := 6. G#L := 2. K#P := 5 + E#O / H#K. K#N := E#O - G#L + E#J. E#O := 1. L#K := 1. E#J := L#K / G#L. J#O := L#K / 5.", "query": "K#N", "answer": 3, "cot": "E#O = 1 -> E#O = 1\nL#K = 1 -> L#K = 1\nG#L = 2 -> G#L = 2\nE#J = L#K / G#L\n = 1 / 2\n 1 / 2 = (1 × 2⁻¹) mod 7 = 4\n => E#J = 4\nK#N = E#O - G#L + E#J\n = 1 - 2 + 4\n 1 - 2 = (1 - 2) mod 7 = 6\n 6 + 4 = (6 + 4) mod 7 = 3\n => K#N = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"E#O": 1, "L#K": 1, "H#K": 1, "G#L": 1, "E#J": 2, "J#O": 2, "K#P": 2, "K#N": 3}, "id": "mod7_arith_d3_0029"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#O := 6 + H#M * E#I. E#I := 2. G#M := H#M / E#I. J#K := 2 - G#M. H#M := 3. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#O := 6 + H#M * E#I. E#I := 2. G#M := H#M / E#I. J#K := 2 - G#M. H#M := 3.", "query": "J#K", "answer": 4, "cot": "H#M = 3 -> H#M = 3\nE#I = 2 -> E#I = 2\nG#M = H#M / E#I\n = 3 / 2\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => G#M = 5\nJ#K = 2 - G#M\n = 2 - 5\n 2 - 5 = (2 - 5) mod 7 = 4\n => J#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#M": 1, "E#I": 1, "L#O": 2, "G#M": 2, "J#K": 3}, "id": "mod7_arith_d3_0030"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := 3. E#L := K#K. K#K := E#N * J#P. E#N := 2. K#M := 5. H#P := H#O + K#M - E#N. J#P := 4. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := 3. E#L := K#K. K#K := E#N * J#P. E#N := 2. K#M := 5. H#P := H#O + K#M - E#N. J#P := 4.", "query": "E#L", "answer": 1, "cot": "J#P = 4 -> J#P = 4\nE#N = 2 -> E#N = 2\nK#K = E#N * J#P\n = 2 * 4\n 2 * 4 = (2 × 4) mod 7 = 1\n => K#K = 1\nE#L = K#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#O": 1, "J#P": 1, "K#M": 1, "E#N": 1, "K#K": 2, "H#P": 2, "E#L": 3}, "id": "mod7_arith_d3_0031"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := G#M. F#L := 5. G#M := L#L / F#L - 4. E#N := L#L. L#L := 2. G#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := G#M. F#L := 5. G#M := L#L / F#L - 4. E#N := L#L. L#L := 2.", "query": "G#N", "answer": 2, "cot": "L#L = 2 -> L#L = 2\nF#L = 5 -> F#L = 5\nG#M = L#L / F#L - 4\n = 2 / 5 - 4\n 2 / 5 = (2 × 5⁻¹) mod 7 = 6\n 6 - 4 = (6 - 4) mod 7 = 2\n => G#M = 2\nG#N = G#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#L": 1, "F#L": 1, "G#M": 2, "E#N": 2, "G#N": 3}, "id": "mod7_arith_d3_0032"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#I := 4. G#M := H#J * 4. I#I := 5. H#J := I#I - F#I - 2. L#M := F#I + I#I. E#K := F#I. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#I := 4. G#M := H#J * 4. I#I := 5. H#J := I#I - F#I - 2. L#M := F#I + I#I. E#K := F#I.", "query": "G#M", "answer": 3, "cot": "F#I = 4 -> F#I = 4\nI#I = 5 -> I#I = 5\nH#J = I#I - F#I - 2\n = 5 - 4 - 2\n 5 - 4 = (5 - 4) mod 7 = 1\n 1 - 2 = (1 - 2) mod 7 = 6\n => H#J = 6\nG#M = H#J * 4\n = 6 * 4\n 6 * 4 = (6 × 4) mod 7 = 3\n => G#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#I": 1, "I#I": 1, "E#K": 2, "L#M": 2, "H#J": 2, "G#M": 3}, "id": "mod7_arith_d3_0033"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := 5. G#J := 6. E#I := E#L + G#J. F#L := 1. H#N := F#P. F#P := 3 * F#L. H#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := 5. G#J := 6. E#I := E#L + G#J. F#L := 1. H#N := F#P. F#P := 3 * F#L.", "query": "H#N", "answer": 3, "cot": "F#L = 1 -> F#L = 1\nF#P = 3 * F#L\n = 3 * 1\n 3 * 1 = (3 × 1) mod 7 = 3\n => F#P = 3\nH#N = F#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#L": 1, "E#L": 1, "G#J": 1, "E#I": 2, "F#P": 2, "H#N": 3}, "id": "mod7_arith_d3_0034"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := 2. E#L := 6. F#I := E#L. K#N := I#K - F#I. I#K := F#P * K#J. F#P := 2. G#I := 3. E#M := 5 - K#J. K#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := 2. E#L := 6. F#I := E#L. K#N := I#K - F#I. I#K := F#P * K#J. F#P := 2. G#I := 3. E#M := 5 - K#J.", "query": "K#N", "answer": 5, "cot": "E#L = 6 -> E#L = 6\nK#J = 2 -> K#J = 2\nF#P = 2 -> F#P = 2\nF#I = E#L = 6\nI#K = F#P * K#J\n = 2 * 2\n 2 * 2 = (2 × 2) mod 7 = 4\n => I#K = 4\nK#N = I#K - F#I\n = 4 - 6\n 4 - 6 = (4 - 6) mod 7 = 5\n => K#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#L": 1, "K#J": 1, "G#I": 1, "F#P": 1, "E#M": 2, "F#I": 2, "I#K": 2, "K#N": 3}, "id": "mod7_arith_d3_0035"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#K := 1. G#J := 6. I#J := H#N + G#J. J#L := I#J - 6 + K#K. I#M := K#K + G#J - H#N. H#N := 1. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#K := 1. G#J := 6. I#J := H#N + G#J. J#L := I#J - 6 + K#K. I#M := K#K + G#J - H#N. H#N := 1.", "query": "J#L", "answer": 2, "cot": "K#K = 1 -> K#K = 1\nG#J = 6 -> G#J = 6\nH#N = 1 -> H#N = 1\nI#J = H#N + G#J\n = 1 + 6\n 1 + 6 = (1 + 6) mod 7 = 0\n => I#J = 0\nJ#L = I#J - 6 + K#K\n = 0 - 6 + 1\n 0 - 6 = (0 - 6) mod 7 = 1\n 1 + 1 = (1 + 1) mod 7 = 2\n => J#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#K": 1, "G#J": 1, "H#N": 1, "I#J": 2, "I#M": 2, "J#L": 3}, "id": "mod7_arith_d3_0036"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#L := 1. F#N := F#O / I#J. F#I := 5. I#J := J#L * F#I. F#O := F#I. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#L := 1. F#N := F#O / I#J. F#I := 5. I#J := J#L * F#I. F#O := F#I.", "query": "F#N", "answer": 1, "cot": "J#L = 1 -> J#L = 1\nF#I = 5 -> F#I = 5\nI#J = J#L * F#I\n = 1 * 5\n 1 * 5 = (1 × 5) mod 7 = 5\n => I#J = 5\nF#O = F#I = 5\nF#N = F#O / I#J\n = 5 / 5\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n => F#N = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#L": 1, "F#I": 1, "I#J": 2, "F#O": 2, "F#N": 3}, "id": "mod7_arith_d3_0037"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := J#K / K#K. J#M := 4. F#J := 3 + L#N. K#K := L#N * J#K. L#L := J#K. L#N := 4. J#K := 3. L#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := J#K / K#K. J#M := 4. F#J := 3 + L#N. K#K := L#N * J#K. L#L := J#K. L#N := 4. J#K := 3.", "query": "L#P", "answer": 2, "cot": "L#N = 4 -> L#N = 4\nJ#K = 3 -> J#K = 3\nK#K = L#N * J#K\n = 4 * 3\n 4 * 3 = (4 × 3) mod 7 = 5\n => K#K = 5\nL#P = J#K / K#K\n = 3 / 5\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n => L#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#N": 1, "J#M": 1, "J#K": 1, "K#K": 2, "F#J": 2, "L#L": 2, "L#P": 3}, "id": "mod7_arith_d3_0038"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#K := 5. F#I := I#N + 6. L#N := K#M / F#I + I#N. J#N := L#M + 2 - 6. I#N := 4. K#M := 4 - I#N. L#M := 6. G#K := 5. L#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#K := 5. F#I := I#N + 6. L#N := K#M / F#I + I#N. J#N := L#M + 2 - 6. I#N := 4. K#M := 4 - I#N. L#M := 6. G#K := 5.", "query": "L#N", "answer": 4, "cot": "I#N = 4 -> I#N = 4\nF#I = I#N + 6\n = 4 + 6\n 4 + 6 = (4 + 6) mod 7 = 3\n => F#I = 3\nK#M = 4 - I#N\n = 4 - 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => K#M = 0\nL#N = K#M / F#I + I#N\n = 0 / 3 + 4\n 0 / 3 = (0 × 3⁻¹) mod 7 = 0\n 0 + 4 = (0 + 4) mod 7 = 4\n => L#N = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#K": 1, "I#N": 1, "I#K": 1, "L#M": 1, "F#I": 2, "J#N": 2, "K#M": 2, "L#N": 3}, "id": "mod7_arith_d3_0039"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#K := 6. G#O := G#N + G#K. G#N := 5. G#P := 2. J#P := K#P. K#N := 2. K#P := G#K * G#N. I#J := 5 * G#K. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#K := 6. G#O := G#N + G#K. G#N := 5. G#P := 2. J#P := K#P. K#N := 2. K#P := G#K * G#N. I#J := 5 * G#K.", "query": "J#P", "answer": 2, "cot": "G#N = 5 -> G#N = 5\nG#K = 6 -> G#K = 6\nK#P = G#K * G#N\n = 6 * 5\n 6 * 5 = (6 × 5) mod 7 = 2\n => K#P = 2\nJ#P = K#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#N": 1, "G#N": 1, "G#K": 1, "G#P": 1, "G#O": 2, "K#P": 2, "I#J": 2, "J#P": 3}, "id": "mod7_arith_d3_0040"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := L#K + 3. G#J := 3. L#K := G#J * K#N. E#N := L#I. L#I := 4. K#N := 2. H#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := L#K + 3. G#J := 3. L#K := G#J * K#N. E#N := L#I. L#I := 4. K#N := 2.", "query": "H#O", "answer": 2, "cot": "K#N = 2 -> K#N = 2\nG#J = 3 -> G#J = 3\nL#K = G#J * K#N\n = 3 * 2\n 3 * 2 = (3 × 2) mod 7 = 6\n => L#K = 6\nH#O = L#K + 3\n = 6 + 3\n 6 + 3 = (6 + 3) mod 7 = 2\n => H#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#N": 1, "G#J": 1, "L#I": 1, "L#K": 2, "E#N": 2, "H#O": 3}, "id": "mod7_arith_d3_0041"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#O := E#L * F#K. E#L := 6. I#K := F#O - E#L. F#K := 4. E#I := F#K - E#L. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#O := E#L * F#K. E#L := 6. I#K := F#O - E#L. F#K := 4. E#I := F#K - E#L.", "query": "I#K", "answer": 4, "cot": "E#L = 6 -> E#L = 6\nF#K = 4 -> F#K = 4\nF#O = E#L * F#K\n = 6 * 4\n 6 * 4 = (6 × 4) mod 7 = 3\n => F#O = 3\nI#K = F#O - E#L\n = 3 - 6\n 3 - 6 = (3 - 6) mod 7 = 4\n => I#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#L": 1, "F#K": 1, "E#I": 2, "F#O": 2, "I#K": 3}, "id": "mod7_arith_d3_0042"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := E#M / H#O. E#I := H#O / E#M. H#O := 3. E#M := 2. K#K := H#O + G#I. L#K := E#I. G#I := 4. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := E#M / H#O. E#I := H#O / E#M. H#O := 3. E#M := 2. K#K := H#O + G#I. L#K := E#I. G#I := 4.", "query": "L#K", "answer": 5, "cot": "E#M = 2 -> E#M = 2\nH#O = 3 -> H#O = 3\nE#I = H#O / E#M\n = 3 / 2\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => E#I = 5\nL#K = E#I = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#M": 1, "H#O": 1, "G#I": 1, "G#N": 2, "E#I": 2, "K#K": 2, "L#K": 3}, "id": "mod7_arith_d3_0043"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := L#N - G#O / F#K. I#N := 1. F#K := 3. K#P := 1. L#N := 4. G#O := K#P + F#K / L#N. F#L := L#N + K#P / F#K. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := L#N - G#O / F#K. I#N := 1. F#K := 3. K#P := 1. L#N := 4. G#O := K#P + F#K / L#N. F#L := L#N + K#P / F#K.", "query": "L#M", "answer": 1, "cot": "K#P = 1 -> K#P = 1\nL#N = 4 -> L#N = 4\nF#K = 3 -> F#K = 3\nG#O = K#P + F#K / L#N\n = 1 + 3 / 4\n 1 + 3 = (1 + 3) mod 7 = 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => G#O = 1\nL#M = L#N - G#O / F#K\n = 4 - 1 / 3\n 4 - 1 = (4 - 1) mod 7 = 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => L#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#P": 1, "I#N": 1, "L#N": 1, "F#K": 1, "G#O": 2, "F#L": 2, "L#M": 3}, "id": "mod7_arith_d3_0044"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#O := F#O / F#P. H#P := E#N - L#K. F#O := 6. F#P := 2. L#K := 4. E#N := F#P. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#O := F#O / F#P. H#P := E#N - L#K. F#O := 6. F#P := 2. L#K := 4. E#N := F#P.", "query": "H#P", "answer": 5, "cot": "F#P = 2 -> F#P = 2\nL#K = 4 -> L#K = 4\nE#N = F#P = 2\nH#P = E#N - L#K\n = 2 - 4\n 2 - 4 = (2 - 4) mod 7 = 5\n => H#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#P": 1, "L#K": 1, "F#O": 1, "L#O": 2, "E#N": 2, "H#P": 3}, "id": "mod7_arith_d3_0045"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := G#I + L#M. G#N := 4 - G#I. I#I := G#I. L#M := 4. I#J := 3 + I#I. I#K := 1. G#I := 5. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := G#I + L#M. G#N := 4 - G#I. I#I := G#I. L#M := 4. I#J := 3 + I#I. I#K := 1. G#I := 5.", "query": "I#J", "answer": 1, "cot": "G#I = 5 -> G#I = 5\nI#I = G#I = 5\nI#J = 3 + I#I\n = 3 + 5\n 3 + 5 = (3 + 5) mod 7 = 1\n => I#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#I": 1, "I#K": 1, "L#M": 1, "G#N": 2, "J#N": 2, "I#I": 2, "I#J": 3}, "id": "mod7_arith_d3_0046"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := 2. K#M := 2. I#M := K#M + F#J. E#L := F#J + K#M. G#K := K#M - E#L. G#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := 2. K#M := 2. I#M := K#M + F#J. E#L := F#J + K#M. G#K := K#M - E#L.", "query": "G#K", "answer": 5, "cot": "K#M = 2 -> K#M = 2\nF#J = 2 -> F#J = 2\nE#L = F#J + K#M\n = 2 + 2\n 2 + 2 = (2 + 2) mod 7 = 4\n => E#L = 4\nG#K = K#M - E#L\n = 2 - 4\n 2 - 4 = (2 - 4) mod 7 = 5\n => G#K = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#M": 1, "F#J": 1, "E#L": 2, "I#M": 2, "G#K": 3}, "id": "mod7_arith_d3_0047"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 5. E#O := I#K / J#N. L#N := J#N - H#I. H#I := J#N. I#K := 2. E#N := 3. I#L := J#N / I#K. L#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 5. E#O := I#K / J#N. L#N := J#N - H#I. H#I := J#N. I#K := 2. E#N := 3. I#L := J#N / I#K.", "query": "L#N", "answer": 0, "cot": "J#N = 5 -> J#N = 5\nH#I = J#N = 5\nL#N = J#N - H#I\n = 5 - 5\n 5 - 5 = (5 - 5) mod 7 = 0\n => L#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#K": 1, "J#N": 1, "E#N": 1, "I#L": 2, "E#O": 2, "H#I": 2, "L#N": 3}, "id": "mod7_arith_d3_0048"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := L#I + G#K / L#O. G#K := 4 - L#P. K#I := H#N * 4 + L#P. H#N := 1. L#P := 2. L#I := 1. L#O := 3. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := L#I + G#K / L#O. G#K := 4 - L#P. K#I := H#N * 4 + L#P. H#N := 1. L#P := 2. L#I := 1. L#O := 3.", "query": "E#I", "answer": 1, "cot": "L#O = 3 -> L#O = 3\nL#I = 1 -> L#I = 1\nL#P = 2 -> L#P = 2\nG#K = 4 - L#P\n = 4 - 2\n 4 - 2 = (4 - 2) mod 7 = 2\n => G#K = 2\nE#I = L#I + G#K / L#O\n = 1 + 2 / 3\n 1 + 2 = (1 + 2) mod 7 = 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => E#I = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#O": 1, "L#I": 1, "H#N": 1, "L#P": 1, "G#K": 2, "K#I": 2, "E#I": 3}, "id": "mod7_arith_d3_0049"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := 6. I#K := G#P * F#O * H#I. H#I := 4. H#K := H#I + G#P. F#O := G#P / H#I / 6. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := 6. I#K := G#P * F#O * H#I. H#I := 4. H#K := H#I + G#P. F#O := G#P / H#I / 6.", "query": "I#K", "answer": 6, "cot": "G#P = 6 -> G#P = 6\nH#I = 4 -> H#I = 4\nF#O = G#P / H#I / 6\n = 6 / 4 / 6\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n 5 / 6 = (5 × 6⁻¹) mod 7 = 2\n => F#O = 2\nI#K = G#P * F#O * H#I\n = 6 * 2 * 4\n 6 * 2 = (6 × 2) mod 7 = 5\n 5 * 4 = (5 × 4) mod 7 = 6\n => I#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#P": 1, "H#I": 1, "H#K": 2, "F#O": 2, "I#K": 3}, "id": "mod7_arith_d3_0050"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#O := G#O * I#N. F#M := I#N * G#O. I#N := 1. G#M := F#M. G#O := 6. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#O := G#O * I#N. F#M := I#N * G#O. I#N := 1. G#M := F#M. G#O := 6.", "query": "G#M", "answer": 6, "cot": "G#O = 6 -> G#O = 6\nI#N = 1 -> I#N = 1\nF#M = I#N * G#O\n = 1 * 6\n 1 * 6 = (1 × 6) mod 7 = 6\n => F#M = 6\nG#M = F#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#O": 1, "I#N": 1, "F#M": 2, "K#O": 2, "G#M": 3}, "id": "mod7_arith_d3_0051"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := H#N. F#J := I#K * H#N - F#O. I#K := 2. F#O := 6. I#I := H#N. H#N := 5. J#L := I#I / 2. E#K := 6. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := H#N. F#J := I#K * H#N - F#O. I#K := 2. F#O := 6. I#I := H#N. H#N := 5. J#L := I#I / 2. E#K := 6.", "query": "J#L", "answer": 6, "cot": "H#N = 5 -> H#N = 5\nI#I = H#N = 5\nJ#L = I#I / 2\n = 5 / 2\n 5 / 2 = (5 × 2⁻¹) mod 7 = 6\n => J#L = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#N": 1, "F#O": 1, "I#K": 1, "E#K": 1, "L#J": 2, "F#J": 2, "I#I": 2, "J#L": 3}, "id": "mod7_arith_d3_0052"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#J := 3. J#J := L#J. J#I := E#J - 4. L#J := E#M / 5 + E#J. E#M := 5. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#J := 3. J#J := L#J. J#I := E#J - 4. L#J := E#M / 5 + E#J. E#M := 5.", "query": "J#J", "answer": 4, "cot": "E#M = 5 -> E#M = 5\nE#J = 3 -> E#J = 3\nL#J = E#M / 5 + E#J\n = 5 / 5 + 3\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n 1 + 3 = (1 + 3) mod 7 = 4\n => L#J = 4\nJ#J = L#J = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#M": 1, "E#J": 1, "L#J": 2, "J#I": 2, "J#J": 3}, "id": "mod7_arith_d3_0053"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := 3. H#I := 4 + 5 * F#K. G#J := 1. K#N := F#K + H#I. L#L := F#K - G#J. K#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := 3. H#I := 4 + 5 * F#K. G#J := 1. K#N := F#K + H#I. L#L := F#K - G#J.", "query": "K#N", "answer": 2, "cot": "F#K = 3 -> F#K = 3\nH#I = 4 + 5 * F#K\n = 4 + 5 * 3\n 4 + 5 = (4 + 5) mod 7 = 2\n 2 * 3 = (2 × 3) mod 7 = 6\n => H#I = 6\nK#N = F#K + H#I\n = 3 + 6\n 3 + 6 = (3 + 6) mod 7 = 2\n => K#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#K": 1, "G#J": 1, "L#L": 2, "H#I": 2, "K#N": 3}, "id": "mod7_arith_d3_0054"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := 4 / G#K / 4. K#L := L#I - L#N. G#K := 1. L#N := G#K / I#I. E#I := G#K * I#I. I#I := 2. K#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := 4 / G#K / 4. K#L := L#I - L#N. G#K := 1. L#N := G#K / I#I. E#I := G#K * I#I. I#I := 2.", "query": "K#L", "answer": 4, "cot": "G#K = 1 -> G#K = 1\nI#I = 2 -> I#I = 2\nL#N = G#K / I#I\n = 1 / 2\n 1 / 2 = (1 × 2⁻¹) mod 7 = 4\n => L#N = 4\nL#I = 4 / G#K / 4\n = 4 / 1 / 4\n 4 / 1 = (4 × 1⁻¹) mod 7 = 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => L#I = 1\nK#L = L#I - L#N\n = 1 - 4\n 1 - 4 = (1 - 4) mod 7 = 4\n => K#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#K": 1, "I#I": 1, "L#N": 2, "E#I": 2, "L#I": 2, "K#L": 3}, "id": "mod7_arith_d3_0055"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := 5. L#M := J#P / F#K. E#L := F#K + H#M - 6. H#M := 2. J#P := 1. G#M := 5 / L#M + F#K. E#N := F#K - H#M. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := 5. L#M := J#P / F#K. E#L := F#K + H#M - 6. H#M := 2. J#P := 1. G#M := 5 / L#M + F#K. E#N := F#K - H#M.", "query": "G#M", "answer": 2, "cot": "J#P = 1 -> J#P = 1\nF#K = 5 -> F#K = 5\nL#M = J#P / F#K\n = 1 / 5\n 1 / 5 = (1 × 5⁻¹) mod 7 = 3\n => L#M = 3\nG#M = 5 / L#M + F#K\n = 5 / 3 + 5\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n 4 + 5 = (4 + 5) mod 7 = 2\n => G#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#P": 1, "F#K": 1, "H#M": 1, "L#M": 2, "E#L": 2, "E#N": 2, "G#M": 3}, "id": "mod7_arith_d3_0056"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#K := 2. F#M := K#M. K#M := 3. I#J := K#M - J#K. G#N := F#M / 5. G#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#K := 2. F#M := K#M. K#M := 3. I#J := K#M - J#K. G#N := F#M / 5.", "query": "G#N", "answer": 2, "cot": "K#M = 3 -> K#M = 3\nF#M = K#M = 3\nG#N = F#M / 5\n = 3 / 5\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n => G#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#M": 1, "J#K": 1, "F#M": 2, "I#J": 2, "G#N": 3}, "id": "mod7_arith_d3_0057"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := J#P + I#N. G#P := J#P - I#N. J#P := 5. H#I := I#N. K#P := K#L. I#N := 4. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := J#P + I#N. G#P := J#P - I#N. J#P := 5. H#I := I#N. K#P := K#L. I#N := 4.", "query": "K#P", "answer": 2, "cot": "I#N = 4 -> I#N = 4\nJ#P = 5 -> J#P = 5\nK#L = J#P + I#N\n = 5 + 4\n 5 + 4 = (5 + 4) mod 7 = 2\n => K#L = 2\nK#P = K#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#N": 1, "J#P": 1, "G#P": 2, "K#L": 2, "H#I": 2, "K#P": 3}, "id": "mod7_arith_d3_0058"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := 5. E#P := E#M. E#M := 4. F#J := E#M + J#O. J#M := E#P * 5. J#O := 1. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := 5. E#P := E#M. E#M := 4. F#J := E#M + J#O. J#M := E#P * 5. J#O := 1.", "query": "J#M", "answer": 6, "cot": "E#M = 4 -> E#M = 4\nE#P = E#M = 4\nJ#M = E#P * 5\n = 4 * 5\n 4 * 5 = (4 × 5) mod 7 = 6\n => J#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#I": 1, "E#M": 1, "J#O": 1, "F#J": 2, "E#P": 2, "J#M": 3}, "id": "mod7_arith_d3_0059"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#K := 6. K#O := G#L + L#K. E#M := L#K * G#L. L#I := K#O - 4. J#M := 3 + L#K. G#L := 4. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#K := 6. K#O := G#L + L#K. E#M := L#K * G#L. L#I := K#O - 4. J#M := 3 + L#K. G#L := 4.", "query": "L#I", "answer": 6, "cot": "G#L = 4 -> G#L = 4\nL#K = 6 -> L#K = 6\nK#O = G#L + L#K\n = 4 + 6\n 4 + 6 = (4 + 6) mod 7 = 3\n => K#O = 3\nL#I = K#O - 4\n = 3 - 4\n 3 - 4 = (3 - 4) mod 7 = 6\n => L#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#L": 1, "L#K": 1, "K#O": 2, "E#M": 2, "J#M": 2, "L#I": 3}, "id": "mod7_arith_d3_0060"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := 6 + K#O + L#P. L#P := E#N. L#I := 6 - F#I / E#N. F#I := 4. K#O := H#I * E#N. H#I := 5. E#N := 4. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := 6 + K#O + L#P. L#P := E#N. L#I := 6 - F#I / E#N. F#I := 4. K#O := H#I * E#N. H#I := 5. E#N := 4.", "query": "E#K", "answer": 2, "cot": "E#N = 4 -> E#N = 4\nH#I = 5 -> H#I = 5\nK#O = H#I * E#N\n = 5 * 4\n 5 * 4 = (5 × 4) mod 7 = 6\n => K#O = 6\nL#P = E#N = 4\nE#K = 6 + K#O + L#P\n = 6 + 6 + 4\n 6 + 6 = (6 + 6) mod 7 = 5\n 5 + 4 = (5 + 4) mod 7 = 2\n => E#K = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#I": 1, "E#N": 1, "H#I": 1, "L#I": 2, "K#O": 2, "L#P": 2, "E#K": 3}, "id": "mod7_arith_d3_0061"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := E#I / F#M. F#M := 2. G#N := F#M + E#I. F#K := G#P / F#M / 2. G#P := F#M. E#I := 1. F#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := E#I / F#M. F#M := 2. G#N := F#M + E#I. F#K := G#P / F#M / 2. G#P := F#M. E#I := 1.", "query": "F#K", "answer": 4, "cot": "F#M = 2 -> F#M = 2\nG#P = F#M = 2\nF#K = G#P / F#M / 2\n = 2 / 2 / 2\n 2 / 2 = (2 × 2⁻¹) mod 7 = 1\n 1 / 2 = (1 × 2⁻¹) mod 7 = 4\n => F#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#M": 1, "E#I": 1, "G#P": 2, "G#J": 2, "G#N": 2, "F#K": 3}, "id": "mod7_arith_d3_0062"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 5. L#O := 6. K#N := L#O + J#N. E#P := 4 / H#M. E#O := 3 + J#N * L#O. H#M := J#N / 3. E#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 5. L#O := 6. K#N := L#O + J#N. E#P := 4 / H#M. E#O := 3 + J#N * L#O. H#M := J#N / 3.", "query": "E#P", "answer": 1, "cot": "J#N = 5 -> J#N = 5\nH#M = J#N / 3\n = 5 / 3\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n => H#M = 4\nE#P = 4 / H#M\n = 4 / 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => E#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#N": 1, "L#O": 1, "K#N": 2, "H#M": 2, "E#O": 2, "E#P": 3}, "id": "mod7_arith_d3_0063"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#O := 6. F#M := I#I - E#O. I#I := 6. G#I := L#N. L#N := 1. E#M := 3. H#M := G#I * 3 + L#L. L#L := E#M - I#I. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#O := 6. F#M := I#I - E#O. I#I := 6. G#I := L#N. L#N := 1. E#M := 3. H#M := G#I * 3 + L#L. L#L := E#M - I#I.", "query": "H#M", "answer": 0, "cot": "I#I = 6 -> I#I = 6\nL#N = 1 -> L#N = 1\nE#M = 3 -> E#M = 3\nG#I = L#N = 1\nL#L = E#M - I#I\n = 3 - 6\n 3 - 6 = (3 - 6) mod 7 = 4\n => L#L = 4\nH#M = G#I * 3 + L#L\n = 1 * 3 + 4\n 1 * 3 = (1 × 3) mod 7 = 3\n 3 + 4 = (3 + 4) mod 7 = 0\n => H#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"I#I": 1, "L#N": 1, "E#O": 1, "E#M": 1, "G#I": 2, "L#L": 2, "F#M": 2, "H#M": 3}, "id": "mod7_arith_d3_0064"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#M := K#M - E#K / 5. G#K := G#J / K#M. K#M := 2. E#K := K#M - G#J. G#J := 2. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#M := K#M - E#K / 5. G#K := G#J / K#M. K#M := 2. E#K := K#M - G#J. G#J := 2.", "query": "J#M", "answer": 6, "cot": "K#M = 2 -> K#M = 2\nG#J = 2 -> G#J = 2\nE#K = K#M - G#J\n = 2 - 2\n 2 - 2 = (2 - 2) mod 7 = 0\n => E#K = 0\nJ#M = K#M - E#K / 5\n = 2 - 0 / 5\n 2 - 0 = (2 - 0) mod 7 = 2\n 2 / 5 = (2 × 5⁻¹) mod 7 = 6\n => J#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#M": 1, "G#J": 1, "G#K": 2, "E#K": 2, "J#M": 3}, "id": "mod7_arith_d3_0065"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#L := 6. H#L := K#I + I#L. K#P := K#J + I#L. K#I := 6. G#N := K#P. K#J := 6. G#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#L := 6. H#L := K#I + I#L. K#P := K#J + I#L. K#I := 6. G#N := K#P. K#J := 6.", "query": "G#N", "answer": 5, "cot": "I#L = 6 -> I#L = 6\nK#J = 6 -> K#J = 6\nK#P = K#J + I#L\n = 6 + 6\n 6 + 6 = (6 + 6) mod 7 = 5\n => K#P = 5\nG#N = K#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#I": 1, "I#L": 1, "K#J": 1, "K#P": 2, "H#L": 2, "G#N": 3}, "id": "mod7_arith_d3_0066"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#L := 1. I#L := 5. G#P := K#N. F#I := 5. J#I := 6. L#K := F#I. K#N := J#I - F#I - J#L. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#L := 1. I#L := 5. G#P := K#N. F#I := 5. J#I := 6. L#K := F#I. K#N := J#I - F#I - J#L.", "query": "G#P", "answer": 0, "cot": "J#L = 1 -> J#L = 1\nF#I = 5 -> F#I = 5\nJ#I = 6 -> J#I = 6\nK#N = J#I - F#I - J#L\n = 6 - 5 - 1\n 6 - 5 = (6 - 5) mod 7 = 1\n 1 - 1 = (1 - 1) mod 7 = 0\n => K#N = 0\nG#P = K#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#L": 1, "F#I": 1, "J#I": 1, "I#L": 1, "L#K": 2, "K#N": 2, "G#P": 3}, "id": "mod7_arith_d3_0067"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := 6. F#K := J#P + J#I. E#J := 1. J#P := 1. H#N := E#K. J#I := E#K - J#P. F#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := 6. F#K := J#P + J#I. E#J := 1. J#P := 1. H#N := E#K. J#I := E#K - J#P.", "query": "F#K", "answer": 6, "cot": "J#P = 1 -> J#P = 1\nE#K = 6 -> E#K = 6\nJ#I = E#K - J#P\n = 6 - 1\n 6 - 1 = (6 - 1) mod 7 = 5\n => J#I = 5\nF#K = J#P + J#I\n = 1 + 5\n 1 + 5 = (1 + 5) mod 7 = 6\n => F#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#J": 1, "J#P": 1, "E#K": 1, "J#I": 2, "H#N": 2, "F#K": 3}, "id": "mod7_arith_d3_0068"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#P := 6 * F#J. F#J := J#I. H#N := 3. H#J := L#O / J#I / 6. H#L := 2. J#I := 5. L#O := 6. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#P := 6 * F#J. F#J := J#I. H#N := 3. H#J := L#O / J#I / 6. H#L := 2. J#I := 5. L#O := 6.", "query": "J#P", "answer": 2, "cot": "J#I = 5 -> J#I = 5\nF#J = J#I = 5\nJ#P = 6 * F#J\n = 6 * 5\n 6 * 5 = (6 × 5) mod 7 = 2\n => J#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#O": 1, "H#L": 1, "H#N": 1, "J#I": 1, "H#J": 2, "F#J": 2, "J#P": 3}, "id": "mod7_arith_d3_0069"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := E#K. F#I := 1. G#P := 4. G#O := K#J / 3 + E#K. E#N := G#P * E#M. E#K := 6. E#J := E#K + G#P. E#M := 5. G#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := E#K. F#I := 1. G#P := 4. G#O := K#J / 3 + E#K. E#N := G#P * E#M. E#K := 6. E#J := E#K + G#P. E#M := 5.", "query": "G#O", "answer": 1, "cot": "E#K = 6 -> E#K = 6\nK#J = E#K = 6\nG#O = K#J / 3 + E#K\n = 6 / 3 + 6\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n 2 + 6 = (2 + 6) mod 7 = 1\n => G#O = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#I": 1, "E#M": 1, "G#P": 1, "E#K": 1, "E#J": 2, "E#N": 2, "K#J": 2, "G#O": 3}, "id": "mod7_arith_d3_0070"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := 4. E#K := L#N * E#I. I#L := 4. H#P := 1. E#M := I#L + E#K + 5. L#N := 3. J#N := H#P. G#M := E#I * I#L + H#P. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := 4. E#K := L#N * E#I. I#L := 4. H#P := 1. E#M := I#L + E#K + 5. L#N := 3. J#N := H#P. G#M := E#I * I#L + H#P.", "query": "E#M", "answer": 0, "cot": "I#L = 4 -> I#L = 4\nL#N = 3 -> L#N = 3\nE#I = 4 -> E#I = 4\nE#K = L#N * E#I\n = 3 * 4\n 3 * 4 = (3 × 4) mod 7 = 5\n => E#K = 5\nE#M = I#L + E#K + 5\n = 4 + 5 + 5\n 4 + 5 = (4 + 5) mod 7 = 2\n 2 + 5 = (2 + 5) mod 7 = 0\n => E#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#L": 1, "H#P": 1, "L#N": 1, "E#I": 1, "J#N": 2, "G#M": 2, "E#K": 2, "E#M": 3}, "id": "mod7_arith_d3_0071"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#L := 1. L#I := I#L + K#O. K#K := 2. H#O := 1. G#I := H#O. E#N := 3 + K#O. F#I := H#O * E#N. K#O := 6. F#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#L := 1. L#I := I#L + K#O. K#K := 2. H#O := 1. G#I := H#O. E#N := 3 + K#O. F#I := H#O * E#N. K#O := 6.", "query": "F#I", "answer": 2, "cot": "K#O = 6 -> K#O = 6\nH#O = 1 -> H#O = 1\nE#N = 3 + K#O\n = 3 + 6\n 3 + 6 = (3 + 6) mod 7 = 2\n => E#N = 2\nF#I = H#O * E#N\n = 1 * 2\n 1 * 2 = (1 × 2) mod 7 = 2\n => F#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#L": 1, "K#K": 1, "K#O": 1, "H#O": 1, "G#I": 2, "E#N": 2, "L#I": 2, "F#I": 3}, "id": "mod7_arith_d3_0072"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 3. K#I := E#J. E#J := 4. I#L := K#I * I#N - J#N. I#N := J#N - E#J. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 3. K#I := E#J. E#J := 4. I#L := K#I * I#N - J#N. I#N := J#N - E#J.", "query": "I#L", "answer": 0, "cot": "J#N = 3 -> J#N = 3\nE#J = 4 -> E#J = 4\nK#I = E#J = 4\nI#N = J#N - E#J\n = 3 - 4\n 3 - 4 = (3 - 4) mod 7 = 6\n => I#N = 6\nI#L = K#I * I#N - J#N\n = 4 * 6 - 3\n 4 * 6 = (4 × 6) mod 7 = 3\n 3 - 3 = (3 - 3) mod 7 = 0\n => I#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#N": 1, "E#J": 1, "K#I": 2, "I#N": 2, "I#L": 3}, "id": "mod7_arith_d3_0073"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := 4. G#P := E#K. G#L := F#O - K#J. J#N := 4. F#O := 5. E#K := F#O * K#J / H#N. H#N := 6. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := 4. G#P := E#K. G#L := F#O - K#J. J#N := 4. F#O := 5. E#K := F#O * K#J / H#N. H#N := 6.", "query": "G#P", "answer": 1, "cot": "F#O = 5 -> F#O = 5\nH#N = 6 -> H#N = 6\nK#J = 4 -> K#J = 4\nE#K = F#O * K#J / H#N\n = 5 * 4 / 6\n 5 * 4 = (5 × 4) mod 7 = 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => E#K = 1\nG#P = E#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#O": 1, "H#N": 1, "K#J": 1, "J#N": 1, "G#L": 2, "E#K": 2, "G#P": 3}, "id": "mod7_arith_d3_0074"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#N := 4. G#P := H#O. J#I := H#M. F#L := K#I * I#N. H#O := H#M * 5. H#M := 5. K#I := 4. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#N := 4. G#P := H#O. J#I := H#M. F#L := K#I * I#N. H#O := H#M * 5. H#M := 5. K#I := 4.", "query": "G#P", "answer": 4, "cot": "H#M = 5 -> H#M = 5\nH#O = H#M * 5\n = 5 * 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => H#O = 4\nG#P = H#O = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#M": 1, "I#N": 1, "K#I": 1, "F#L": 2, "J#I": 2, "H#O": 2, "G#P": 3}, "id": "mod7_arith_d3_0075"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := 6. I#J := E#P - F#N. E#P := 1. K#N := L#I * F#N + 6. L#I := 4. F#N := 1. I#L := K#N * I#J. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := 6. I#J := E#P - F#N. E#P := 1. K#N := L#I * F#N + 6. L#I := 4. F#N := 1. I#L := K#N * I#J.", "query": "I#L", "answer": 0, "cot": "F#N = 1 -> F#N = 1\nL#I = 4 -> L#I = 4\nE#P = 1 -> E#P = 1\nK#N = L#I * F#N + 6\n = 4 * 1 + 6\n 4 * 1 = (4 × 1) mod 7 = 4\n 4 + 6 = (4 + 6) mod 7 = 3\n => K#N = 3\nI#J = E#P - F#N\n = 1 - 1\n 1 - 1 = (1 - 1) mod 7 = 0\n => I#J = 0\nI#L = K#N * I#J\n = 3 * 0\n 3 * 0 = (3 × 0) mod 7 = 0\n => I#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#N": 1, "L#I": 1, "L#J": 1, "E#P": 1, "K#N": 2, "I#J": 2, "I#L": 3}, "id": "mod7_arith_d3_0076"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#L := 5 + G#K * H#M. H#M := 6. J#L := H#N - H#M / G#K. I#P := G#K. H#N := 2. G#K := 4. J#P := 4 - H#L / J#L. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#L := 5 + G#K * H#M. H#M := 6. J#L := H#N - H#M / G#K. I#P := G#K. H#N := 2. G#K := 4. J#P := 4 - H#L / J#L.", "query": "J#P", "answer": 1, "cot": "G#K = 4 -> G#K = 4\nH#N = 2 -> H#N = 2\nH#M = 6 -> H#M = 6\nJ#L = H#N - H#M / G#K\n = 2 - 6 / 4\n 2 - 6 = (2 - 6) mod 7 = 3\n 3 / 4 = (3 × 4⁻¹) mod 7 = 6\n => J#L = 6\nH#L = 5 + G#K * H#M\n = 5 + 4 * 6\n 5 + 4 = (5 + 4) mod 7 = 2\n 2 * 6 = (2 × 6) mod 7 = 5\n => H#L = 5\nJ#P = 4 - H#L / J#L\n = 4 - 5 / 6\n 4 - 5 = (4 - 5) mod 7 = 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => J#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#K": 1, "H#N": 1, "H#M": 1, "J#L": 2, "I#P": 2, "H#L": 2, "J#P": 3}, "id": "mod7_arith_d3_0077"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := L#L / F#N * 6. J#O := F#N / 3 - I#P. I#P := F#N - G#M * L#L. G#M := 2. F#N := 1. L#L := 4. J#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := L#L / F#N * 6. J#O := F#N / 3 - I#P. I#P := F#N - G#M * L#L. G#M := 2. F#N := 1. L#L := 4.", "query": "J#O", "answer": 2, "cot": "G#M = 2 -> G#M = 2\nL#L = 4 -> L#L = 4\nF#N = 1 -> F#N = 1\nI#P = F#N - G#M * L#L\n = 1 - 2 * 4\n 1 - 2 = (1 - 2) mod 7 = 6\n 6 * 4 = (6 × 4) mod 7 = 3\n => I#P = 3\nJ#O = F#N / 3 - I#P\n = 1 / 3 - 3\n 1 / 3 = (1 × 3⁻¹) mod 7 = 5\n 5 - 3 = (5 - 3) mod 7 = 2\n => J#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#M": 1, "L#L": 1, "F#N": 1, "H#J": 2, "I#P": 2, "J#O": 3}, "id": "mod7_arith_d3_0078"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := G#P. E#L := K#M * K#J. L#J := 4. G#P := 6. K#M := G#P. K#N := L#J - G#P. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := G#P. E#L := K#M * K#J. L#J := 4. G#P := 6. K#M := G#P. K#N := L#J - G#P.", "query": "E#L", "answer": 1, "cot": "G#P = 6 -> G#P = 6\nK#J = G#P = 6\nK#M = G#P = 6\nE#L = K#M * K#J\n = 6 * 6\n 6 * 6 = (6 × 6) mod 7 = 1\n => E#L = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#P": 1, "L#J": 1, "K#J": 2, "K#M": 2, "K#N": 2, "E#L": 3}, "id": "mod7_arith_d3_0079"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#N := 5. L#L := J#L. H#N := 5. I#I := H#N / I#N. I#P := H#N * I#N. J#L := H#N. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#N := 5. L#L := J#L. H#N := 5. I#I := H#N / I#N. I#P := H#N * I#N. J#L := H#N.", "query": "L#L", "answer": 5, "cot": "H#N = 5 -> H#N = 5\nJ#L = H#N = 5\nL#L = J#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#N": 1, "I#N": 1, "I#I": 2, "I#P": 2, "J#L": 2, "L#L": 3}, "id": "mod7_arith_d3_0080"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := K#K + I#J. E#L := G#O / I#J * K#K. K#J := H#L * G#O - 6. K#K := 1. H#L := G#O + I#J + K#K. G#O := 5. I#J := 1. K#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := K#K + I#J. E#L := G#O / I#J * K#K. K#J := H#L * G#O - 6. K#K := 1. H#L := G#O + I#J + K#K. G#O := 5. I#J := 1.", "query": "K#J", "answer": 1, "cot": "I#J = 1 -> I#J = 1\nK#K = 1 -> K#K = 1\nG#O = 5 -> G#O = 5\nH#L = G#O + I#J + K#K\n = 5 + 1 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n 6 + 1 = (6 + 1) mod 7 = 0\n => H#L = 0\nK#J = H#L * G#O - 6\n = 0 * 5 - 6\n 0 * 5 = (0 × 5) mod 7 = 0\n 0 - 6 = (0 - 6) mod 7 = 1\n => K#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#J": 1, "K#K": 1, "G#O": 1, "G#N": 2, "H#L": 2, "E#L": 2, "K#J": 3}, "id": "mod7_arith_d3_0081"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#L := 3. J#P := 5. F#P := H#L. I#I := H#L. G#L := F#P - I#I. G#O := 5. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#L := 3. J#P := 5. F#P := H#L. I#I := H#L. G#L := F#P - I#I. G#O := 5.", "query": "G#L", "answer": 0, "cot": "H#L = 3 -> H#L = 3\nI#I = H#L = 3\nF#P = H#L = 3\nG#L = F#P - I#I\n = 3 - 3\n 3 - 3 = (3 - 3) mod 7 = 0\n => G#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#O": 1, "H#L": 1, "J#P": 1, "I#I": 2, "F#P": 2, "G#L": 3}, "id": "mod7_arith_d3_0082"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := 6. F#M := 3. F#L := I#N + L#K. I#N := F#M * H#I. L#K := G#J + F#M. H#I := 5. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := 6. F#M := 3. F#L := I#N + L#K. I#N := F#M * H#I. L#K := G#J + F#M. H#I := 5.", "query": "F#L", "answer": 3, "cot": "F#M = 3 -> F#M = 3\nH#I = 5 -> H#I = 5\nG#J = 6 -> G#J = 6\nI#N = F#M * H#I\n = 3 * 5\n 3 * 5 = (3 × 5) mod 7 = 1\n => I#N = 1\nL#K = G#J + F#M\n = 6 + 3\n 6 + 3 = (6 + 3) mod 7 = 2\n => L#K = 2\nF#L = I#N + L#K\n = 1 + 2\n 1 + 2 = (1 + 2) mod 7 = 3\n => F#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#M": 1, "H#I": 1, "G#J": 1, "I#N": 2, "L#K": 2, "F#L": 3}, "id": "mod7_arith_d3_0083"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := 1. K#N := F#O * J#O. K#L := F#O. I#O := K#L. F#O := 4. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := 1. K#N := F#O * J#O. K#L := F#O. I#O := K#L. F#O := 4.", "query": "I#O", "answer": 4, "cot": "F#O = 4 -> F#O = 4\nK#L = F#O = 4\nI#O = K#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#O": 1, "J#O": 1, "K#N": 2, "K#L": 2, "I#O": 3}, "id": "mod7_arith_d3_0084"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#M := H#O + G#M. E#L := H#O / I#I. H#O := 3. I#I := H#O * G#M * 2. I#J := 6. G#O := H#O. G#M := 5. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#M := H#O + G#M. E#L := H#O / I#I. H#O := 3. I#I := H#O * G#M * 2. I#J := 6. G#O := H#O. G#M := 5.", "query": "E#L", "answer": 5, "cot": "G#M = 5 -> G#M = 5\nH#O = 3 -> H#O = 3\nI#I = H#O * G#M * 2\n = 3 * 5 * 2\n 3 * 5 = (3 × 5) mod 7 = 1\n 1 * 2 = (1 × 2) mod 7 = 2\n => I#I = 2\nE#L = H#O / I#I\n = 3 / 2\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => E#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#J": 1, "G#M": 1, "H#O": 1, "G#O": 2, "I#I": 2, "F#M": 2, "E#L": 3}, "id": "mod7_arith_d3_0085"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := G#J. I#P := 1. G#L := 1. I#N := 5. G#J := 4. K#O := I#L / I#P. I#L := G#L / I#P. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := G#J. I#P := 1. G#L := 1. I#N := 5. G#J := 4. K#O := I#L / I#P. I#L := G#L / I#P.", "query": "K#O", "answer": 1, "cot": "G#L = 1 -> G#L = 1\nI#P = 1 -> I#P = 1\nI#L = G#L / I#P\n = 1 / 1\n 1 / 1 = (1 × 1⁻¹) mod 7 = 1\n => I#L = 1\nK#O = I#L / I#P\n = 1 / 1\n 1 / 1 = (1 × 1⁻¹) mod 7 = 1\n => K#O = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#L": 1, "I#N": 1, "G#J": 1, "I#P": 1, "F#P": 2, "I#L": 2, "K#O": 3}, "id": "mod7_arith_d3_0086"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#K := 1. K#J := K#K. H#I := 1. K#I := K#J. G#K := 1. J#N := 6 / K#K. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#K := 1. K#J := K#K. H#I := 1. K#I := K#J. G#K := 1. J#N := 6 / K#K.", "query": "K#I", "answer": 1, "cot": "K#K = 1 -> K#K = 1\nK#J = K#K = 1\nK#I = K#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#K": 1, "G#K": 1, "H#I": 1, "J#N": 2, "K#J": 2, "K#I": 3}, "id": "mod7_arith_d3_0087"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#K := 1. J#P := K#K + E#M. H#O := 4. I#J := K#K / 5. E#M := K#K. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#K := 1. J#P := K#K + E#M. H#O := 4. I#J := K#K / 5. E#M := K#K.", "query": "J#P", "answer": 2, "cot": "K#K = 1 -> K#K = 1\nE#M = K#K = 1\nJ#P = K#K + E#M\n = 1 + 1\n 1 + 1 = (1 + 1) mod 7 = 2\n => J#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#K": 1, "H#O": 1, "E#M": 2, "I#J": 2, "J#P": 3}, "id": "mod7_arith_d3_0088"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := 4. L#N := 1. E#N := 5. I#O := E#N * K#L. F#O := E#N + K#L / L#N. K#I := F#O. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := 4. L#N := 1. E#N := 5. I#O := E#N * K#L. F#O := E#N + K#L / L#N. K#I := F#O.", "query": "K#I", "answer": 2, "cot": "L#N = 1 -> L#N = 1\nE#N = 5 -> E#N = 5\nK#L = 4 -> K#L = 4\nF#O = E#N + K#L / L#N\n = 5 + 4 / 1\n 5 + 4 = (5 + 4) mod 7 = 2\n 2 / 1 = (2 × 1⁻¹) mod 7 = 2\n => F#O = 2\nK#I = F#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#N": 1, "E#N": 1, "K#L": 1, "I#O": 2, "F#O": 2, "K#I": 3}, "id": "mod7_arith_d3_0089"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := I#M. E#P := 6. F#M := 6. J#K := 5. J#N := 6. K#J := 3 - F#M. I#M := J#N * F#M * E#P. K#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := I#M. E#P := 6. F#M := 6. J#K := 5. J#N := 6. K#J := 3 - F#M. I#M := J#N * F#M * E#P.", "query": "K#M", "answer": 6, "cot": "E#P = 6 -> E#P = 6\nF#M = 6 -> F#M = 6\nJ#N = 6 -> J#N = 6\nI#M = J#N * F#M * E#P\n = 6 * 6 * 6\n 6 * 6 = (6 × 6) mod 7 = 1\n 1 * 6 = (1 × 6) mod 7 = 6\n => I#M = 6\nK#M = I#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"E#P": 1, "F#M": 1, "J#N": 1, "J#K": 1, "I#M": 2, "K#J": 2, "K#M": 3}, "id": "mod7_arith_d3_0090"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := 4. K#M := F#I * E#I. J#I := 4 - F#I. G#J := 1. F#I := 2. H#O := K#M. H#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := 4. K#M := F#I * E#I. J#I := 4 - F#I. G#J := 1. F#I := 2. H#O := K#M.", "query": "H#O", "answer": 1, "cot": "E#I = 4 -> E#I = 4\nF#I = 2 -> F#I = 2\nK#M = F#I * E#I\n = 2 * 4\n 2 * 4 = (2 × 4) mod 7 = 1\n => K#M = 1\nH#O = K#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#I": 1, "G#J": 1, "F#I": 1, "J#I": 2, "K#M": 2, "H#O": 3}, "id": "mod7_arith_d3_0091"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := 2 / L#P. F#J := 3. L#P := 1. I#I := E#I. H#L := F#J * L#P. G#O := F#J * 5. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := 2 / L#P. F#J := 3. L#P := 1. I#I := E#I. H#L := F#J * L#P. G#O := F#J * 5.", "query": "I#I", "answer": 2, "cot": "L#P = 1 -> L#P = 1\nE#I = 2 / L#P\n = 2 / 1\n 2 / 1 = (2 × 1⁻¹) mod 7 = 2\n => E#I = 2\nI#I = E#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#P": 1, "F#J": 1, "H#L": 2, "E#I": 2, "G#O": 2, "I#I": 3}, "id": "mod7_arith_d3_0092"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#N := I#L. H#O := 2. J#P := 1. L#I := J#P + H#O. I#L := J#P - J#M + 5. J#M := 2. L#L := 4. H#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#N := I#L. H#O := 2. J#P := 1. L#I := J#P + H#O. I#L := J#P - J#M + 5. J#M := 2. L#L := 4.", "query": "H#N", "answer": 4, "cot": "J#M = 2 -> J#M = 2\nJ#P = 1 -> J#P = 1\nI#L = J#P - J#M + 5\n = 1 - 2 + 5\n 1 - 2 = (1 - 2) mod 7 = 6\n 6 + 5 = (6 + 5) mod 7 = 4\n => I#L = 4\nH#N = I#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#M": 1, "L#L": 1, "J#P": 1, "H#O": 1, "I#L": 2, "L#I": 2, "H#N": 3}, "id": "mod7_arith_d3_0093"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#I := 3. H#P := G#I * G#J / F#L. G#J := 4 / G#I - L#I. K#P := I#L * G#I. L#I := 1. I#N := 2. F#L := G#I. I#L := 3. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#I := 3. H#P := G#I * G#J / F#L. G#J := 4 / G#I - L#I. K#P := I#L * G#I. L#I := 1. I#N := 2. F#L := G#I. I#L := 3.", "query": "H#P", "answer": 5, "cot": "G#I = 3 -> G#I = 3\nL#I = 1 -> L#I = 1\nF#L = G#I = 3\nG#J = 4 / G#I - L#I\n = 4 / 3 - 1\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n 6 - 1 = (6 - 1) mod 7 = 5\n => G#J = 5\nH#P = G#I * G#J / F#L\n = 3 * 5 / 3\n 3 * 5 = (3 × 5) mod 7 = 1\n 1 / 3 = (1 × 3⁻¹) mod 7 = 5\n => H#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#L": 1, "I#N": 1, "G#I": 1, "L#I": 1, "F#L": 2, "K#P": 2, "G#J": 2, "H#P": 3}, "id": "mod7_arith_d3_0094"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := F#O * E#O - H#L. F#O := 2. H#N := J#O + F#O. H#L := 6. E#O := 4. J#O := 2. H#O := H#N. H#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := F#O * E#O - H#L. F#O := 2. H#N := J#O + F#O. H#L := 6. E#O := 4. J#O := 2. H#O := H#N.", "query": "H#O", "answer": 4, "cot": "F#O = 2 -> F#O = 2\nJ#O = 2 -> J#O = 2\nH#N = J#O + F#O\n = 2 + 2\n 2 + 2 = (2 + 2) mod 7 = 4\n => H#N = 4\nH#O = H#N = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#L": 1, "F#O": 1, "E#O": 1, "J#O": 1, "H#N": 2, "E#N": 2, "H#O": 3}, "id": "mod7_arith_d3_0095"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := 5. E#L := F#P. G#K := I#P * 6. G#I := G#K - E#J. E#J := F#K. I#P := 5. F#K := 5. F#P := 6. G#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := 5. E#L := F#P. G#K := I#P * 6. G#I := G#K - E#J. E#J := F#K. I#P := 5. F#K := 5. F#P := 6.", "query": "G#I", "answer": 4, "cot": "I#P = 5 -> I#P = 5\nF#K = 5 -> F#K = 5\nE#J = F#K = 5\nG#K = I#P * 6\n = 5 * 6\n 5 * 6 = (5 × 6) mod 7 = 2\n => G#K = 2\nG#I = G#K - E#J\n = 2 - 5\n 2 - 5 = (2 - 5) mod 7 = 4\n => G#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#I": 1, "F#P": 1, "I#P": 1, "F#K": 1, "E#L": 2, "E#J": 2, "G#K": 2, "G#I": 3}, "id": "mod7_arith_d3_0096"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := 2. F#O := J#O. F#J := F#O - 2. L#M := 6 - J#O. E#O := J#O. H#J := 6. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := 2. F#O := J#O. F#J := F#O - 2. L#M := 6 - J#O. E#O := J#O. H#J := 6.", "query": "F#J", "answer": 0, "cot": "J#O = 2 -> J#O = 2\nF#O = J#O = 2\nF#J = F#O - 2\n = 2 - 2\n 2 - 2 = (2 - 2) mod 7 = 0\n => F#J = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#J": 1, "J#O": 1, "L#M": 2, "E#O": 2, "F#O": 2, "F#J": 3}, "id": "mod7_arith_d3_0097"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#O := 3. K#P := 4. F#M := J#P. G#N := 2. F#L := K#P - 5 / G#N. F#J := 2. J#P := K#P / F#O + F#J. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#O := 3. K#P := 4. F#M := J#P. G#N := 2. F#L := K#P - 5 / G#N. F#J := 2. J#P := K#P / F#O + F#J.", "query": "F#M", "answer": 1, "cot": "F#J = 2 -> F#J = 2\nF#O = 3 -> F#O = 3\nK#P = 4 -> K#P = 4\nJ#P = K#P / F#O + F#J\n = 4 / 3 + 2\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n 6 + 2 = (6 + 2) mod 7 = 1\n => J#P = 1\nF#M = J#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#J": 1, "F#O": 1, "G#N": 1, "K#P": 1, "J#P": 2, "F#L": 2, "F#M": 3}, "id": "mod7_arith_d3_0098"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := H#P * H#M. H#M := 5. H#P := 1. F#I := 3. H#L := H#M + F#I. E#O := G#I + H#L. I#N := 4. G#I := H#M * H#P. E#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := H#P * H#M. H#M := 5. H#P := 1. F#I := 3. H#L := H#M + F#I. E#O := G#I + H#L. I#N := 4. G#I := H#M * H#P.", "query": "E#O", "answer": 6, "cot": "H#P = 1 -> H#P = 1\nH#M = 5 -> H#M = 5\nF#I = 3 -> F#I = 3\nG#I = H#M * H#P\n = 5 * 1\n 5 * 1 = (5 × 1) mod 7 = 5\n => G#I = 5\nH#L = H#M + F#I\n = 5 + 3\n 5 + 3 = (5 + 3) mod 7 = 1\n => H#L = 1\nE#O = G#I + H#L\n = 5 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n => E#O = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"H#P": 1, "H#M": 1, "F#I": 1, "I#N": 1, "G#N": 2, "G#I": 2, "H#L": 2, "E#O": 3}, "id": "mod7_arith_d3_0099"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#I := 6 / L#I - E#N. K#M := 1. E#N := K#M * J#L. J#L := 3. F#P := 3. F#K := J#L - 5 + K#M. L#I := 5 + K#M + 6. J#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#I := 6 / L#I - E#N. K#M := 1. E#N := K#M * J#L. J#L := 3. F#P := 3. F#K := J#L - 5 + K#M. L#I := 5 + K#M + 6.", "query": "J#I", "answer": 1, "cot": "K#M = 1 -> K#M = 1\nJ#L = 3 -> J#L = 3\nE#N = K#M * J#L\n = 1 * 3\n 1 * 3 = (1 × 3) mod 7 = 3\n => E#N = 3\nL#I = 5 + K#M + 6\n = 5 + 1 + 6\n 5 + 1 = (5 + 1) mod 7 = 6\n 6 + 6 = (6 + 6) mod 7 = 5\n => L#I = 5\nJ#I = 6 / L#I - E#N\n = 6 / 5 - 3\n 6 / 5 = (6 × 5⁻¹) mod 7 = 4\n 4 - 3 = (4 - 3) mod 7 = 1\n => J#I = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#P": 1, "K#M": 1, "J#L": 1, "E#N": 2, "F#K": 2, "L#I": 2, "J#I": 3}, "id": "mod7_arith_d3_0100"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#M := 5. I#M := 6. H#P := L#J / H#M * 3. L#J := H#M * 3 / 3. K#L := I#M * H#M / 3. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#M := 5. I#M := 6. H#P := L#J / H#M * 3. L#J := H#M * 3 / 3. K#L := I#M * H#M / 3.", "query": "H#P", "answer": 3, "cot": "H#M = 5 -> H#M = 5\nL#J = H#M * 3 / 3\n = 5 * 3 / 3\n 5 * 3 = (5 × 3) mod 7 = 1\n 1 / 3 = (1 × 3⁻¹) mod 7 = 5\n => L#J = 5\nH#P = L#J / H#M * 3\n = 5 / 5 * 3\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n 1 * 3 = (1 × 3) mod 7 = 3\n => H#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#M": 1, "H#M": 1, "L#J": 2, "K#L": 2, "H#P": 3}, "id": "mod7_arith_d3_0101"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := 1. F#N := G#P / F#P - K#N. K#N := 6. I#P := K#N * G#P. L#N := I#P * F#N. F#P := 3. L#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := 1. F#N := G#P / F#P - K#N. K#N := 6. I#P := K#N * G#P. L#N := I#P * F#N. F#P := 3.", "query": "L#N", "answer": 1, "cot": "G#P = 1 -> G#P = 1\nF#P = 3 -> F#P = 3\nK#N = 6 -> K#N = 6\nI#P = K#N * G#P\n = 6 * 1\n 6 * 1 = (6 × 1) mod 7 = 6\n => I#P = 6\nF#N = G#P / F#P - K#N\n = 1 / 3 - 6\n 1 / 3 = (1 × 3⁻¹) mod 7 = 5\n 5 - 6 = (5 - 6) mod 7 = 6\n => F#N = 6\nL#N = I#P * F#N\n = 6 * 6\n 6 * 6 = (6 × 6) mod 7 = 1\n => L#N = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#P": 1, "F#P": 1, "K#N": 1, "I#P": 2, "F#N": 2, "L#N": 3}, "id": "mod7_arith_d3_0102"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := 5 * I#N + F#L. L#P := I#O. I#K := 3. G#K := 6. F#L := 1. I#O := 2. I#N := F#L * I#K. E#N := 5 * 3 + G#K. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := 5 * I#N + F#L. L#P := I#O. I#K := 3. G#K := 6. F#L := 1. I#O := 2. I#N := F#L * I#K. E#N := 5 * 3 + G#K.", "query": "L#I", "answer": 2, "cot": "F#L = 1 -> F#L = 1\nI#K = 3 -> I#K = 3\nI#N = F#L * I#K\n = 1 * 3\n 1 * 3 = (1 × 3) mod 7 = 3\n => I#N = 3\nL#I = 5 * I#N + F#L\n = 5 * 3 + 1\n 5 * 3 = (5 × 3) mod 7 = 1\n 1 + 1 = (1 + 1) mod 7 = 2\n => L#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#L": 1, "G#K": 1, "I#O": 1, "I#K": 1, "I#N": 2, "E#N": 2, "L#P": 2, "L#I": 3}, "id": "mod7_arith_d3_0103"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := J#N. E#P := 2 + J#N. K#I := 5. J#N := 3. J#K := 4. H#P := 3. L#M := 6 - J#K - E#P. I#N := 5 - J#N + 3. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := J#N. E#P := 2 + J#N. K#I := 5. J#N := 3. J#K := 4. H#P := 3. L#M := 6 - J#K - E#P. I#N := 5 - J#N + 3.", "query": "L#M", "answer": 4, "cot": "J#N = 3 -> J#N = 3\nJ#K = 4 -> J#K = 4\nE#P = 2 + J#N\n = 2 + 3\n 2 + 3 = (2 + 3) mod 7 = 5\n => E#P = 5\nL#M = 6 - J#K - E#P\n = 6 - 4 - 5\n 6 - 4 = (6 - 4) mod 7 = 2\n 2 - 5 = (2 - 5) mod 7 = 4\n => L#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#N": 1, "K#I": 1, "J#K": 1, "H#P": 1, "H#O": 2, "E#P": 2, "I#N": 2, "L#M": 3}, "id": "mod7_arith_d3_0104"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#L := 5. J#M := L#K * L#L. L#K := 3. J#J := L#L - L#K. H#O := J#M. H#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#L := 5. J#M := L#K * L#L. L#K := 3. J#J := L#L - L#K. H#O := J#M.", "query": "H#O", "answer": 1, "cot": "L#K = 3 -> L#K = 3\nL#L = 5 -> L#L = 5\nJ#M = L#K * L#L\n = 3 * 5\n 3 * 5 = (3 × 5) mod 7 = 1\n => J#M = 1\nH#O = J#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#K": 1, "L#L": 1, "J#J": 2, "J#M": 2, "H#O": 3}, "id": "mod7_arith_d3_0105"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := E#J. L#K := 6. I#P := K#J * F#L. F#N := 2. K#J := L#K / 3 / E#J. J#M := 5. E#J := 4. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := E#J. L#K := 6. I#P := K#J * F#L. F#N := 2. K#J := L#K / 3 / E#J. J#M := 5. E#J := 4.", "query": "I#P", "answer": 2, "cot": "L#K = 6 -> L#K = 6\nE#J = 4 -> E#J = 4\nK#J = L#K / 3 / E#J\n = 6 / 3 / 4\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n => K#J = 4\nF#L = E#J = 4\nI#P = K#J * F#L\n = 4 * 4\n 4 * 4 = (4 × 4) mod 7 = 2\n => I#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#M": 1, "F#N": 1, "L#K": 1, "E#J": 1, "K#J": 2, "F#L": 2, "I#P": 3}, "id": "mod7_arith_d3_0106"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#L := F#L - J#K / I#L. I#L := 3. F#L := 3. K#J := I#L + 6 - J#K. J#K := 2. I#J := 2 - K#J. L#M := I#L. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#L := F#L - J#K / I#L. I#L := 3. F#L := 3. K#J := I#L + 6 - J#K. J#K := 2. I#J := 2 - K#J. L#M := I#L.", "query": "I#J", "answer": 2, "cot": "I#L = 3 -> I#L = 3\nJ#K = 2 -> J#K = 2\nK#J = I#L + 6 - J#K\n = 3 + 6 - 2\n 3 + 6 = (3 + 6) mod 7 = 2\n 2 - 2 = (2 - 2) mod 7 = 0\n => K#J = 0\nI#J = 2 - K#J\n = 2 - 0\n 2 - 0 = (2 - 0) mod 7 = 2\n => I#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#L": 1, "F#L": 1, "J#K": 1, "K#J": 2, "L#M": 2, "L#L": 2, "I#J": 3}, "id": "mod7_arith_d3_0107"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := 1. K#N := H#L / H#M. H#M := G#K. G#K := 2. J#L := G#K. H#L := E#K. K#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := 1. K#N := H#L / H#M. H#M := G#K. G#K := 2. J#L := G#K. H#L := E#K.", "query": "K#N", "answer": 4, "cot": "E#K = 1 -> E#K = 1\nG#K = 2 -> G#K = 2\nH#M = G#K = 2\nH#L = E#K = 1\nK#N = H#L / H#M\n = 1 / 2\n 1 / 2 = (1 × 2⁻¹) mod 7 = 4\n => K#N = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"E#K": 1, "G#K": 1, "H#M": 2, "J#L": 2, "H#L": 2, "K#N": 3}, "id": "mod7_arith_d3_0108"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := I#O * H#P. J#I := I#O. H#P := 4. G#M := I#O / H#P. I#P := J#I / 5. I#O := 4. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := I#O * H#P. J#I := I#O. H#P := 4. G#M := I#O / H#P. I#P := J#I / 5. I#O := 4.", "query": "I#P", "answer": 5, "cot": "I#O = 4 -> I#O = 4\nJ#I = I#O = 4\nI#P = J#I / 5\n = 4 / 5\n 4 / 5 = (4 × 5⁻¹) mod 7 = 5\n => I#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#O": 1, "H#P": 1, "G#M": 2, "J#I": 2, "H#K": 2, "I#P": 3}, "id": "mod7_arith_d3_0109"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := 6. F#L := 5. F#P := K#M * F#L. H#I := F#L / F#K. F#K := G#I. G#I := 2. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := 6. F#L := 5. F#P := K#M * F#L. H#I := F#L / F#K. F#K := G#I. G#I := 2.", "query": "H#I", "answer": 6, "cot": "F#L = 5 -> F#L = 5\nG#I = 2 -> G#I = 2\nF#K = G#I = 2\nH#I = F#L / F#K\n = 5 / 2\n 5 / 2 = (5 × 2⁻¹) mod 7 = 6\n => H#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#L": 1, "K#M": 1, "G#I": 1, "F#K": 2, "F#P": 2, "H#I": 3}, "id": "mod7_arith_d3_0110"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#L := L#O. E#O := E#L. L#N := 1. H#K := L#N * E#L. E#L := 1. L#O := E#L. H#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#L := L#O. E#O := E#L. L#N := 1. H#K := L#N * E#L. E#L := 1. L#O := E#L.", "query": "H#L", "answer": 1, "cot": "E#L = 1 -> E#L = 1\nL#O = E#L = 1\nH#L = L#O = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#L": 1, "L#N": 1, "L#O": 2, "H#K": 2, "E#O": 2, "H#L": 3}, "id": "mod7_arith_d3_0111"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := 6. I#O := F#O - K#I. K#I := F#O + E#P. I#K := L#L. E#P := 2. L#L := 2. F#O := 5. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := 6. I#O := F#O - K#I. K#I := F#O + E#P. I#K := L#L. E#P := 2. L#L := 2. F#O := 5.", "query": "I#O", "answer": 5, "cot": "E#P = 2 -> E#P = 2\nF#O = 5 -> F#O = 5\nK#I = F#O + E#P\n = 5 + 2\n 5 + 2 = (5 + 2) mod 7 = 0\n => K#I = 0\nI#O = F#O - K#I\n = 5 - 0\n 5 - 0 = (5 - 0) mod 7 = 5\n => I#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#P": 1, "F#O": 1, "L#L": 1, "E#N": 1, "I#K": 2, "K#I": 2, "I#O": 3}, "id": "mod7_arith_d3_0112"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := 3. F#J := L#P - E#I. E#I := 1. I#J := F#J. F#N := 3. G#O := F#N. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := 3. F#J := L#P - E#I. E#I := 1. I#J := F#J. F#N := 3. G#O := F#N.", "query": "I#J", "answer": 2, "cot": "E#I = 1 -> E#I = 1\nL#P = 3 -> L#P = 3\nF#J = L#P - E#I\n = 3 - 1\n 3 - 1 = (3 - 1) mod 7 = 2\n => F#J = 2\nI#J = F#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#N": 1, "E#I": 1, "L#P": 1, "G#O": 2, "F#J": 2, "I#J": 3}, "id": "mod7_arith_d3_0113"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 3. G#I := E#N - L#J. E#N := 5. L#M := J#N * E#N. L#P := L#J - J#N + E#I. E#I := 1. I#I := E#I + 4 * G#I. L#J := 5. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 3. G#I := E#N - L#J. E#N := 5. L#M := J#N * E#N. L#P := L#J - J#N + E#I. E#I := 1. I#I := E#I + 4 * G#I. L#J := 5.", "query": "I#I", "answer": 0, "cot": "L#J = 5 -> L#J = 5\nE#N = 5 -> E#N = 5\nE#I = 1 -> E#I = 1\nG#I = E#N - L#J\n = 5 - 5\n 5 - 5 = (5 - 5) mod 7 = 0\n => G#I = 0\nI#I = E#I + 4 * G#I\n = 1 + 4 * 0\n 1 + 4 = (1 + 4) mod 7 = 5\n 5 * 0 = (5 × 0) mod 7 = 0\n => I#I = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#J": 1, "J#N": 1, "E#N": 1, "E#I": 1, "L#P": 2, "G#I": 2, "L#M": 2, "I#I": 3}, "id": "mod7_arith_d3_0114"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#P := 2. G#J := F#L. J#I := 6 * K#J. H#K := 5. F#L := H#K. I#M := H#K * I#L. I#L := 2. K#J := 3. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#P := 2. G#J := F#L. J#I := 6 * K#J. H#K := 5. F#L := H#K. I#M := H#K * I#L. I#L := 2. K#J := 3.", "query": "G#J", "answer": 5, "cot": "H#K = 5 -> H#K = 5\nF#L = H#K = 5\nG#J = F#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#J": 1, "H#K": 1, "K#P": 1, "I#L": 1, "F#L": 2, "J#I": 2, "I#M": 2, "G#J": 3}, "id": "mod7_arith_d3_0115"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#I := 1. L#I := 4 - I#M. H#I := 2. I#M := F#I. E#J := H#I. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#I := 1. L#I := 4 - I#M. H#I := 2. I#M := F#I. E#J := H#I.", "query": "L#I", "answer": 3, "cot": "F#I = 1 -> F#I = 1\nI#M = F#I = 1\nL#I = 4 - I#M\n = 4 - 1\n 4 - 1 = (4 - 1) mod 7 = 3\n => L#I = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#I": 1, "H#I": 1, "E#J": 2, "I#M": 2, "L#I": 3}, "id": "mod7_arith_d3_0116"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := 2. K#J := F#P + F#M. I#M := F#M / F#P / K#J. F#I := F#P - F#M. F#M := 3. I#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := 2. K#J := F#P + F#M. I#M := F#M / F#P / K#J. F#I := F#P - F#M. F#M := 3.", "query": "I#M", "answer": 1, "cot": "F#P = 2 -> F#P = 2\nF#M = 3 -> F#M = 3\nK#J = F#P + F#M\n = 2 + 3\n 2 + 3 = (2 + 3) mod 7 = 5\n => K#J = 5\nI#M = F#M / F#P / K#J\n = 3 / 2 / 5\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n => I#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#P": 1, "F#M": 1, "F#I": 2, "K#J": 2, "I#M": 3}, "id": "mod7_arith_d3_0117"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := G#I / J#I. K#K := 4 + G#I. H#O := J#I. I#L := J#O. J#I := 6. G#I := 3. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := G#I / J#I. K#K := 4 + G#I. H#O := J#I. I#L := J#O. J#I := 6. G#I := 3.", "query": "I#L", "answer": 4, "cot": "G#I = 3 -> G#I = 3\nJ#I = 6 -> J#I = 6\nJ#O = G#I / J#I\n = 3 / 6\n 3 / 6 = (3 × 6⁻¹) mod 7 = 4\n => J#O = 4\nI#L = J#O = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#I": 1, "J#I": 1, "K#K": 2, "J#O": 2, "H#O": 2, "I#L": 3}, "id": "mod7_arith_d3_0118"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#J := 3. H#K := 4. G#J := 5. F#J := G#J / H#K - J#J. H#I := L#P / J#J. L#P := H#K - 4. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#J := 3. H#K := 4. G#J := 5. F#J := G#J / H#K - J#J. H#I := L#P / J#J. L#P := H#K - 4.", "query": "H#I", "answer": 0, "cot": "J#J = 3 -> J#J = 3\nH#K = 4 -> H#K = 4\nL#P = H#K - 4\n = 4 - 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => L#P = 0\nH#I = L#P / J#J\n = 0 / 3\n 0 / 3 = (0 × 3⁻¹) mod 7 = 0\n => H#I = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#J": 1, "J#J": 1, "H#K": 1, "F#J": 2, "L#P": 2, "H#I": 3}, "id": "mod7_arith_d3_0119"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := 5. I#L := G#P. K#O := L#K * 4. L#K := 2. K#L := E#L - I#L. J#M := 4. E#L := 3. K#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := 5. I#L := G#P. K#O := L#K * 4. L#K := 2. K#L := E#L - I#L. J#M := 4. E#L := 3.", "query": "K#L", "answer": 5, "cot": "G#P = 5 -> G#P = 5\nE#L = 3 -> E#L = 3\nI#L = G#P = 5\nK#L = E#L - I#L\n = 3 - 5\n 3 - 5 = (3 - 5) mod 7 = 5\n => K#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#K": 1, "G#P": 1, "E#L": 1, "J#M": 1, "K#O": 2, "I#L": 2, "K#L": 3}, "id": "mod7_arith_d3_0120"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#I := 4. F#P := 2. G#J := 1. J#O := E#P. E#P := G#J. F#M := G#J. J#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#I := 4. F#P := 2. G#J := 1. J#O := E#P. E#P := G#J. F#M := G#J.", "query": "J#O", "answer": 1, "cot": "G#J = 1 -> G#J = 1\nE#P = G#J = 1\nJ#O = E#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#I": 1, "F#P": 1, "G#J": 1, "F#M": 2, "E#P": 2, "J#O": 3}, "id": "mod7_arith_d3_0121"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := 1. G#O := L#P. F#P := 4. K#I := L#P + F#P. E#J := F#L. F#L := 4. I#O := E#J * 5 * K#I. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := 1. G#O := L#P. F#P := 4. K#I := L#P + F#P. E#J := F#L. F#L := 4. I#O := E#J * 5 * K#I.", "query": "I#O", "answer": 2, "cot": "F#L = 4 -> F#L = 4\nL#P = 1 -> L#P = 1\nF#P = 4 -> F#P = 4\nK#I = L#P + F#P\n = 1 + 4\n 1 + 4 = (1 + 4) mod 7 = 5\n => K#I = 5\nE#J = F#L = 4\nI#O = E#J * 5 * K#I\n = 4 * 5 * 5\n 4 * 5 = (4 × 5) mod 7 = 6\n 6 * 5 = (6 × 5) mod 7 = 2\n => I#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#L": 1, "L#P": 1, "F#P": 1, "K#I": 2, "E#J": 2, "G#O": 2, "I#O": 3}, "id": "mod7_arith_d3_0122"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#O := 6. K#I := K#M / 5 + I#J. K#N := 4. H#L := I#J + L#J. G#N := F#O. K#M := 4. I#J := 2. L#J := F#O. H#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#O := 6. K#I := K#M / 5 + I#J. K#N := 4. H#L := I#J + L#J. G#N := F#O. K#M := 4. I#J := 2. L#J := F#O.", "query": "H#L", "answer": 1, "cot": "I#J = 2 -> I#J = 2\nF#O = 6 -> F#O = 6\nL#J = F#O = 6\nH#L = I#J + L#J\n = 2 + 6\n 2 + 6 = (2 + 6) mod 7 = 1\n => H#L = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#J": 1, "F#O": 1, "K#M": 1, "K#N": 1, "K#I": 2, "G#N": 2, "L#J": 2, "H#L": 3}, "id": "mod7_arith_d3_0123"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := K#L. H#N := 2. E#O := 2. I#K := E#O * H#O - 2. L#M := H#N + E#O. H#O := 1. K#L := E#O / H#N + H#O. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := K#L. H#N := 2. E#O := 2. I#K := E#O * H#O - 2. L#M := H#N + E#O. H#O := 1. K#L := E#O / H#N + H#O.", "query": "F#L", "answer": 2, "cot": "H#O = 1 -> H#O = 1\nH#N = 2 -> H#N = 2\nE#O = 2 -> E#O = 2\nK#L = E#O / H#N + H#O\n = 2 / 2 + 1\n 2 / 2 = (2 × 2⁻¹) mod 7 = 1\n 1 + 1 = (1 + 1) mod 7 = 2\n => K#L = 2\nF#L = K#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#O": 1, "H#N": 1, "E#O": 1, "I#K": 2, "K#L": 2, "L#M": 2, "F#L": 3}, "id": "mod7_arith_d3_0124"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#K := E#L + 6. H#N := 4. H#J := I#L + H#N. E#M := I#L - H#K. I#L := 4. E#L := 5. K#P := H#J. H#K := 1. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#K := E#L + 6. H#N := 4. H#J := I#L + H#N. E#M := I#L - H#K. I#L := 4. E#L := 5. K#P := H#J. H#K := 1.", "query": "K#P", "answer": 1, "cot": "I#L = 4 -> I#L = 4\nH#N = 4 -> H#N = 4\nH#J = I#L + H#N\n = 4 + 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => H#J = 1\nK#P = H#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#L": 1, "H#N": 1, "E#L": 1, "H#K": 1, "J#K": 2, "H#J": 2, "E#M": 2, "K#P": 3}, "id": "mod7_arith_d3_0125"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#O := 1. H#J := L#M * I#O. H#K := K#L + I#O * H#J. L#M := 3. K#L := 4 / I#O. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#O := 1. H#J := L#M * I#O. H#K := K#L + I#O * H#J. L#M := 3. K#L := 4 / I#O.", "query": "H#K", "answer": 1, "cot": "L#M = 3 -> L#M = 3\nI#O = 1 -> I#O = 1\nH#J = L#M * I#O\n = 3 * 1\n 3 * 1 = (3 × 1) mod 7 = 3\n => H#J = 3\nK#L = 4 / I#O\n = 4 / 1\n 4 / 1 = (4 × 1⁻¹) mod 7 = 4\n => K#L = 4\nH#K = K#L + I#O * H#J\n = 4 + 1 * 3\n 4 + 1 = (4 + 1) mod 7 = 5\n 5 * 3 = (5 × 3) mod 7 = 1\n => H#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#M": 1, "I#O": 1, "H#J": 2, "K#L": 2, "H#K": 3}, "id": "mod7_arith_d3_0126"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#I := J#J - F#N * F#M. F#M := 6. E#K := 2 * 4 * F#M. F#P := J#J / F#M. F#N := J#J * F#M. J#J := 3. F#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#I := J#J - F#N * F#M. F#M := 6. E#K := 2 * 4 * F#M. F#P := J#J / F#M. F#N := J#J * F#M. J#J := 3.", "query": "F#I", "answer": 1, "cot": "F#M = 6 -> F#M = 6\nJ#J = 3 -> J#J = 3\nF#N = J#J * F#M\n = 3 * 6\n 3 * 6 = (3 × 6) mod 7 = 4\n => F#N = 4\nF#I = J#J - F#N * F#M\n = 3 - 4 * 6\n 3 - 4 = (3 - 4) mod 7 = 6\n 6 * 6 = (6 × 6) mod 7 = 1\n => F#I = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#M": 1, "J#J": 1, "F#P": 2, "F#N": 2, "E#K": 2, "F#I": 3}, "id": "mod7_arith_d3_0127"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := 4. J#P := H#N + F#P. H#N := 4. F#K := F#P - H#N. L#L := H#N * F#K. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := 4. J#P := H#N + F#P. H#N := 4. F#K := F#P - H#N. L#L := H#N * F#K.", "query": "L#L", "answer": 0, "cot": "F#P = 4 -> F#P = 4\nH#N = 4 -> H#N = 4\nF#K = F#P - H#N\n = 4 - 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => F#K = 0\nL#L = H#N * F#K\n = 4 * 0\n 4 * 0 = (4 × 0) mod 7 = 0\n => L#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#P": 1, "H#N": 1, "J#P": 2, "F#K": 2, "L#L": 3}, "id": "mod7_arith_d3_0128"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := 5. J#J := 4. F#P := 6. J#M := 2. H#P := L#I * J#M. H#M := 5 + J#J * 6. L#O := J#J * L#I. K#N := L#O. K#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := 5. J#J := 4. F#P := 6. J#M := 2. H#P := L#I * J#M. H#M := 5 + J#J * 6. L#O := J#J * L#I. K#N := L#O.", "query": "K#N", "answer": 6, "cot": "L#I = 5 -> L#I = 5\nJ#J = 4 -> J#J = 4\nL#O = J#J * L#I\n = 4 * 5\n 4 * 5 = (4 × 5) mod 7 = 6\n => L#O = 6\nK#N = L#O = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#M": 1, "L#I": 1, "J#J": 1, "F#P": 1, "L#O": 2, "H#M": 2, "H#P": 2, "K#N": 3}, "id": "mod7_arith_d3_0129"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := 2. G#P := I#J + G#N. G#N := 6. K#I := J#M. J#M := 4 / I#J - 5. L#I := G#N + I#J + 4. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := 2. G#P := I#J + G#N. G#N := 6. K#I := J#M. J#M := 4 / I#J - 5. L#I := G#N + I#J + 4.", "query": "K#I", "answer": 4, "cot": "I#J = 2 -> I#J = 2\nJ#M = 4 / I#J - 5\n = 4 / 2 - 5\n 4 / 2 = (4 × 2⁻¹) mod 7 = 2\n 2 - 5 = (2 - 5) mod 7 = 4\n => J#M = 4\nK#I = J#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#N": 1, "I#J": 1, "G#P": 2, "J#M": 2, "L#I": 2, "K#I": 3}, "id": "mod7_arith_d3_0130"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#I := K#I - J#K. K#I := H#M / 3. H#M := 6. J#K := G#N / 4 / H#M. G#N := 5. F#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#I := K#I - J#K. K#I := H#M / 3. H#M := 6. J#K := G#N / 4 / H#M. G#N := 5.", "query": "F#I", "answer": 5, "cot": "H#M = 6 -> H#M = 6\nG#N = 5 -> G#N = 5\nK#I = H#M / 3\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => K#I = 2\nJ#K = G#N / 4 / H#M\n = 5 / 4 / 6\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n 3 / 6 = (3 × 6⁻¹) mod 7 = 4\n => J#K = 4\nF#I = K#I - J#K\n = 2 - 4\n 2 - 4 = (2 - 4) mod 7 = 5\n => F#I = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#M": 1, "G#N": 1, "K#I": 2, "J#K": 2, "F#I": 3}, "id": "mod7_arith_d3_0131"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := H#N. H#M := 1. L#O := 4. L#J := 2. F#I := 3. H#N := 5 / H#M. H#K := 5 / L#J. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := H#N. H#M := 1. L#O := 4. L#J := 2. F#I := 3. H#N := 5 / H#M. H#K := 5 / L#J.", "query": "I#J", "answer": 5, "cot": "H#M = 1 -> H#M = 1\nH#N = 5 / H#M\n = 5 / 1\n 5 / 1 = (5 × 1⁻¹) mod 7 = 5\n => H#N = 5\nI#J = H#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#I": 1, "L#O": 1, "L#J": 1, "H#M": 1, "H#K": 2, "H#N": 2, "I#J": 3}, "id": "mod7_arith_d3_0132"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#P := E#I - G#N - E#K. K#O := E#I / E#K * G#N. E#I := 4. E#K := 3. I#I := J#P + E#J / E#K. E#J := G#N + E#I. G#N := 3. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#P := E#I - G#N - E#K. K#O := E#I / E#K * G#N. E#I := 4. E#K := 3. I#I := J#P + E#J / E#K. E#J := G#N + E#I. G#N := 3.", "query": "I#I", "answer": 4, "cot": "E#K = 3 -> E#K = 3\nE#I = 4 -> E#I = 4\nG#N = 3 -> G#N = 3\nE#J = G#N + E#I\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => E#J = 0\nJ#P = E#I - G#N - E#K\n = 4 - 3 - 3\n 4 - 3 = (4 - 3) mod 7 = 1\n 1 - 3 = (1 - 3) mod 7 = 5\n => J#P = 5\nI#I = J#P + E#J / E#K\n = 5 + 0 / 3\n 5 + 0 = (5 + 0) mod 7 = 5\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n => I#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#K": 1, "E#I": 1, "G#N": 1, "E#J": 2, "J#P": 2, "K#O": 2, "I#I": 3}, "id": "mod7_arith_d3_0133"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#M := I#O * E#K. I#O := 6. E#K := 6. H#I := I#O / G#L. G#L := 3. K#M := G#L. H#J := H#M. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#M := I#O * E#K. I#O := 6. E#K := 6. H#I := I#O / G#L. G#L := 3. K#M := G#L. H#J := H#M.", "query": "H#J", "answer": 1, "cot": "E#K = 6 -> E#K = 6\nI#O = 6 -> I#O = 6\nH#M = I#O * E#K\n = 6 * 6\n 6 * 6 = (6 × 6) mod 7 = 1\n => H#M = 1\nH#J = H#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#K": 1, "I#O": 1, "G#L": 1, "H#M": 2, "K#M": 2, "H#I": 2, "H#J": 3}, "id": "mod7_arith_d3_0134"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#M := 2. F#M := 4. E#O := 6. K#P := 6. I#L := F#M / E#O. H#O := J#M * F#M. H#M := I#L * E#O. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#M := 2. F#M := 4. E#O := 6. K#P := 6. I#L := F#M / E#O. H#O := J#M * F#M. H#M := I#L * E#O.", "query": "H#M", "answer": 4, "cot": "F#M = 4 -> F#M = 4\nE#O = 6 -> E#O = 6\nI#L = F#M / E#O\n = 4 / 6\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n => I#L = 3\nH#M = I#L * E#O\n = 3 * 6\n 3 * 6 = (3 × 6) mod 7 = 4\n => H#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#M": 1, "K#P": 1, "F#M": 1, "E#O": 1, "H#O": 2, "I#L": 2, "H#M": 3}, "id": "mod7_arith_d3_0135"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := L#L + F#I + 4. L#L := 1. L#O := 6. F#L := F#I. K#O := 5. L#K := K#O / L#J. F#I := 3. K#J := L#L. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := L#L + F#I + 4. L#L := 1. L#O := 6. F#L := F#I. K#O := 5. L#K := K#O / L#J. F#I := 3. K#J := L#L.", "query": "L#K", "answer": 5, "cot": "L#L = 1 -> L#L = 1\nF#I = 3 -> F#I = 3\nK#O = 5 -> K#O = 5\nL#J = L#L + F#I + 4\n = 1 + 3 + 4\n 1 + 3 = (1 + 3) mod 7 = 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => L#J = 1\nL#K = K#O / L#J\n = 5 / 1\n 5 / 1 = (5 × 1⁻¹) mod 7 = 5\n => L#K = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#L": 1, "L#O": 1, "F#I": 1, "K#O": 1, "F#L": 2, "L#J": 2, "K#J": 2, "L#K": 3}, "id": "mod7_arith_d3_0136"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#L := 6 - H#I. I#K := 2. H#N := I#K + G#P - I#L. I#L := 1. H#I := I#L / G#P - 3. G#P := 6. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#L := 6 - H#I. I#K := 2. H#N := I#K + G#P - I#L. I#L := 1. H#I := I#L / G#P - 3. G#P := 6.", "query": "J#L", "answer": 3, "cot": "G#P = 6 -> G#P = 6\nI#L = 1 -> I#L = 1\nH#I = I#L / G#P - 3\n = 1 / 6 - 3\n 1 / 6 = (1 × 6⁻¹) mod 7 = 6\n 6 - 3 = (6 - 3) mod 7 = 3\n => H#I = 3\nJ#L = 6 - H#I\n = 6 - 3\n 6 - 3 = (6 - 3) mod 7 = 3\n => J#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#P": 1, "I#L": 1, "I#K": 1, "H#I": 2, "H#N": 2, "J#L": 3}, "id": "mod7_arith_d3_0137"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#M := 6 - 4 * H#M. H#M := 1. J#O := H#L - H#M. H#L := 3. J#K := L#K + J#O. L#K := H#L + H#M. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#M := 6 - 4 * H#M. H#M := 1. J#O := H#L - H#M. H#L := 3. J#K := L#K + J#O. L#K := H#L + H#M.", "query": "J#K", "answer": 6, "cot": "H#M = 1 -> H#M = 1\nH#L = 3 -> H#L = 3\nJ#O = H#L - H#M\n = 3 - 1\n 3 - 1 = (3 - 1) mod 7 = 2\n => J#O = 2\nL#K = H#L + H#M\n = 3 + 1\n 3 + 1 = (3 + 1) mod 7 = 4\n => L#K = 4\nJ#K = L#K + J#O\n = 4 + 2\n 4 + 2 = (4 + 2) mod 7 = 6\n => J#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#M": 1, "H#L": 1, "G#M": 2, "J#O": 2, "L#K": 2, "J#K": 3}, "id": "mod7_arith_d3_0138"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := L#N. I#K := 3. L#O := I#K / G#J. G#J := 1. L#N := G#J / I#K. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := L#N. I#K := 3. L#O := I#K / G#J. G#J := 1. L#N := G#J / I#K.", "query": "H#J", "answer": 5, "cot": "G#J = 1 -> G#J = 1\nI#K = 3 -> I#K = 3\nL#N = G#J / I#K\n = 1 / 3\n 1 / 3 = (1 × 3⁻¹) mod 7 = 5\n => L#N = 5\nH#J = L#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#J": 1, "I#K": 1, "L#O": 2, "L#N": 2, "H#J": 3}, "id": "mod7_arith_d3_0139"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := I#M. L#J := E#P - I#M. I#M := 4. J#P := I#M - E#P. G#L := L#J * E#P. E#P := 5. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := I#M. L#J := E#P - I#M. I#M := 4. J#P := I#M - E#P. G#L := L#J * E#P. E#P := 5.", "query": "G#L", "answer": 5, "cot": "E#P = 5 -> E#P = 5\nI#M = 4 -> I#M = 4\nL#J = E#P - I#M\n = 5 - 4\n 5 - 4 = (5 - 4) mod 7 = 1\n => L#J = 1\nG#L = L#J * E#P\n = 1 * 5\n 1 * 5 = (1 × 5) mod 7 = 5\n => G#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#P": 1, "I#M": 1, "L#J": 2, "J#P": 2, "L#I": 2, "G#L": 3}, "id": "mod7_arith_d3_0140"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := 4. K#P := 1. L#K := K#P / I#M. I#J := 6. I#M := J#I - 5. J#I := 1. K#L := I#P + J#I * 4. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := 4. K#P := 1. L#K := K#P / I#M. I#J := 6. I#M := J#I - 5. J#I := 1. K#L := I#P + J#I * 4.", "query": "L#K", "answer": 5, "cot": "K#P = 1 -> K#P = 1\nJ#I = 1 -> J#I = 1\nI#M = J#I - 5\n = 1 - 5\n 1 - 5 = (1 - 5) mod 7 = 3\n => I#M = 3\nL#K = K#P / I#M\n = 1 / 3\n 1 / 3 = (1 × 3⁻¹) mod 7 = 5\n => L#K = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#P": 1, "J#I": 1, "I#P": 1, "I#J": 1, "I#M": 2, "K#L": 2, "L#K": 3}, "id": "mod7_arith_d3_0141"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#O := K#P. L#K := 3. I#K := 1. F#P := L#K * I#K. L#J := 1. H#M := 4. K#P := 2 - I#K. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#O := K#P. L#K := 3. I#K := 1. F#P := L#K * I#K. L#J := 1. H#M := 4. K#P := 2 - I#K.", "query": "I#O", "answer": 1, "cot": "I#K = 1 -> I#K = 1\nK#P = 2 - I#K\n = 2 - 1\n 2 - 1 = (2 - 1) mod 7 = 1\n => K#P = 1\nI#O = K#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"L#K": 1, "I#K": 1, "L#J": 1, "H#M": 1, "F#P": 2, "K#P": 2, "I#O": 3}, "id": "mod7_arith_d3_0142"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := 5 / L#J. I#L := L#J - F#O. E#P := F#O + L#J. F#O := 1. L#J := 3. L#N := I#L. L#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := 5 / L#J. I#L := L#J - F#O. E#P := F#O + L#J. F#O := 1. L#J := 3. L#N := I#L.", "query": "L#N", "answer": 2, "cot": "L#J = 3 -> L#J = 3\nF#O = 1 -> F#O = 1\nI#L = L#J - F#O\n = 3 - 1\n 3 - 1 = (3 - 1) mod 7 = 2\n => I#L = 2\nL#N = I#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#J": 1, "F#O": 1, "I#L": 2, "G#N": 2, "E#P": 2, "L#N": 3}, "id": "mod7_arith_d3_0143"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#N := K#L * H#O. H#O := 5. G#M := K#I - 4. F#J := H#O - E#M. K#I := K#L - H#O. E#M := 6. K#L := 6. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#N := K#L * H#O. H#O := 5. G#M := K#I - 4. F#J := H#O - E#M. K#I := K#L - H#O. E#M := 6. K#L := 6.", "query": "G#M", "answer": 4, "cot": "K#L = 6 -> K#L = 6\nH#O = 5 -> H#O = 5\nK#I = K#L - H#O\n = 6 - 5\n 6 - 5 = (6 - 5) mod 7 = 1\n => K#I = 1\nG#M = K#I - 4\n = 1 - 4\n 1 - 4 = (1 - 4) mod 7 = 4\n => G#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#M": 1, "K#L": 1, "H#O": 1, "K#N": 2, "F#J": 2, "K#I": 2, "G#M": 3}, "id": "mod7_arith_d3_0144"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#I := H#K / I#L. I#J := 3. H#K := 2 - H#P. J#O := 4. J#K := 2. L#K := I#J / J#K * 3. H#P := 3. I#L := I#J + J#K. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#I := H#K / I#L. I#J := 3. H#K := 2 - H#P. J#O := 4. J#K := 2. L#K := I#J / J#K * 3. H#P := 3. I#L := I#J + J#K.", "query": "H#I", "answer": 4, "cot": "H#P = 3 -> H#P = 3\nJ#K = 2 -> J#K = 2\nI#J = 3 -> I#J = 3\nI#L = I#J + J#K\n = 3 + 2\n 3 + 2 = (3 + 2) mod 7 = 5\n => I#L = 5\nH#K = 2 - H#P\n = 2 - 3\n 2 - 3 = (2 - 3) mod 7 = 6\n => H#K = 6\nH#I = H#K / I#L\n = 6 / 5\n 6 / 5 = (6 × 5⁻¹) mod 7 = 4\n => H#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#O": 1, "H#P": 1, "J#K": 1, "I#J": 1, "I#L": 2, "L#K": 2, "H#K": 2, "H#I": 3}, "id": "mod7_arith_d3_0145"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#I := F#L - K#M. J#P := 3. F#L := 5. G#K := F#L * K#M. H#I := 3 - G#K. E#N := F#L. K#M := 5. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#I := F#L - K#M. J#P := 3. F#L := 5. G#K := F#L * K#M. H#I := 3 - G#K. E#N := F#L. K#M := 5.", "query": "H#I", "answer": 6, "cot": "F#L = 5 -> F#L = 5\nK#M = 5 -> K#M = 5\nG#K = F#L * K#M\n = 5 * 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => G#K = 4\nH#I = 3 - G#K\n = 3 - 4\n 3 - 4 = (3 - 4) mod 7 = 6\n => H#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#P": 1, "F#L": 1, "K#M": 1, "G#K": 2, "F#I": 2, "E#N": 2, "H#I": 3}, "id": "mod7_arith_d3_0146"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := 5. I#M := G#O. F#N := 5. L#K := F#N / 3 + G#N. G#N := G#O * F#N. G#M := F#N + G#O. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := 5. I#M := G#O. F#N := 5. L#K := F#N / 3 + G#N. G#N := G#O * F#N. G#M := F#N + G#O.", "query": "L#K", "answer": 1, "cot": "G#O = 5 -> G#O = 5\nF#N = 5 -> F#N = 5\nG#N = G#O * F#N\n = 5 * 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => G#N = 4\nL#K = F#N / 3 + G#N\n = 5 / 3 + 4\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => L#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#O": 1, "F#N": 1, "I#M": 2, "G#N": 2, "G#M": 2, "L#K": 3}, "id": "mod7_arith_d3_0147"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := 2. F#O := 1. G#L := 1. H#O := J#K * G#L / E#L. G#K := G#L. H#J := 5 * G#K. J#K := 2. J#I := F#O. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := 2. F#O := 1. G#L := 1. H#O := J#K * G#L / E#L. G#K := G#L. H#J := 5 * G#K. J#K := 2. J#I := F#O.", "query": "H#J", "answer": 5, "cot": "G#L = 1 -> G#L = 1\nG#K = G#L = 1\nH#J = 5 * G#K\n = 5 * 1\n 5 * 1 = (5 × 1) mod 7 = 5\n => H#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#L": 1, "F#O": 1, "J#K": 1, "G#L": 1, "H#O": 2, "J#I": 2, "G#K": 2, "H#J": 3}, "id": "mod7_arith_d3_0148"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#N := H#P - H#O + G#J. I#M := G#J - H#O. H#O := 5. H#P := G#J / H#O. G#J := 6. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#N := H#P - H#O + G#J. I#M := G#J - H#O. H#O := 5. H#P := G#J / H#O. G#J := 6.", "query": "I#N", "answer": 5, "cot": "H#O = 5 -> H#O = 5\nG#J = 6 -> G#J = 6\nH#P = G#J / H#O\n = 6 / 5\n 6 / 5 = (6 × 5⁻¹) mod 7 = 4\n => H#P = 4\nI#N = H#P - H#O + G#J\n = 4 - 5 + 6\n 4 - 5 = (4 - 5) mod 7 = 6\n 6 + 6 = (6 + 6) mod 7 = 5\n => I#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#O": 1, "G#J": 1, "I#M": 2, "H#P": 2, "I#N": 3}, "id": "mod7_arith_d3_0149"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := G#J * 4 + G#N. G#J := 1. E#N := G#N / G#J. G#N := 6. J#M := 4 - E#N. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := G#J * 4 + G#N. G#J := 1. E#N := G#N / G#J. G#N := 6. J#M := 4 - E#N.", "query": "J#M", "answer": 5, "cot": "G#J = 1 -> G#J = 1\nG#N = 6 -> G#N = 6\nE#N = G#N / G#J\n = 6 / 1\n 6 / 1 = (6 × 1⁻¹) mod 7 = 6\n => E#N = 6\nJ#M = 4 - E#N\n = 4 - 6\n 4 - 6 = (4 - 6) mod 7 = 5\n => J#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#J": 1, "G#N": 1, "K#L": 2, "E#N": 2, "J#M": 3}, "id": "mod7_arith_d3_0150"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#K := K#L. G#J := 2. K#M := 1. G#L := K#M. K#L := 2. K#K := K#M - K#L - G#L. K#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#K := K#L. G#J := 2. K#M := 1. G#L := K#M. K#L := 2. K#K := K#M - K#L - G#L.", "query": "K#K", "answer": 5, "cot": "K#M = 1 -> K#M = 1\nK#L = 2 -> K#L = 2\nG#L = K#M = 1\nK#K = K#M - K#L - G#L\n = 1 - 2 - 1\n 1 - 2 = (1 - 2) mod 7 = 6\n 6 - 1 = (6 - 1) mod 7 = 5\n => K#K = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#M": 1, "K#L": 1, "G#J": 1, "G#L": 2, "G#K": 2, "K#K": 3}, "id": "mod7_arith_d3_0151"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := 5. G#P := 5. L#J := F#L / G#P. H#L := K#K / F#M * 4. F#M := 6. E#O := H#L. K#K := 4. E#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := 5. G#P := 5. L#J := F#L / G#P. H#L := K#K / F#M * 4. F#M := 6. E#O := H#L. K#K := 4.", "query": "E#O", "answer": 5, "cot": "F#M = 6 -> F#M = 6\nK#K = 4 -> K#K = 4\nH#L = K#K / F#M * 4\n = 4 / 6 * 4\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n 3 * 4 = (3 × 4) mod 7 = 5\n => H#L = 5\nE#O = H#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#M": 1, "F#L": 1, "K#K": 1, "G#P": 1, "H#L": 2, "L#J": 2, "E#O": 3}, "id": "mod7_arith_d3_0152"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#L := 3. F#N := G#P. F#M := G#L. G#J := 4. J#L := 1. G#P := J#L + 4 + G#L. K#I := 5. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#L := 3. F#N := G#P. F#M := G#L. G#J := 4. J#L := 1. G#P := J#L + 4 + G#L. K#I := 5.", "query": "F#N", "answer": 1, "cot": "G#L = 3 -> G#L = 3\nJ#L = 1 -> J#L = 1\nG#P = J#L + 4 + G#L\n = 1 + 4 + 3\n 1 + 4 = (1 + 4) mod 7 = 5\n 5 + 3 = (5 + 3) mod 7 = 1\n => G#P = 1\nF#N = G#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#J": 1, "G#L": 1, "K#I": 1, "J#L": 1, "F#M": 2, "G#P": 2, "F#N": 3}, "id": "mod7_arith_d3_0153"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#M := 2. J#P := I#N + F#M. H#N := F#M. E#I := H#N + J#P. I#N := 5. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#M := 2. J#P := I#N + F#M. H#N := F#M. E#I := H#N + J#P. I#N := 5.", "query": "E#I", "answer": 2, "cot": "I#N = 5 -> I#N = 5\nF#M = 2 -> F#M = 2\nH#N = F#M = 2\nJ#P = I#N + F#M\n = 5 + 2\n 5 + 2 = (5 + 2) mod 7 = 0\n => J#P = 0\nE#I = H#N + J#P\n = 2 + 0\n 2 + 0 = (2 + 0) mod 7 = 2\n => E#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#N": 1, "F#M": 1, "H#N": 2, "J#P": 2, "E#I": 3}, "id": "mod7_arith_d3_0154"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := H#M. H#I := G#N. F#N := 6. L#K := 1. H#M := L#K * F#N. G#N := 3. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := H#M. H#I := G#N. F#N := 6. L#K := 1. H#M := L#K * F#N. G#N := 3.", "query": "H#J", "answer": 6, "cot": "F#N = 6 -> F#N = 6\nL#K = 1 -> L#K = 1\nH#M = L#K * F#N\n = 1 * 6\n 1 * 6 = (1 × 6) mod 7 = 6\n => H#M = 6\nH#J = H#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#N": 1, "G#N": 1, "L#K": 1, "H#M": 2, "H#I": 2, "H#J": 3}, "id": "mod7_arith_d3_0155"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#M := 5. K#O := 6. G#N := 5. K#J := G#N * 2 + E#M. G#K := G#N - K#O. G#M := G#K - K#O / J#K. J#K := G#N - E#M - K#O. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#M := 5. K#O := 6. G#N := 5. K#J := G#N * 2 + E#M. G#K := G#N - K#O. G#M := G#K - K#O / J#K. J#K := G#N - E#M - K#O.", "query": "G#M", "answer": 0, "cot": "G#N = 5 -> G#N = 5\nK#O = 6 -> K#O = 6\nE#M = 5 -> E#M = 5\nJ#K = G#N - E#M - K#O\n = 5 - 5 - 6\n 5 - 5 = (5 - 5) mod 7 = 0\n 0 - 6 = (0 - 6) mod 7 = 1\n => J#K = 1\nG#K = G#N - K#O\n = 5 - 6\n 5 - 6 = (5 - 6) mod 7 = 6\n => G#K = 6\nG#M = G#K - K#O / J#K\n = 6 - 6 / 1\n 6 - 6 = (6 - 6) mod 7 = 0\n 0 / 1 = (0 × 1⁻¹) mod 7 = 0\n => G#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#N": 1, "K#O": 1, "E#M": 1, "K#J": 2, "J#K": 2, "G#K": 2, "G#M": 3}, "id": "mod7_arith_d3_0156"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#P := I#K. H#J := I#K + H#L. I#K := 5. H#L := 3. K#I := 6. H#N := K#I * K#P + H#J. H#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#P := I#K. H#J := I#K + H#L. I#K := 5. H#L := 3. K#I := 6. H#N := K#I * K#P + H#J.", "query": "H#N", "answer": 3, "cot": "K#I = 6 -> K#I = 6\nH#L = 3 -> H#L = 3\nI#K = 5 -> I#K = 5\nH#J = I#K + H#L\n = 5 + 3\n 5 + 3 = (5 + 3) mod 7 = 1\n => H#J = 1\nK#P = I#K = 5\nH#N = K#I * K#P + H#J\n = 6 * 5 + 1\n 6 * 5 = (6 × 5) mod 7 = 2\n 2 + 1 = (2 + 1) mod 7 = 3\n => H#N = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#I": 1, "H#L": 1, "I#K": 1, "H#J": 2, "K#P": 2, "H#N": 3}, "id": "mod7_arith_d3_0157"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#P := E#N / K#K. H#K := E#J * E#N. E#J := 6. K#K := E#J + 4. E#N := 3. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#P := E#N / K#K. H#K := E#J * E#N. E#J := 6. K#K := E#J + 4. E#N := 3.", "query": "K#P", "answer": 1, "cot": "E#J = 6 -> E#J = 6\nE#N = 3 -> E#N = 3\nK#K = E#J + 4\n = 6 + 4\n 6 + 4 = (6 + 4) mod 7 = 3\n => K#K = 3\nK#P = E#N / K#K\n = 3 / 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => K#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#J": 1, "E#N": 1, "H#K": 2, "K#K": 2, "K#P": 3}, "id": "mod7_arith_d3_0158"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := J#P. G#J := F#L / 6. I#P := 2. H#N := I#P + J#P. E#P := J#P. J#P := 3. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := J#P. G#J := F#L / 6. I#P := 2. H#N := I#P + J#P. E#P := J#P. J#P := 3.", "query": "G#J", "answer": 4, "cot": "J#P = 3 -> J#P = 3\nF#L = J#P = 3\nG#J = F#L / 6\n = 3 / 6\n 3 / 6 = (3 × 6⁻¹) mod 7 = 4\n => G#J = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#P": 1, "J#P": 1, "E#P": 2, "H#N": 2, "F#L": 2, "G#J": 3}, "id": "mod7_arith_d3_0159"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#M := 2 - I#I. K#L := 5. I#I := J#P * 5. H#L := K#L. J#I := 5. H#N := 2. J#P := 6. L#N := J#I * H#N. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#M := 2 - I#I. K#L := 5. I#I := J#P * 5. H#L := K#L. J#I := 5. H#N := 2. J#P := 6. L#N := J#I * H#N.", "query": "G#M", "answer": 0, "cot": "J#P = 6 -> J#P = 6\nI#I = J#P * 5\n = 6 * 5\n 6 * 5 = (6 × 5) mod 7 = 2\n => I#I = 2\nG#M = 2 - I#I\n = 2 - 2\n 2 - 2 = (2 - 2) mod 7 = 0\n => G#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#L": 1, "J#I": 1, "J#P": 1, "H#N": 1, "H#L": 2, "I#I": 2, "L#N": 2, "G#M": 3}, "id": "mod7_arith_d3_0160"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := K#N * H#J. K#K := 6. K#N := 6. H#I := 2. G#M := H#K / K#N. F#O := H#I * K#K. H#J := 3. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := K#N * H#J. K#K := 6. K#N := 6. H#I := 2. G#M := H#K / K#N. F#O := H#I * K#K. H#J := 3.", "query": "G#M", "answer": 3, "cot": "H#J = 3 -> H#J = 3\nK#N = 6 -> K#N = 6\nH#K = K#N * H#J\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => H#K = 4\nG#M = H#K / K#N\n = 4 / 6\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n => G#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#J": 1, "K#K": 1, "H#I": 1, "K#N": 1, "F#O": 2, "H#K": 2, "G#M": 3}, "id": "mod7_arith_d3_0161"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := 3 - H#P * J#J. K#K := F#N + 3. H#P := 1. J#J := 2. I#J := H#P + J#J. K#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := 3 - H#P * J#J. K#K := F#N + 3. H#P := 1. J#J := 2. I#J := H#P + J#J.", "query": "K#K", "answer": 0, "cot": "J#J = 2 -> J#J = 2\nH#P = 1 -> H#P = 1\nF#N = 3 - H#P * J#J\n = 3 - 1 * 2\n 3 - 1 = (3 - 1) mod 7 = 2\n 2 * 2 = (2 × 2) mod 7 = 4\n => F#N = 4\nK#K = F#N + 3\n = 4 + 3\n 4 + 3 = (4 + 3) mod 7 = 0\n => K#K = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#J": 1, "H#P": 1, "I#J": 2, "F#N": 2, "K#K": 3}, "id": "mod7_arith_d3_0162"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 2. I#M := L#P / J#N. G#O := L#P + J#N. L#M := 3 + 4 - I#M. H#K := L#P. L#P := 3. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 2. I#M := L#P / J#N. G#O := L#P + J#N. L#M := 3 + 4 - I#M. H#K := L#P. L#P := 3.", "query": "L#M", "answer": 2, "cot": "J#N = 2 -> J#N = 2\nL#P = 3 -> L#P = 3\nI#M = L#P / J#N\n = 3 / 2\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => I#M = 5\nL#M = 3 + 4 - I#M\n = 3 + 4 - 5\n 3 + 4 = (3 + 4) mod 7 = 0\n 0 - 5 = (0 - 5) mod 7 = 2\n => L#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#N": 1, "L#P": 1, "I#M": 2, "H#K": 2, "G#O": 2, "L#M": 3}, "id": "mod7_arith_d3_0163"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := 2. L#K := L#N * L#P. F#O := 4. L#N := 5. L#L := 4 + F#O. J#K := L#L * E#O. E#O := 3. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := 2. L#K := L#N * L#P. F#O := 4. L#N := 5. L#L := 4 + F#O. J#K := L#L * E#O. E#O := 3.", "query": "J#K", "answer": 3, "cot": "E#O = 3 -> E#O = 3\nF#O = 4 -> F#O = 4\nL#L = 4 + F#O\n = 4 + 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => L#L = 1\nJ#K = L#L * E#O\n = 1 * 3\n 1 * 3 = (1 × 3) mod 7 = 3\n => J#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#N": 1, "E#O": 1, "F#O": 1, "L#P": 1, "L#L": 2, "L#K": 2, "J#K": 3}, "id": "mod7_arith_d3_0164"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#P := 3. K#I := 6. G#N := 2. G#L := E#N. E#N := K#I / J#P. K#P := J#P + G#N. H#O := 1. H#J := G#N - 5. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#P := 3. K#I := 6. G#N := 2. G#L := E#N. E#N := K#I / J#P. K#P := J#P + G#N. H#O := 1. H#J := G#N - 5.", "query": "G#L", "answer": 2, "cot": "J#P = 3 -> J#P = 3\nK#I = 6 -> K#I = 6\nE#N = K#I / J#P\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => E#N = 2\nG#L = E#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#N": 1, "H#O": 1, "J#P": 1, "K#I": 1, "K#P": 2, "E#N": 2, "H#J": 2, "G#L": 3}, "id": "mod7_arith_d3_0165"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#O := 1. K#I := 2. F#N := K#I. J#P := 1. F#P := K#I - L#O. E#P := 3. F#I := L#O - F#P. F#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#O := 1. K#I := 2. F#N := K#I. J#P := 1. F#P := K#I - L#O. E#P := 3. F#I := L#O - F#P.", "query": "F#I", "answer": 0, "cot": "L#O = 1 -> L#O = 1\nK#I = 2 -> K#I = 2\nF#P = K#I - L#O\n = 2 - 1\n 2 - 1 = (2 - 1) mod 7 = 1\n => F#P = 1\nF#I = L#O - F#P\n = 1 - 1\n 1 - 1 = (1 - 1) mod 7 = 0\n => F#I = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#P": 1, "E#P": 1, "L#O": 1, "K#I": 1, "F#P": 2, "F#N": 2, "F#I": 3}, "id": "mod7_arith_d3_0166"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#K := 3. H#L := 6 * H#P. H#P := E#M. F#K := 2 + I#K / E#M. E#M := 4. H#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#K := 3. H#L := 6 * H#P. H#P := E#M. F#K := 2 + I#K / E#M. E#M := 4.", "query": "H#L", "answer": 3, "cot": "E#M = 4 -> E#M = 4\nH#P = E#M = 4\nH#L = 6 * H#P\n = 6 * 4\n 6 * 4 = (6 × 4) mod 7 = 3\n => H#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"I#K": 1, "E#M": 1, "F#K": 2, "H#P": 2, "H#L": 3}, "id": "mod7_arith_d3_0167"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := G#N + I#O. F#M := G#P - 3. G#N := 5. I#O := 5. F#L := 5 - G#N. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := G#N + I#O. F#M := G#P - 3. G#N := 5. I#O := 5. F#L := 5 - G#N.", "query": "F#M", "answer": 0, "cot": "I#O = 5 -> I#O = 5\nG#N = 5 -> G#N = 5\nG#P = G#N + I#O\n = 5 + 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => G#P = 3\nF#M = G#P - 3\n = 3 - 3\n 3 - 3 = (3 - 3) mod 7 = 0\n => F#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#O": 1, "G#N": 1, "F#L": 2, "G#P": 2, "F#M": 3}, "id": "mod7_arith_d3_0168"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#I := K#N / 3 - E#P. E#P := 3. K#N := 3. J#N := 2. L#L := K#N / E#P / J#N. F#K := F#I. H#I := J#N + 4. F#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#I := K#N / 3 - E#P. E#P := 3. K#N := 3. J#N := 2. L#L := K#N / E#P / J#N. F#K := F#I. H#I := J#N + 4.", "query": "F#K", "answer": 5, "cot": "E#P = 3 -> E#P = 3\nK#N = 3 -> K#N = 3\nF#I = K#N / 3 - E#P\n = 3 / 3 - 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n 1 - 3 = (1 - 3) mod 7 = 5\n => F#I = 5\nF#K = F#I = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#P": 1, "J#N": 1, "K#N": 1, "F#I": 2, "H#I": 2, "L#L": 2, "F#K": 3}, "id": "mod7_arith_d3_0169"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#O := L#K * 6 - I#N. L#K := 5. I#P := I#N * 5. H#I := L#K / I#N. I#N := 5. F#J := 6. H#J := I#P * I#O + H#I. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#O := L#K * 6 - I#N. L#K := 5. I#P := I#N * 5. H#I := L#K / I#N. I#N := 5. F#J := 6. H#J := I#P * I#O + H#I.", "query": "H#J", "answer": 3, "cot": "I#N = 5 -> I#N = 5\nL#K = 5 -> L#K = 5\nH#I = L#K / I#N\n = 5 / 5\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n => H#I = 1\nI#P = I#N * 5\n = 5 * 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => I#P = 4\nI#O = L#K * 6 - I#N\n = 5 * 6 - 5\n 5 * 6 = (5 × 6) mod 7 = 2\n 2 - 5 = (2 - 5) mod 7 = 4\n => I#O = 4\nH#J = I#P * I#O + H#I\n = 4 * 4 + 1\n 4 * 4 = (4 × 4) mod 7 = 2\n 2 + 1 = (2 + 1) mod 7 = 3\n => H#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#J": 1, "I#N": 1, "L#K": 1, "H#I": 2, "I#P": 2, "I#O": 2, "H#J": 3}, "id": "mod7_arith_d3_0170"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := L#K - K#K / F#P. K#K := 6. H#K := L#J + K#K. L#J := F#P * L#K. F#P := 1. L#I := 1. L#K := 4. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := L#K - K#K / F#P. K#K := 6. H#K := L#J + K#K. L#J := F#P * L#K. F#P := 1. L#I := 1. L#K := 4.", "query": "H#K", "answer": 3, "cot": "F#P = 1 -> F#P = 1\nK#K = 6 -> K#K = 6\nL#K = 4 -> L#K = 4\nL#J = F#P * L#K\n = 1 * 4\n 1 * 4 = (1 × 4) mod 7 = 4\n => L#J = 4\nH#K = L#J + K#K\n = 4 + 6\n 4 + 6 = (4 + 6) mod 7 = 3\n => H#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#P": 1, "L#I": 1, "K#K": 1, "L#K": 1, "G#J": 2, "L#J": 2, "H#K": 3}, "id": "mod7_arith_d3_0171"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := 5. J#P := F#P - L#M. F#O := 5 - K#N. F#P := K#N. E#I := 4 * K#N. K#N := 6. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := 5. J#P := F#P - L#M. F#O := 5 - K#N. F#P := K#N. E#I := 4 * K#N. K#N := 6.", "query": "J#P", "answer": 1, "cot": "L#M = 5 -> L#M = 5\nK#N = 6 -> K#N = 6\nF#P = K#N = 6\nJ#P = F#P - L#M\n = 6 - 5\n 6 - 5 = (6 - 5) mod 7 = 1\n => J#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#M": 1, "K#N": 1, "F#P": 2, "F#O": 2, "E#I": 2, "J#P": 3}, "id": "mod7_arith_d3_0172"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := 5. F#P := K#P / E#L. K#I := K#M + 5. K#P := 6. K#M := K#P + E#L. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := 5. F#P := K#P / E#L. K#I := K#M + 5. K#P := 6. K#M := K#P + E#L.", "query": "K#I", "answer": 2, "cot": "E#L = 5 -> E#L = 5\nK#P = 6 -> K#P = 6\nK#M = K#P + E#L\n = 6 + 5\n 6 + 5 = (6 + 5) mod 7 = 4\n => K#M = 4\nK#I = K#M + 5\n = 4 + 5\n 4 + 5 = (4 + 5) mod 7 = 2\n => K#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#L": 1, "K#P": 1, "K#M": 2, "F#P": 2, "K#I": 3}, "id": "mod7_arith_d3_0173"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := F#O * I#N. G#L := E#O / E#K - 2. I#J := 4. E#O := F#O - I#J. K#K := 2. F#O := 1. I#N := 5. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := F#O * I#N. G#L := E#O / E#K - 2. I#J := 4. E#O := F#O - I#J. K#K := 2. F#O := 1. I#N := 5.", "query": "G#L", "answer": 3, "cot": "I#N = 5 -> I#N = 5\nI#J = 4 -> I#J = 4\nF#O = 1 -> F#O = 1\nE#K = F#O * I#N\n = 1 * 5\n 1 * 5 = (1 × 5) mod 7 = 5\n => E#K = 5\nE#O = F#O - I#J\n = 1 - 4\n 1 - 4 = (1 - 4) mod 7 = 4\n => E#O = 4\nG#L = E#O / E#K - 2\n = 4 / 5 - 2\n 4 / 5 = (4 × 5⁻¹) mod 7 = 5\n 5 - 2 = (5 - 2) mod 7 = 3\n => G#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"I#N": 1, "I#J": 1, "K#K": 1, "F#O": 1, "E#K": 2, "E#O": 2, "G#L": 3}, "id": "mod7_arith_d3_0174"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := 6. I#O := 6. E#N := J#I. J#N := 5. I#K := F#P / 4 - I#O. H#M := E#N + I#K. J#I := 3. I#L := F#P * J#N + J#I. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := 6. I#O := 6. E#N := J#I. J#N := 5. I#K := F#P / 4 - I#O. H#M := E#N + I#K. J#I := 3. I#L := F#P * J#N + J#I.", "query": "H#M", "answer": 2, "cot": "I#O = 6 -> I#O = 6\nJ#I = 3 -> J#I = 3\nF#P = 6 -> F#P = 6\nI#K = F#P / 4 - I#O\n = 6 / 4 - 6\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n 5 - 6 = (5 - 6) mod 7 = 6\n => I#K = 6\nE#N = J#I = 3\nH#M = E#N + I#K\n = 3 + 6\n 3 + 6 = (3 + 6) mod 7 = 2\n => H#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"I#O": 1, "J#I": 1, "J#N": 1, "F#P": 1, "I#K": 2, "E#N": 2, "I#L": 2, "H#M": 3}, "id": "mod7_arith_d3_0175"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := G#O / F#L. I#J := F#L + G#O. E#P := F#L - G#O. J#J := I#J / K#M. G#O := 5. F#L := 3. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := G#O / F#L. I#J := F#L + G#O. E#P := F#L - G#O. J#J := I#J / K#M. G#O := 5. F#L := 3.", "query": "J#J", "answer": 2, "cot": "F#L = 3 -> F#L = 3\nG#O = 5 -> G#O = 5\nK#M = G#O / F#L\n = 5 / 3\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n => K#M = 4\nI#J = F#L + G#O\n = 3 + 5\n 3 + 5 = (3 + 5) mod 7 = 1\n => I#J = 1\nJ#J = I#J / K#M\n = 1 / 4\n 1 / 4 = (1 × 4⁻¹) mod 7 = 2\n => J#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#L": 1, "G#O": 1, "K#M": 2, "I#J": 2, "E#P": 2, "J#J": 3}, "id": "mod7_arith_d3_0176"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := F#O. K#P := 2. E#J := K#P. F#O := K#P. L#I := 5. G#I := 2. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := F#O. K#P := 2. E#J := K#P. F#O := K#P. L#I := 5. G#I := 2.", "query": "K#I", "answer": 2, "cot": "K#P = 2 -> K#P = 2\nF#O = K#P = 2\nK#I = F#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#P": 1, "G#I": 1, "L#I": 1, "F#O": 2, "E#J": 2, "K#I": 3}, "id": "mod7_arith_d3_0177"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := J#K - I#N. J#K := 1. F#K := J#N. J#N := J#K / I#N. I#N := 3. F#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := J#K - I#N. J#K := 1. F#K := J#N. J#N := J#K / I#N. I#N := 3.", "query": "F#K", "answer": 5, "cot": "I#N = 3 -> I#N = 3\nJ#K = 1 -> J#K = 1\nJ#N = J#K / I#N\n = 1 / 3\n 1 / 3 = (1 × 3⁻¹) mod 7 = 5\n => J#N = 5\nF#K = J#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#N": 1, "J#K": 1, "E#K": 2, "J#N": 2, "F#K": 3}, "id": "mod7_arith_d3_0178"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#N := 1. I#L := E#O + H#N. J#J := 4 / H#N. F#J := I#L + 2 / 5. H#L := 1. E#O := 1. I#M := 1. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#N := 1. I#L := E#O + H#N. J#J := 4 / H#N. F#J := I#L + 2 / 5. H#L := 1. E#O := 1. I#M := 1.", "query": "F#J", "answer": 5, "cot": "E#O = 1 -> E#O = 1\nH#N = 1 -> H#N = 1\nI#L = E#O + H#N\n = 1 + 1\n 1 + 1 = (1 + 1) mod 7 = 2\n => I#L = 2\nF#J = I#L + 2 / 5\n = 2 + 2 / 5\n 2 + 2 = (2 + 2) mod 7 = 4\n 4 / 5 = (4 × 5⁻¹) mod 7 = 5\n => F#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#M": 1, "E#O": 1, "H#N": 1, "H#L": 1, "I#L": 2, "J#J": 2, "F#J": 3}, "id": "mod7_arith_d3_0179"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#I := 5. G#J := 3 + H#I. I#M := I#N * I#K + L#M. I#K := 5. L#M := 5. I#N := 6 + I#K. K#I := I#K. I#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#I := 5. G#J := 3 + H#I. I#M := I#N * I#K + L#M. I#K := 5. L#M := 5. I#N := 6 + I#K. K#I := I#K.", "query": "I#M", "answer": 4, "cot": "L#M = 5 -> L#M = 5\nI#K = 5 -> I#K = 5\nI#N = 6 + I#K\n = 6 + 5\n 6 + 5 = (6 + 5) mod 7 = 4\n => I#N = 4\nI#M = I#N * I#K + L#M\n = 4 * 5 + 5\n 4 * 5 = (4 × 5) mod 7 = 6\n 6 + 5 = (6 + 5) mod 7 = 4\n => I#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#M": 1, "H#I": 1, "I#K": 1, "K#I": 2, "I#N": 2, "G#J": 2, "I#M": 3}, "id": "mod7_arith_d3_0180"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#L := G#O + L#M. L#J := G#O - 6 / L#M. E#O := 3. G#O := 1. L#M := 6. E#I := G#O. L#K := E#I. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#L := G#O + L#M. L#J := G#O - 6 / L#M. E#O := 3. G#O := 1. L#M := 6. E#I := G#O. L#K := E#I.", "query": "L#K", "answer": 1, "cot": "G#O = 1 -> G#O = 1\nE#I = G#O = 1\nL#K = E#I = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"G#O": 1, "L#M": 1, "E#O": 1, "E#I": 2, "J#L": 2, "L#J": 2, "L#K": 3}, "id": "mod7_arith_d3_0181"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := G#L + I#J. I#J := E#N / K#P + 3. E#N := 5. G#L := L#M * 3 * K#P. L#M := 2. K#P := 2. K#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := G#L + I#J. I#J := E#N / K#P + 3. E#N := 5. G#L := L#M * 3 * K#P. L#M := 2. K#P := 2.", "query": "K#M", "answer": 0, "cot": "E#N = 5 -> E#N = 5\nL#M = 2 -> L#M = 2\nK#P = 2 -> K#P = 2\nI#J = E#N / K#P + 3\n = 5 / 2 + 3\n 5 / 2 = (5 × 2⁻¹) mod 7 = 6\n 6 + 3 = (6 + 3) mod 7 = 2\n => I#J = 2\nG#L = L#M * 3 * K#P\n = 2 * 3 * 2\n 2 * 3 = (2 × 3) mod 7 = 6\n 6 * 2 = (6 × 2) mod 7 = 5\n => G#L = 5\nK#M = G#L + I#J\n = 5 + 2\n 5 + 2 = (5 + 2) mod 7 = 0\n => K#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#N": 1, "L#M": 1, "K#P": 1, "I#J": 2, "G#L": 2, "K#M": 3}, "id": "mod7_arith_d3_0182"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#M := I#J. L#P := F#J + E#N. K#M := F#J - 6 + J#M. F#J := 6. I#J := 3. E#N := 2. J#I := 4. I#I := J#I - E#N / 4. K#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#M := I#J. L#P := F#J + E#N. K#M := F#J - 6 + J#M. F#J := 6. I#J := 3. E#N := 2. J#I := 4. I#I := J#I - E#N / 4.", "query": "K#M", "answer": 3, "cot": "F#J = 6 -> F#J = 6\nI#J = 3 -> I#J = 3\nJ#M = I#J = 3\nK#M = F#J - 6 + J#M\n = 6 - 6 + 3\n 6 - 6 = (6 - 6) mod 7 = 0\n 0 + 3 = (0 + 3) mod 7 = 3\n => K#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#I": 1, "F#J": 1, "I#J": 1, "E#N": 1, "J#M": 2, "L#P": 2, "I#I": 2, "K#M": 3}, "id": "mod7_arith_d3_0183"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#J := 6 / H#I. E#N := 5. I#M := 5. H#I := I#M + 6 / E#N. L#I := 2. H#P := E#N * L#I - I#M. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#J := 6 / H#I. E#N := 5. I#M := 5. H#I := I#M + 6 / E#N. L#I := 2. H#P := E#N * L#I - I#M.", "query": "E#J", "answer": 4, "cot": "I#M = 5 -> I#M = 5\nE#N = 5 -> E#N = 5\nH#I = I#M + 6 / E#N\n = 5 + 6 / 5\n 5 + 6 = (5 + 6) mod 7 = 4\n 4 / 5 = (4 × 5⁻¹) mod 7 = 5\n => H#I = 5\nE#J = 6 / H#I\n = 6 / 5\n 6 / 5 = (6 × 5⁻¹) mod 7 = 4\n => E#J = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#I": 1, "I#M": 1, "E#N": 1, "H#I": 2, "H#P": 2, "E#J": 3}, "id": "mod7_arith_d3_0184"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#L := 2. J#L := 4. F#O := G#L + J#L * 4. E#P := 1. J#I := 1. E#J := J#L / 5 * E#P. F#J := F#O + E#J. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#L := 2. J#L := 4. F#O := G#L + J#L * 4. E#P := 1. J#I := 1. E#J := J#L / 5 * E#P. F#J := F#O + E#J.", "query": "F#J", "answer": 1, "cot": "G#L = 2 -> G#L = 2\nE#P = 1 -> E#P = 1\nJ#L = 4 -> J#L = 4\nF#O = G#L + J#L * 4\n = 2 + 4 * 4\n 2 + 4 = (2 + 4) mod 7 = 6\n 6 * 4 = (6 × 4) mod 7 = 3\n => F#O = 3\nE#J = J#L / 5 * E#P\n = 4 / 5 * 1\n 4 / 5 = (4 × 5⁻¹) mod 7 = 5\n 5 * 1 = (5 × 1) mod 7 = 5\n => E#J = 5\nF#J = F#O + E#J\n = 3 + 5\n 3 + 5 = (3 + 5) mod 7 = 1\n => F#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#L": 1, "E#P": 1, "J#L": 1, "J#I": 1, "F#O": 2, "E#J": 2, "F#J": 3}, "id": "mod7_arith_d3_0185"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := 2. F#J := E#I + J#P. E#I := E#N. J#P := G#J. G#J := 5. E#J := G#J + 4. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := 2. F#J := E#I + J#P. E#I := E#N. J#P := G#J. G#J := 5. E#J := G#J + 4.", "query": "F#J", "answer": 0, "cot": "E#N = 2 -> E#N = 2\nG#J = 5 -> G#J = 5\nJ#P = G#J = 5\nE#I = E#N = 2\nF#J = E#I + J#P\n = 2 + 5\n 2 + 5 = (2 + 5) mod 7 = 0\n => F#J = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"E#N": 1, "G#J": 1, "J#P": 2, "E#J": 2, "E#I": 2, "F#J": 3}, "id": "mod7_arith_d3_0186"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := L#N + 3. I#N := 1. H#I := L#N / I#N / 3. J#K := 6. G#P := H#I / 5 + F#J. L#N := 3. E#K := I#N. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := L#N + 3. I#N := 1. H#I := L#N / I#N / 3. J#K := 6. G#P := H#I / 5 + F#J. L#N := 3. E#K := I#N.", "query": "G#P", "answer": 2, "cot": "I#N = 1 -> I#N = 1\nL#N = 3 -> L#N = 3\nF#J = L#N + 3\n = 3 + 3\n 3 + 3 = (3 + 3) mod 7 = 6\n => F#J = 6\nH#I = L#N / I#N / 3\n = 3 / 1 / 3\n 3 / 1 = (3 × 1⁻¹) mod 7 = 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => H#I = 1\nG#P = H#I / 5 + F#J\n = 1 / 5 + 6\n 1 / 5 = (1 × 5⁻¹) mod 7 = 3\n 3 + 6 = (3 + 6) mod 7 = 2\n => G#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#N": 1, "L#N": 1, "J#K": 1, "E#K": 2, "F#J": 2, "H#I": 2, "G#P": 3}, "id": "mod7_arith_d3_0187"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := 5. I#I := 3. E#N := E#O / G#N. E#O := I#I - 5 * H#O. G#N := I#I - H#O. E#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := 5. I#I := 3. E#N := E#O / G#N. E#O := I#I - 5 * H#O. G#N := I#I - H#O.", "query": "E#N", "answer": 5, "cot": "I#I = 3 -> I#I = 3\nH#O = 5 -> H#O = 5\nG#N = I#I - H#O\n = 3 - 5\n 3 - 5 = (3 - 5) mod 7 = 5\n => G#N = 5\nE#O = I#I - 5 * H#O\n = 3 - 5 * 5\n 3 - 5 = (3 - 5) mod 7 = 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => E#O = 4\nE#N = E#O / G#N\n = 4 / 5\n 4 / 5 = (4 × 5⁻¹) mod 7 = 5\n => E#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#I": 1, "H#O": 1, "G#N": 2, "E#O": 2, "E#N": 3}, "id": "mod7_arith_d3_0188"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := L#N - G#L. L#N := 4. K#L := I#I - 4. K#M := 3. G#L := 2. G#P := L#N. K#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := L#N - G#L. L#N := 4. K#L := I#I - 4. K#M := 3. G#L := 2. G#P := L#N.", "query": "K#L", "answer": 5, "cot": "L#N = 4 -> L#N = 4\nG#L = 2 -> G#L = 2\nI#I = L#N - G#L\n = 4 - 2\n 4 - 2 = (4 - 2) mod 7 = 2\n => I#I = 2\nK#L = I#I - 4\n = 2 - 4\n 2 - 4 = (2 - 4) mod 7 = 5\n => K#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#N": 1, "G#L": 1, "K#M": 1, "I#I": 2, "G#P": 2, "K#L": 3}, "id": "mod7_arith_d3_0189"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#M := F#O * 3. F#M := F#L. F#O := 2. J#K := F#M. F#L := 1. E#I := F#O * F#L. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#M := F#O * 3. F#M := F#L. F#O := 2. J#K := F#M. F#L := 1. E#I := F#O * F#L.", "query": "J#K", "answer": 1, "cot": "F#L = 1 -> F#L = 1\nF#M = F#L = 1\nJ#K = F#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"F#L": 1, "F#O": 1, "E#M": 2, "E#I": 2, "F#M": 2, "J#K": 3}, "id": "mod7_arith_d3_0190"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#K := E#O + J#O. E#M := F#I - J#O + E#O. E#O := 1. I#P := 6. J#O := 6. I#M := 2. F#I := E#O. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#K := E#O + J#O. E#M := F#I - J#O + E#O. E#O := 1. I#P := 6. J#O := 6. I#M := 2. F#I := E#O.", "query": "E#M", "answer": 3, "cot": "J#O = 6 -> J#O = 6\nE#O = 1 -> E#O = 1\nF#I = E#O = 1\nE#M = F#I - J#O + E#O\n = 1 - 6 + 1\n 1 - 6 = (1 - 6) mod 7 = 2\n 2 + 1 = (2 + 1) mod 7 = 3\n => E#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#O": 1, "I#P": 1, "I#M": 1, "E#O": 1, "F#I": 2, "L#K": 2, "E#M": 3}, "id": "mod7_arith_d3_0191"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := F#O / H#M. G#M := 4 * E#O. G#O := 2. F#O := 2. L#I := 1. E#O := G#O - F#O. H#M := 3. L#P := H#M / G#O. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := F#O / H#M. G#M := 4 * E#O. G#O := 2. F#O := 2. L#I := 1. E#O := G#O - F#O. H#M := 3. L#P := H#M / G#O.", "query": "G#M", "answer": 0, "cot": "G#O = 2 -> G#O = 2\nF#O = 2 -> F#O = 2\nE#O = G#O - F#O\n = 2 - 2\n 2 - 2 = (2 - 2) mod 7 = 0\n => E#O = 0\nG#M = 4 * E#O\n = 4 * 0\n 4 * 0 = (4 × 0) mod 7 = 0\n => G#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#M": 1, "L#I": 1, "G#O": 1, "F#O": 1, "L#P": 2, "H#K": 2, "E#O": 2, "G#M": 3}, "id": "mod7_arith_d3_0192"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := 2. G#P := K#J - 2. K#M := K#J + K#I. E#K := K#I. K#J := 3. F#O := G#P * K#M. F#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := 2. G#P := K#J - 2. K#M := K#J + K#I. E#K := K#I. K#J := 3. F#O := G#P * K#M.", "query": "F#O", "answer": 5, "cot": "K#J = 3 -> K#J = 3\nK#I = 2 -> K#I = 2\nG#P = K#J - 2\n = 3 - 2\n 3 - 2 = (3 - 2) mod 7 = 1\n => G#P = 1\nK#M = K#J + K#I\n = 3 + 2\n 3 + 2 = (3 + 2) mod 7 = 5\n => K#M = 5\nF#O = G#P * K#M\n = 1 * 5\n 1 * 5 = (1 × 5) mod 7 = 5\n => F#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#J": 1, "K#I": 1, "G#P": 2, "K#M": 2, "E#K": 2, "F#O": 3}, "id": "mod7_arith_d3_0193"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#I := 2. I#L := E#M * J#I + 4. F#J := I#L - E#M + F#O. E#M := 6. F#O := 5 * E#M. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#I := 2. I#L := E#M * J#I + 4. F#J := I#L - E#M + F#O. E#M := 6. F#O := 5 * E#M.", "query": "F#J", "answer": 5, "cot": "E#M = 6 -> E#M = 6\nJ#I = 2 -> J#I = 2\nF#O = 5 * E#M\n = 5 * 6\n 5 * 6 = (5 × 6) mod 7 = 2\n => F#O = 2\nI#L = E#M * J#I + 4\n = 6 * 2 + 4\n 6 * 2 = (6 × 2) mod 7 = 5\n 5 + 4 = (5 + 4) mod 7 = 2\n => I#L = 2\nF#J = I#L - E#M + F#O\n = 2 - 6 + 2\n 2 - 6 = (2 - 6) mod 7 = 3\n 3 + 2 = (3 + 2) mod 7 = 5\n => F#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 5, "num_distractors": 0, "var_layer": {"E#M": 1, "J#I": 1, "F#O": 2, "I#L": 2, "F#J": 3}, "id": "mod7_arith_d3_0194"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := 4. E#O := H#M * E#N. G#N := J#K - L#M. E#N := L#M * J#K. J#K := 6. H#M := 6 / L#M. E#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := 4. E#O := H#M * E#N. G#N := J#K - L#M. E#N := L#M * J#K. J#K := 6. H#M := 6 / L#M.", "query": "E#O", "answer": 1, "cot": "J#K = 6 -> J#K = 6\nL#M = 4 -> L#M = 4\nH#M = 6 / L#M\n = 6 / 4\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n => H#M = 5\nE#N = L#M * J#K\n = 4 * 6\n 4 * 6 = (4 × 6) mod 7 = 3\n => E#N = 3\nE#O = H#M * E#N\n = 5 * 3\n 5 * 3 = (5 × 3) mod 7 = 1\n => E#O = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#K": 1, "L#M": 1, "G#N": 2, "H#M": 2, "E#N": 2, "E#O": 3}, "id": "mod7_arith_d3_0195"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#I := E#L / H#L. E#L := 6. K#K := 1. I#P := H#I. H#L := 6. F#J := E#L. I#N := 1. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#I := E#L / H#L. E#L := 6. K#K := 1. I#P := H#I. H#L := 6. F#J := E#L. I#N := 1.", "query": "I#P", "answer": 1, "cot": "H#L = 6 -> H#L = 6\nE#L = 6 -> E#L = 6\nH#I = E#L / H#L\n = 6 / 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => H#I = 1\nI#P = H#I = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#K": 1, "I#N": 1, "H#L": 1, "E#L": 1, "F#J": 2, "H#I": 2, "I#P": 3}, "id": "mod7_arith_d3_0196"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#P := 5. J#J := 1. G#L := L#J / K#J - 5. K#J := 3. J#L := J#P - K#K. L#J := 5 / K#K / J#J. K#K := 4. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#P := 5. J#J := 1. G#L := L#J / K#J - 5. K#J := 3. J#L := J#P - K#K. L#J := 5 / K#K / J#J. K#K := 4.", "query": "G#L", "answer": 3, "cot": "K#J = 3 -> K#J = 3\nJ#J = 1 -> J#J = 1\nK#K = 4 -> K#K = 4\nL#J = 5 / K#K / J#J\n = 5 / 4 / 1\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n 3 / 1 = (3 × 1⁻¹) mod 7 = 3\n => L#J = 3\nG#L = L#J / K#J - 5\n = 3 / 3 - 5\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n 1 - 5 = (1 - 5) mod 7 = 3\n => G#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#J": 1, "J#J": 1, "K#K": 1, "J#P": 1, "J#L": 2, "L#J": 2, "G#L": 3}, "id": "mod7_arith_d3_0197"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := I#L / 2 + K#K. K#O := I#L + L#N / K#K. L#N := 1. K#N := 5 - I#L. G#M := 5 + K#J * K#N. K#K := 4. I#L := 6. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := I#L / 2 + K#K. K#O := I#L + L#N / K#K. L#N := 1. K#N := 5 - I#L. G#M := 5 + K#J * K#N. K#K := 4. I#L := 6.", "query": "G#M", "answer": 2, "cot": "I#L = 6 -> I#L = 6\nK#K = 4 -> K#K = 4\nK#N = 5 - I#L\n = 5 - 6\n 5 - 6 = (5 - 6) mod 7 = 6\n => K#N = 6\nK#J = I#L / 2 + K#K\n = 6 / 2 + 4\n 6 / 2 = (6 × 2⁻¹) mod 7 = 3\n 3 + 4 = (3 + 4) mod 7 = 0\n => K#J = 0\nG#M = 5 + K#J * K#N\n = 5 + 0 * 6\n 5 + 0 = (5 + 0) mod 7 = 5\n 5 * 6 = (5 × 6) mod 7 = 2\n => G#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#L": 1, "K#K": 1, "L#N": 1, "K#O": 2, "K#N": 2, "K#J": 2, "G#M": 3}, "id": "mod7_arith_d3_0198"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#O := F#M. G#K := 2. F#K := I#P + E#I / G#K. E#N := 4. I#J := G#K * E#I / I#P. F#M := E#N + 3. I#P := 1. E#I := 3. E#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#O := F#M. G#K := 2. F#K := I#P + E#I / G#K. E#N := 4. I#J := G#K * E#I / I#P. F#M := E#N + 3. I#P := 1. E#I := 3.", "query": "E#O", "answer": 0, "cot": "E#N = 4 -> E#N = 4\nF#M = E#N + 3\n = 4 + 3\n 4 + 3 = (4 + 3) mod 7 = 0\n => F#M = 0\nE#O = F#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#I": 1, "E#N": 1, "I#P": 1, "G#K": 1, "I#J": 2, "F#M": 2, "F#K": 2, "E#O": 3}, "id": "mod7_arith_d3_0199"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := 1. J#K := 5 - F#K. E#I := 4. G#L := H#I. H#I := I#P + 3. F#K := 1. K#J := H#K. I#P := 3. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := 1. J#K := 5 - F#K. E#I := 4. G#L := H#I. H#I := I#P + 3. F#K := 1. K#J := H#K. I#P := 3.", "query": "G#L", "answer": 6, "cot": "I#P = 3 -> I#P = 3\nH#I = I#P + 3\n = 3 + 3\n 3 + 3 = (3 + 3) mod 7 = 6\n => H#I = 6\nG#L = H#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#K": 1, "I#P": 1, "E#I": 1, "F#K": 1, "H#I": 2, "J#K": 2, "K#J": 2, "G#L": 3}, "id": "mod7_arith_d3_0200"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := 4. F#L := 5. L#P := F#L - L#M * 3. E#P := L#M / F#L. G#L := E#P. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := 4. F#L := 5. L#P := F#L - L#M * 3. E#P := L#M / F#L. G#L := E#P.", "query": "G#L", "answer": 5, "cot": "F#L = 5 -> F#L = 5\nL#M = 4 -> L#M = 4\nE#P = L#M / F#L\n = 4 / 5\n 4 / 5 = (4 × 5⁻¹) mod 7 = 5\n => E#P = 5\nG#L = E#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#L": 1, "L#M": 1, "E#P": 2, "L#P": 2, "G#L": 3}, "id": "mod7_arith_d3_0201"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#P := H#O - F#O. F#O := 4. K#I := H#O + K#M * 5. H#O := 3. K#M := 1. J#J := K#M - F#O. G#L := H#O - J#J * F#O. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#P := H#O - F#O. F#O := 4. K#I := H#O + K#M * 5. H#O := 3. K#M := 1. J#J := K#M - F#O. G#L := H#O - J#J * F#O.", "query": "G#L", "answer": 3, "cot": "K#M = 1 -> K#M = 1\nF#O = 4 -> F#O = 4\nH#O = 3 -> H#O = 3\nJ#J = K#M - F#O\n = 1 - 4\n 1 - 4 = (1 - 4) mod 7 = 4\n => J#J = 4\nG#L = H#O - J#J * F#O\n = 3 - 4 * 4\n 3 - 4 = (3 - 4) mod 7 = 6\n 6 * 4 = (6 × 4) mod 7 = 3\n => G#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#M": 1, "F#O": 1, "H#O": 1, "J#P": 2, "K#I": 2, "J#J": 2, "G#L": 3}, "id": "mod7_arith_d3_0202"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := 3. H#I := 5. J#M := I#O - L#J - G#N. J#O := L#J + I#M. I#O := 2 + G#N. G#N := 2. I#M := 2. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := 3. H#I := 5. J#M := I#O - L#J - G#N. J#O := L#J + I#M. I#O := 2 + G#N. G#N := 2. I#M := 2.", "query": "J#M", "answer": 6, "cot": "G#N = 2 -> G#N = 2\nL#J = 3 -> L#J = 3\nI#O = 2 + G#N\n = 2 + 2\n 2 + 2 = (2 + 2) mod 7 = 4\n => I#O = 4\nJ#M = I#O - L#J - G#N\n = 4 - 3 - 2\n 4 - 3 = (4 - 3) mod 7 = 1\n 1 - 2 = (1 - 2) mod 7 = 6\n => J#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#N": 1, "I#M": 1, "L#J": 1, "H#I": 1, "I#O": 2, "J#O": 2, "J#M": 3}, "id": "mod7_arith_d3_0203"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := L#J. H#M := 4 / H#L. L#J := 2 + H#L. G#O := 6 + H#L. H#L := 1. J#M := 1. K#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := L#J. H#M := 4 / H#L. L#J := 2 + H#L. G#O := 6 + H#L. H#L := 1. J#M := 1.", "query": "K#M", "answer": 3, "cot": "H#L = 1 -> H#L = 1\nL#J = 2 + H#L\n = 2 + 1\n 2 + 1 = (2 + 1) mod 7 = 3\n => L#J = 3\nK#M = L#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"H#L": 1, "J#M": 1, "L#J": 2, "H#M": 2, "G#O": 2, "K#M": 3}, "id": "mod7_arith_d3_0204"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#P := G#O + J#M. E#J := I#O. J#M := 1. G#O := 3. L#M := G#O. I#O := J#M + G#O. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#P := G#O + J#M. E#J := I#O. J#M := 1. G#O := 3. L#M := G#O. I#O := J#M + G#O.", "query": "E#J", "answer": 4, "cot": "G#O = 3 -> G#O = 3\nJ#M = 1 -> J#M = 1\nI#O = J#M + G#O\n = 1 + 3\n 1 + 3 = (1 + 3) mod 7 = 4\n => I#O = 4\nE#J = I#O = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#O": 1, "J#M": 1, "L#M": 2, "J#P": 2, "I#O": 2, "E#J": 3}, "id": "mod7_arith_d3_0205"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#M := L#P. L#M := 1. L#J := 6. I#N := L#J. L#P := L#J - I#J / 2. I#J := 3. E#P := 6. J#J := E#P. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#M := L#P. L#M := 1. L#J := 6. I#N := L#J. L#P := L#J - I#J / 2. I#J := 3. E#P := 6. J#J := E#P.", "query": "G#M", "answer": 5, "cot": "L#J = 6 -> L#J = 6\nI#J = 3 -> I#J = 3\nL#P = L#J - I#J / 2\n = 6 - 3 / 2\n 6 - 3 = (6 - 3) mod 7 = 3\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => L#P = 5\nG#M = L#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#M": 1, "L#J": 1, "E#P": 1, "I#J": 1, "J#J": 2, "I#N": 2, "L#P": 2, "G#M": 3}, "id": "mod7_arith_d3_0206"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#K := 3 - E#J - E#K. F#N := 4 - J#K. G#K := E#J * E#K. E#K := 5. E#J := 1. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#K := 3 - E#J - E#K. F#N := 4 - J#K. G#K := E#J * E#K. E#K := 5. E#J := 1.", "query": "F#N", "answer": 0, "cot": "E#J = 1 -> E#J = 1\nE#K = 5 -> E#K = 5\nJ#K = 3 - E#J - E#K\n = 3 - 1 - 5\n 3 - 1 = (3 - 1) mod 7 = 2\n 2 - 5 = (2 - 5) mod 7 = 4\n => J#K = 4\nF#N = 4 - J#K\n = 4 - 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => F#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#J": 1, "E#K": 1, "J#K": 2, "G#K": 2, "F#N": 3}, "id": "mod7_arith_d3_0207"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#M := 4. F#P := E#L * 6. E#L := 4. K#L := E#L. I#I := 4 / F#P. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#M := 4. F#P := E#L * 6. E#L := 4. K#L := E#L. I#I := 4 / F#P.", "query": "I#I", "answer": 6, "cot": "E#L = 4 -> E#L = 4\nF#P = E#L * 6\n = 4 * 6\n 4 * 6 = (4 × 6) mod 7 = 3\n => F#P = 3\nI#I = 4 / F#P\n = 4 / 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => I#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#M": 1, "E#L": 1, "K#L": 2, "F#P": 2, "I#I": 3}, "id": "mod7_arith_d3_0208"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#J := K#K. J#P := 6. E#N := K#K / J#P * J#M. E#L := J#M. J#M := 3. K#K := 2. K#N := E#L. K#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#J := K#K. J#P := 6. E#N := K#K / J#P * J#M. E#L := J#M. J#M := 3. K#K := 2. K#N := E#L.", "query": "K#N", "answer": 3, "cot": "J#M = 3 -> J#M = 3\nE#L = J#M = 3\nK#N = E#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"K#K": 1, "J#M": 1, "J#P": 1, "E#L": 2, "E#N": 2, "E#J": 2, "K#N": 3}, "id": "mod7_arith_d3_0209"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#J := I#P - F#O. E#O := F#O / K#I. I#P := 6. L#I := 2. K#I := 1. F#O := 6. J#M := J#J - E#O. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#J := I#P - F#O. E#O := F#O / K#I. I#P := 6. L#I := 2. K#I := 1. F#O := 6. J#M := J#J - E#O.", "query": "J#M", "answer": 1, "cot": "F#O = 6 -> F#O = 6\nK#I = 1 -> K#I = 1\nI#P = 6 -> I#P = 6\nJ#J = I#P - F#O\n = 6 - 6\n 6 - 6 = (6 - 6) mod 7 = 0\n => J#J = 0\nE#O = F#O / K#I\n = 6 / 1\n 6 / 1 = (6 × 1⁻¹) mod 7 = 6\n => E#O = 6\nJ#M = J#J - E#O\n = 0 - 6\n 0 - 6 = (0 - 6) mod 7 = 1\n => J#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#O": 1, "K#I": 1, "I#P": 1, "L#I": 1, "J#J": 2, "E#O": 2, "J#M": 3}, "id": "mod7_arith_d3_0210"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := 6. E#L := K#P * K#J. K#P := 5. K#N := K#P. I#K := K#J. H#J := E#L * K#N. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := 6. E#L := K#P * K#J. K#P := 5. K#N := K#P. I#K := K#J. H#J := E#L * K#N.", "query": "H#J", "answer": 3, "cot": "K#J = 6 -> K#J = 6\nK#P = 5 -> K#P = 5\nK#N = K#P = 5\nE#L = K#P * K#J\n = 5 * 6\n 5 * 6 = (5 × 6) mod 7 = 2\n => E#L = 2\nH#J = E#L * K#N\n = 2 * 5\n 2 * 5 = (2 × 5) mod 7 = 3\n => H#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#J": 1, "K#P": 1, "I#K": 2, "K#N": 2, "E#L": 2, "H#J": 3}, "id": "mod7_arith_d3_0211"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#N := I#P / I#I - H#N. I#P := 5. I#I := H#J - 4. H#J := 5. H#N := 6 - H#J + I#P. L#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#N := I#P / I#I - H#N. I#P := 5. I#I := H#J - 4. H#J := 5. H#N := 6 - H#J + I#P.", "query": "L#N", "answer": 6, "cot": "I#P = 5 -> I#P = 5\nH#J = 5 -> H#J = 5\nI#I = H#J - 4\n = 5 - 4\n 5 - 4 = (5 - 4) mod 7 = 1\n => I#I = 1\nH#N = 6 - H#J + I#P\n = 6 - 5 + 5\n 6 - 5 = (6 - 5) mod 7 = 1\n 1 + 5 = (1 + 5) mod 7 = 6\n => H#N = 6\nL#N = I#P / I#I - H#N\n = 5 / 1 - 6\n 5 / 1 = (5 × 1⁻¹) mod 7 = 5\n 5 - 6 = (5 - 6) mod 7 = 6\n => L#N = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#P": 1, "H#J": 1, "I#I": 2, "H#N": 2, "L#N": 3}, "id": "mod7_arith_d3_0212"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#M := 6 / G#P. J#N := G#P + J#O. J#P := J#O / G#P * 2. K#K := 2. F#I := 4. J#J := 5 + J#P + F#I. J#O := 2. G#P := 6. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#M := 6 / G#P. J#N := G#P + J#O. J#P := J#O / G#P * 2. K#K := 2. F#I := 4. J#J := 5 + J#P + F#I. J#O := 2. G#P := 6.", "query": "J#J", "answer": 5, "cot": "J#O = 2 -> J#O = 2\nG#P = 6 -> G#P = 6\nF#I = 4 -> F#I = 4\nJ#P = J#O / G#P * 2\n = 2 / 6 * 2\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n 5 * 2 = (5 × 2) mod 7 = 3\n => J#P = 3\nJ#J = 5 + J#P + F#I\n = 5 + 3 + 4\n 5 + 3 = (5 + 3) mod 7 = 1\n 1 + 4 = (1 + 4) mod 7 = 5\n => J#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#O": 1, "G#P": 1, "K#K": 1, "F#I": 1, "J#N": 2, "J#P": 2, "I#M": 2, "J#J": 3}, "id": "mod7_arith_d3_0213"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := J#N / 2. F#P := J#N * H#O. J#M := E#K - J#N. J#N := 4. H#O := 2. F#O := H#O * J#N. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := J#N / 2. F#P := J#N * H#O. J#M := E#K - J#N. J#N := 4. H#O := 2. F#O := H#O * J#N.", "query": "J#M", "answer": 5, "cot": "J#N = 4 -> J#N = 4\nE#K = J#N / 2\n = 4 / 2\n 4 / 2 = (4 × 2⁻¹) mod 7 = 2\n => E#K = 2\nJ#M = E#K - J#N\n = 2 - 4\n 2 - 4 = (2 - 4) mod 7 = 5\n => J#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#N": 1, "H#O": 1, "E#K": 2, "F#O": 2, "F#P": 2, "J#M": 3}, "id": "mod7_arith_d3_0214"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#L := H#M. I#M := 1. I#L := 1. F#O := I#M + I#L. E#J := I#M * I#L. H#M := I#L / I#M - 6. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#L := H#M. I#M := 1. I#L := 1. F#O := I#M + I#L. E#J := I#M * I#L. H#M := I#L / I#M - 6.", "query": "G#L", "answer": 2, "cot": "I#L = 1 -> I#L = 1\nI#M = 1 -> I#M = 1\nH#M = I#L / I#M - 6\n = 1 / 1 - 6\n 1 / 1 = (1 × 1⁻¹) mod 7 = 1\n 1 - 6 = (1 - 6) mod 7 = 2\n => H#M = 2\nG#L = H#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#L": 1, "I#M": 1, "H#M": 2, "E#J": 2, "F#O": 2, "G#L": 3}, "id": "mod7_arith_d3_0215"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#O := E#L / F#K. F#K := 3. F#M := L#K * F#I. J#O := F#K - E#L. L#K := 3. E#L := 3. F#I := 6 * 3 * L#K. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#O := E#L / F#K. F#K := 3. F#M := L#K * F#I. J#O := F#K - E#L. L#K := 3. E#L := 3. F#I := 6 * 3 * L#K.", "query": "F#M", "answer": 1, "cot": "L#K = 3 -> L#K = 3\nF#I = 6 * 3 * L#K\n = 6 * 3 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n 4 * 3 = (4 × 3) mod 7 = 5\n => F#I = 5\nF#M = L#K * F#I\n = 3 * 5\n 3 * 5 = (3 × 5) mod 7 = 1\n => F#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#L": 1, "L#K": 1, "F#K": 1, "E#O": 2, "F#I": 2, "J#O": 2, "F#M": 3}, "id": "mod7_arith_d3_0216"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := L#K + G#I. G#I := 3. G#M := G#I - L#K. L#K := 5. I#O := G#M. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := L#K + G#I. G#I := 3. G#M := G#I - L#K. L#K := 5. I#O := G#M.", "query": "I#O", "answer": 5, "cot": "L#K = 5 -> L#K = 5\nG#I = 3 -> G#I = 3\nG#M = G#I - L#K\n = 3 - 5\n 3 - 5 = (3 - 5) mod 7 = 5\n => G#M = 5\nI#O = G#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#K": 1, "G#I": 1, "G#M": 2, "F#N": 2, "I#O": 3}, "id": "mod7_arith_d3_0217"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := I#P / H#O. G#N := I#P / 5. I#P := 5. L#N := H#O - I#P. E#L := F#N. H#O := 6. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := I#P / H#O. G#N := I#P / 5. I#P := 5. L#N := H#O - I#P. E#L := F#N. H#O := 6.", "query": "E#L", "answer": 2, "cot": "I#P = 5 -> I#P = 5\nH#O = 6 -> H#O = 6\nF#N = I#P / H#O\n = 5 / 6\n 5 / 6 = (5 × 6⁻¹) mod 7 = 2\n => F#N = 2\nE#L = F#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#P": 1, "H#O": 1, "F#N": 2, "G#N": 2, "L#N": 2, "E#L": 3}, "id": "mod7_arith_d3_0218"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#K := 4. G#M := K#J / L#K / 2. K#J := 5. J#N := L#K * 5 / K#J. F#N := G#M. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#K := 4. G#M := K#J / L#K / 2. K#J := 5. J#N := L#K * 5 / K#J. F#N := G#M.", "query": "F#N", "answer": 5, "cot": "K#J = 5 -> K#J = 5\nL#K = 4 -> L#K = 4\nG#M = K#J / L#K / 2\n = 5 / 4 / 2\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => G#M = 5\nF#N = G#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#J": 1, "L#K": 1, "G#M": 2, "J#N": 2, "F#N": 3}, "id": "mod7_arith_d3_0219"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := G#M * H#K. G#M := 4. H#K := 3. L#P := G#M. K#O := 4 / H#K. J#P := 5. H#I := H#O. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := G#M * H#K. G#M := 4. H#K := 3. L#P := G#M. K#O := 4 / H#K. J#P := 5. H#I := H#O.", "query": "H#I", "answer": 5, "cot": "H#K = 3 -> H#K = 3\nG#M = 4 -> G#M = 4\nH#O = G#M * H#K\n = 4 * 3\n 4 * 3 = (4 × 3) mod 7 = 5\n => H#O = 5\nH#I = H#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#P": 1, "H#K": 1, "G#M": 1, "K#O": 2, "L#P": 2, "H#O": 2, "H#I": 3}, "id": "mod7_arith_d3_0220"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := E#K - L#J. J#L := 1. H#M := 5 + J#L. E#K := 5. K#M := H#M + 6. L#J := 5. K#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := E#K - L#J. J#L := 1. H#M := 5 + J#L. E#K := 5. K#M := H#M + 6. L#J := 5.", "query": "K#M", "answer": 5, "cot": "J#L = 1 -> J#L = 1\nH#M = 5 + J#L\n = 5 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n => H#M = 6\nK#M = H#M + 6\n = 6 + 6\n 6 + 6 = (6 + 6) mod 7 = 5\n => K#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#L": 1, "L#J": 1, "E#K": 1, "K#L": 2, "H#M": 2, "K#M": 3}, "id": "mod7_arith_d3_0221"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#L := 6. I#L := 3. H#O := G#M * I#L. G#M := 2. H#N := H#O. E#O := K#N / J#L. K#N := 6. H#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#L := 6. I#L := 3. H#O := G#M * I#L. G#M := 2. H#N := H#O. E#O := K#N / J#L. K#N := 6.", "query": "H#N", "answer": 6, "cot": "I#L = 3 -> I#L = 3\nG#M = 2 -> G#M = 2\nH#O = G#M * I#L\n = 2 * 3\n 2 * 3 = (2 × 3) mod 7 = 6\n => H#O = 6\nH#N = H#O = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#N": 1, "I#L": 1, "J#L": 1, "G#M": 1, "E#O": 2, "H#O": 2, "H#N": 3}, "id": "mod7_arith_d3_0222"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := 4. H#P := 3 - L#P. H#L := J#K + 6 / E#M. E#M := 5. F#L := H#L + 3 / H#P. J#K := 1. H#J := 4 * L#P + E#M. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := 4. H#P := 3 - L#P. H#L := J#K + 6 / E#M. E#M := 5. F#L := H#L + 3 / H#P. J#K := 1. H#J := 4 * L#P + E#M.", "query": "F#L", "answer": 4, "cot": "J#K = 1 -> J#K = 1\nL#P = 4 -> L#P = 4\nE#M = 5 -> E#M = 5\nH#L = J#K + 6 / E#M\n = 1 + 6 / 5\n 1 + 6 = (1 + 6) mod 7 = 0\n 0 / 5 = (0 × 5⁻¹) mod 7 = 0\n => H#L = 0\nH#P = 3 - L#P\n = 3 - 4\n 3 - 4 = (3 - 4) mod 7 = 6\n => H#P = 6\nF#L = H#L + 3 / H#P\n = 0 + 3 / 6\n 0 + 3 = (0 + 3) mod 7 = 3\n 3 / 6 = (3 × 6⁻¹) mod 7 = 4\n => F#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#K": 1, "L#P": 1, "E#M": 1, "H#L": 2, "H#P": 2, "H#J": 2, "F#L": 3}, "id": "mod7_arith_d3_0223"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := 2. H#N := L#M - L#J. H#O := L#M / G#O. F#M := L#J * H#N - G#O. G#O := 5. L#J := 5. H#M := G#O - 6 + L#J. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := 2. H#N := L#M - L#J. H#O := L#M / G#O. F#M := L#J * H#N - G#O. G#O := 5. L#J := 5. H#M := G#O - 6 + L#J.", "query": "F#M", "answer": 1, "cot": "L#J = 5 -> L#J = 5\nG#O = 5 -> G#O = 5\nL#M = 2 -> L#M = 2\nH#N = L#M - L#J\n = 2 - 5\n 2 - 5 = (2 - 5) mod 7 = 4\n => H#N = 4\nF#M = L#J * H#N - G#O\n = 5 * 4 - 5\n 5 * 4 = (5 × 4) mod 7 = 6\n 6 - 5 = (6 - 5) mod 7 = 1\n => F#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#J": 1, "G#O": 1, "L#M": 1, "H#N": 2, "H#M": 2, "H#O": 2, "F#M": 3}, "id": "mod7_arith_d3_0224"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#L := 6. K#O := L#L. L#N := L#L / H#P. H#P := 3. H#K := H#P / L#L. I#J := L#N / H#P. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#L := 6. K#O := L#L. L#N := L#L / H#P. H#P := 3. H#K := H#P / L#L. I#J := L#N / H#P.", "query": "I#J", "answer": 3, "cot": "L#L = 6 -> L#L = 6\nH#P = 3 -> H#P = 3\nL#N = L#L / H#P\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => L#N = 2\nI#J = L#N / H#P\n = 2 / 3\n 2 / 3 = (2 × 3⁻¹) mod 7 = 3\n => I#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#L": 1, "H#P": 1, "H#K": 2, "L#N": 2, "K#O": 2, "I#J": 3}, "id": "mod7_arith_d3_0225"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := 3. J#N := 3. J#O := E#N. L#I := J#O. E#K := 1. G#J := E#K. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := 3. J#N := 3. J#O := E#N. L#I := J#O. E#K := 1. G#J := E#K.", "query": "L#I", "answer": 3, "cot": "E#N = 3 -> E#N = 3\nJ#O = E#N = 3\nL#I = J#O = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 3, "num_distractors": 0, "var_layer": {"E#K": 1, "J#N": 1, "E#N": 1, "G#J": 2, "J#O": 2, "L#I": 3}, "id": "mod7_arith_d3_0226"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#K := 4. L#I := 4 - K#M * J#K. K#M := 3. J#I := K#M + K#K. K#K := K#M + J#K. G#I := K#M. J#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#K := 4. L#I := 4 - K#M * J#K. K#M := 3. J#I := K#M + K#K. K#K := K#M + J#K. G#I := K#M.", "query": "J#I", "answer": 3, "cot": "K#M = 3 -> K#M = 3\nJ#K = 4 -> J#K = 4\nK#K = K#M + J#K\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => K#K = 0\nJ#I = K#M + K#K\n = 3 + 0\n 3 + 0 = (3 + 0) mod 7 = 3\n => J#I = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#M": 1, "J#K": 1, "K#K": 2, "G#I": 2, "L#I": 2, "J#I": 3}, "id": "mod7_arith_d3_0227"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#J := 2. L#N := 4. E#P := 3 * J#J. F#J := 6. E#J := 5. F#L := E#I + L#N. E#I := E#J. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#J := 2. L#N := 4. E#P := 3 * J#J. F#J := 6. E#J := 5. F#L := E#I + L#N. E#I := E#J.", "query": "F#L", "answer": 2, "cot": "L#N = 4 -> L#N = 4\nE#J = 5 -> E#J = 5\nE#I = E#J = 5\nF#L = E#I + L#N\n = 5 + 4\n 5 + 4 = (5 + 4) mod 7 = 2\n => F#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#J": 1, "L#N": 1, "F#J": 1, "E#J": 1, "E#I": 2, "E#P": 2, "F#L": 3}, "id": "mod7_arith_d3_0228"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#M := J#K. J#K := 4. K#J := J#K. I#N := 4. J#J := K#J. F#M := 2. G#M := 2. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#M := J#K. J#K := 4. K#J := J#K. I#N := 4. J#J := K#J. F#M := 2. G#M := 2.", "query": "J#J", "answer": 4, "cot": "J#K = 4 -> J#K = 4\nK#J = J#K = 4\nJ#J = K#J = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 3, "num_distractors": 0, "var_layer": {"J#K": 1, "I#N": 1, "F#M": 1, "G#M": 1, "J#M": 2, "K#J": 2, "J#J": 3}, "id": "mod7_arith_d3_0229"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := 2. H#N := 3. F#P := H#J + K#J. K#J := 1. F#J := L#I - 6. L#I := H#N + F#L. K#N := H#J / K#J. F#L := 6. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := 2. H#N := 3. F#P := H#J + K#J. K#J := 1. F#J := L#I - 6. L#I := H#N + F#L. K#N := H#J / K#J. F#L := 6.", "query": "F#J", "answer": 3, "cot": "H#N = 3 -> H#N = 3\nF#L = 6 -> F#L = 6\nL#I = H#N + F#L\n = 3 + 6\n 3 + 6 = (3 + 6) mod 7 = 2\n => L#I = 2\nF#J = L#I - 6\n = 2 - 6\n 2 - 6 = (2 - 6) mod 7 = 3\n => F#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#J": 1, "H#N": 1, "F#L": 1, "H#J": 1, "F#P": 2, "L#I": 2, "K#N": 2, "F#J": 3}, "id": "mod7_arith_d3_0230"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := I#N - G#L. G#J := G#M / I#P - 4. I#P := 1. J#N := 2. G#L := I#P / 5. I#N := J#N - I#P. L#N := 1. G#M := 2. F#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := I#N - G#L. G#J := G#M / I#P - 4. I#P := 1. J#N := 2. G#L := I#P / 5. I#N := J#N - I#P. L#N := 1. G#M := 2.", "query": "F#K", "answer": 5, "cot": "I#P = 1 -> I#P = 1\nJ#N = 2 -> J#N = 2\nI#N = J#N - I#P\n = 2 - 1\n 2 - 1 = (2 - 1) mod 7 = 1\n => I#N = 1\nG#L = I#P / 5\n = 1 / 5\n 1 / 5 = (1 × 5⁻¹) mod 7 = 3\n => G#L = 3\nF#K = I#N - G#L\n = 1 - 3\n 1 - 3 = (1 - 3) mod 7 = 5\n => F#K = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#N": 1, "G#M": 1, "I#P": 1, "J#N": 1, "I#N": 2, "G#L": 2, "G#J": 2, "F#K": 3}, "id": "mod7_arith_d3_0231"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#I := L#J * F#M. F#M := 5. K#N := F#M + L#J. L#J := 2. J#K := J#I + K#N * L#J. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#I := L#J * F#M. F#M := 5. K#N := F#M + L#J. L#J := 2. J#K := J#I + K#N * L#J.", "query": "J#K", "answer": 6, "cot": "F#M = 5 -> F#M = 5\nL#J = 2 -> L#J = 2\nJ#I = L#J * F#M\n = 2 * 5\n 2 * 5 = (2 × 5) mod 7 = 3\n => J#I = 3\nK#N = F#M + L#J\n = 5 + 2\n 5 + 2 = (5 + 2) mod 7 = 0\n => K#N = 0\nJ#K = J#I + K#N * L#J\n = 3 + 0 * 2\n 3 + 0 = (3 + 0) mod 7 = 3\n 3 * 2 = (3 × 2) mod 7 = 6\n => J#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#M": 1, "L#J": 1, "J#I": 2, "K#N": 2, "J#K": 3}, "id": "mod7_arith_d3_0232"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := F#M + J#N. G#N := L#M - F#M. J#N := 3. K#N := J#N + F#M. F#M := 2. K#P := J#N - F#M. G#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := F#M + J#N. G#N := L#M - F#M. J#N := 3. K#N := J#N + F#M. F#M := 2. K#P := J#N - F#M.", "query": "G#N", "answer": 3, "cot": "J#N = 3 -> J#N = 3\nF#M = 2 -> F#M = 2\nL#M = F#M + J#N\n = 2 + 3\n 2 + 3 = (2 + 3) mod 7 = 5\n => L#M = 5\nG#N = L#M - F#M\n = 5 - 2\n 5 - 2 = (5 - 2) mod 7 = 3\n => G#N = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#N": 1, "F#M": 1, "K#N": 2, "L#M": 2, "K#P": 2, "G#N": 3}, "id": "mod7_arith_d3_0233"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := 6. H#O := 5 * 5 / K#I. H#I := G#O + E#M. E#O := 3. G#O := E#M * E#O. E#M := 2. H#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := 6. H#O := 5 * 5 / K#I. H#I := G#O + E#M. E#O := 3. G#O := E#M * E#O. E#M := 2.", "query": "H#I", "answer": 1, "cot": "E#M = 2 -> E#M = 2\nE#O = 3 -> E#O = 3\nG#O = E#M * E#O\n = 2 * 3\n 2 * 3 = (2 × 3) mod 7 = 6\n => G#O = 6\nH#I = G#O + E#M\n = 6 + 2\n 6 + 2 = (6 + 2) mod 7 = 1\n => H#I = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#I": 1, "E#M": 1, "E#O": 1, "H#O": 2, "G#O": 2, "H#I": 3}, "id": "mod7_arith_d3_0234"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := 5. F#N := L#O + J#O. H#P := 4. L#O := 3. F#J := F#N + 5 / 2. L#K := L#O * 5. J#O := 4. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := 5. F#N := L#O + J#O. H#P := 4. L#O := 3. F#J := F#N + 5 / 2. L#K := L#O * 5. J#O := 4.", "query": "F#J", "answer": 6, "cot": "L#O = 3 -> L#O = 3\nJ#O = 4 -> J#O = 4\nF#N = L#O + J#O\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => F#N = 0\nF#J = F#N + 5 / 2\n = 0 + 5 / 2\n 0 + 5 = (0 + 5) mod 7 = 5\n 5 / 2 = (5 × 2⁻¹) mod 7 = 6\n => F#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#P": 1, "L#O": 1, "K#J": 1, "J#O": 1, "L#K": 2, "F#N": 2, "F#J": 3}, "id": "mod7_arith_d3_0235"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := E#K * J#P. K#K := 1. J#L := J#P * K#K. E#K := 2. L#J := 3. J#P := 2. H#K := J#L - L#M. E#P := L#J / J#P. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := E#K * J#P. K#K := 1. J#L := J#P * K#K. E#K := 2. L#J := 3. J#P := 2. H#K := J#L - L#M. E#P := L#J / J#P.", "query": "H#K", "answer": 5, "cot": "J#P = 2 -> J#P = 2\nK#K = 1 -> K#K = 1\nE#K = 2 -> E#K = 2\nJ#L = J#P * K#K\n = 2 * 1\n 2 * 1 = (2 × 1) mod 7 = 2\n => J#L = 2\nL#M = E#K * J#P\n = 2 * 2\n 2 * 2 = (2 × 2) mod 7 = 4\n => L#M = 4\nH#K = J#L - L#M\n = 2 - 4\n 2 - 4 = (2 - 4) mod 7 = 5\n => H#K = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"L#J": 1, "J#P": 1, "K#K": 1, "E#K": 1, "J#L": 2, "E#P": 2, "L#M": 2, "H#K": 3}, "id": "mod7_arith_d3_0236"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 3. K#L := 2. K#J := K#L. I#O := F#L / K#J. F#L := 4 + J#N / K#L. G#K := K#L. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 3. K#L := 2. K#J := K#L. I#O := F#L / K#J. F#L := 4 + J#N / K#L. G#K := K#L.", "query": "I#O", "answer": 0, "cot": "K#L = 2 -> K#L = 2\nJ#N = 3 -> J#N = 3\nK#J = K#L = 2\nF#L = 4 + J#N / K#L\n = 4 + 3 / 2\n 4 + 3 = (4 + 3) mod 7 = 0\n 0 / 2 = (0 × 2⁻¹) mod 7 = 0\n => F#L = 0\nI#O = F#L / K#J\n = 0 / 2\n 0 / 2 = (0 × 2⁻¹) mod 7 = 0\n => I#O = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#L": 1, "J#N": 1, "K#J": 2, "F#L": 2, "G#K": 2, "I#O": 3}, "id": "mod7_arith_d3_0237"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := F#K * J#I. J#I := 1. F#K := 2. K#M := F#K * J#I. G#P := H#J. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := F#K * J#I. J#I := 1. F#K := 2. K#M := F#K * J#I. G#P := H#J.", "query": "G#P", "answer": 2, "cot": "F#K = 2 -> F#K = 2\nJ#I = 1 -> J#I = 1\nH#J = F#K * J#I\n = 2 * 1\n 2 * 1 = (2 × 1) mod 7 = 2\n => H#J = 2\nG#P = H#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#K": 1, "J#I": 1, "K#M": 2, "H#J": 2, "G#P": 3}, "id": "mod7_arith_d3_0238"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := L#P + F#K. G#K := I#O - F#K + 4. I#L := 3. L#P := I#L + F#K. F#K := 6. I#O := 2. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := L#P + F#K. G#K := I#O - F#K + 4. I#L := 3. L#P := I#L + F#K. F#K := 6. I#O := 2.", "query": "G#J", "answer": 1, "cot": "F#K = 6 -> F#K = 6\nI#L = 3 -> I#L = 3\nL#P = I#L + F#K\n = 3 + 6\n 3 + 6 = (3 + 6) mod 7 = 2\n => L#P = 2\nG#J = L#P + F#K\n = 2 + 6\n 2 + 6 = (2 + 6) mod 7 = 1\n => G#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#K": 1, "I#L": 1, "I#O": 1, "L#P": 2, "G#K": 2, "G#J": 3}, "id": "mod7_arith_d3_0239"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 2. F#P := G#M - E#P - J#N. G#M := 2. K#M := J#N / G#M. G#N := J#N. E#P := 5. H#L := F#P + G#N / 3. H#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 2. F#P := G#M - E#P - J#N. G#M := 2. K#M := J#N / G#M. G#N := J#N. E#P := 5. H#L := F#P + G#N / 3.", "query": "H#L", "answer": 6, "cot": "G#M = 2 -> G#M = 2\nE#P = 5 -> E#P = 5\nJ#N = 2 -> J#N = 2\nG#N = J#N = 2\nF#P = G#M - E#P - J#N\n = 2 - 5 - 2\n 2 - 5 = (2 - 5) mod 7 = 4\n 4 - 2 = (4 - 2) mod 7 = 2\n => F#P = 2\nH#L = F#P + G#N / 3\n = 2 + 2 / 3\n 2 + 2 = (2 + 2) mod 7 = 4\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => H#L = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#M": 1, "E#P": 1, "J#N": 1, "G#N": 2, "K#M": 2, "F#P": 2, "H#L": 3}, "id": "mod7_arith_d3_0240"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := F#M - K#O + G#P. G#P := 2. K#O := G#P * F#M. F#M := 2. L#I := F#M * G#P. K#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := F#M - K#O + G#P. G#P := 2. K#O := G#P * F#M. F#M := 2. L#I := F#M * G#P.", "query": "K#J", "answer": 0, "cot": "F#M = 2 -> F#M = 2\nG#P = 2 -> G#P = 2\nK#O = G#P * F#M\n = 2 * 2\n 2 * 2 = (2 × 2) mod 7 = 4\n => K#O = 4\nK#J = F#M - K#O + G#P\n = 2 - 4 + 2\n 2 - 4 = (2 - 4) mod 7 = 5\n 5 + 2 = (5 + 2) mod 7 = 0\n => K#J = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#M": 1, "G#P": 1, "L#I": 2, "K#O": 2, "K#J": 3}, "id": "mod7_arith_d3_0241"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#P := 6. K#K := J#P. G#O := I#L - 3. G#J := 1. I#O := J#P / I#L + 6. J#O := K#K / G#O / H#J. I#L := 4. H#J := 6. J#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#P := 6. K#K := J#P. G#O := I#L - 3. G#J := 1. I#O := J#P / I#L + 6. J#O := K#K / G#O / H#J. I#L := 4. H#J := 6.", "query": "J#O", "answer": 1, "cot": "H#J = 6 -> H#J = 6\nI#L = 4 -> I#L = 4\nJ#P = 6 -> J#P = 6\nG#O = I#L - 3\n = 4 - 3\n 4 - 3 = (4 - 3) mod 7 = 1\n => G#O = 1\nK#K = J#P = 6\nJ#O = K#K / G#O / H#J\n = 6 / 1 / 6\n 6 / 1 = (6 × 1⁻¹) mod 7 = 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => J#O = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"H#J": 1, "I#L": 1, "G#J": 1, "J#P": 1, "G#O": 2, "I#O": 2, "K#K": 2, "J#O": 3}, "id": "mod7_arith_d3_0242"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := 1. G#I := I#L + I#P. I#L := I#P - F#M. H#K := I#P - F#M. F#M := 6. G#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := 1. G#I := I#L + I#P. I#L := I#P - F#M. H#K := I#P - F#M. F#M := 6.", "query": "G#I", "answer": 3, "cot": "F#M = 6 -> F#M = 6\nI#P = 1 -> I#P = 1\nI#L = I#P - F#M\n = 1 - 6\n 1 - 6 = (1 - 6) mod 7 = 2\n => I#L = 2\nG#I = I#L + I#P\n = 2 + 1\n 2 + 1 = (2 + 1) mod 7 = 3\n => G#I = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#M": 1, "I#P": 1, "H#K": 2, "I#L": 2, "G#I": 3}, "id": "mod7_arith_d3_0243"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := 2 / K#M + 4. F#L := K#M - L#P. L#K := 4. L#P := L#K. K#M := 3. G#K := 1. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := 2 / K#M + 4. F#L := K#M - L#P. L#K := 4. L#P := L#K. K#M := 3. G#K := 1.", "query": "F#L", "answer": 6, "cot": "K#M = 3 -> K#M = 3\nL#K = 4 -> L#K = 4\nL#P = L#K = 4\nF#L = K#M - L#P\n = 3 - 4\n 3 - 4 = (3 - 4) mod 7 = 6\n => F#L = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#K": 1, "K#M": 1, "L#K": 1, "L#P": 2, "F#P": 2, "F#L": 3}, "id": "mod7_arith_d3_0244"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#K := G#O + I#I. E#N := 2. G#O := F#N - H#L. F#N := 5. H#L := 2. I#I := E#N / H#L. G#J := 3. K#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#K := G#O + I#I. E#N := 2. G#O := F#N - H#L. F#N := 5. H#L := 2. I#I := E#N / H#L. G#J := 3.", "query": "K#K", "answer": 4, "cot": "F#N = 5 -> F#N = 5\nH#L = 2 -> H#L = 2\nE#N = 2 -> E#N = 2\nG#O = F#N - H#L\n = 5 - 2\n 5 - 2 = (5 - 2) mod 7 = 3\n => G#O = 3\nI#I = E#N / H#L\n = 2 / 2\n 2 / 2 = (2 × 2⁻¹) mod 7 = 1\n => I#I = 1\nK#K = G#O + I#I\n = 3 + 1\n 3 + 1 = (3 + 1) mod 7 = 4\n => K#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#N": 1, "H#L": 1, "G#J": 1, "E#N": 1, "G#O": 2, "I#I": 2, "K#K": 3}, "id": "mod7_arith_d3_0245"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := 5. L#M := 3. H#P := 6 / H#K. K#N := L#M * I#I. G#O := 5. G#L := G#O * L#M + I#I. I#I := 3. H#K := G#O + H#J. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := 5. L#M := 3. H#P := 6 / H#K. K#N := L#M * I#I. G#O := 5. G#L := G#O * L#M + I#I. I#I := 3. H#K := G#O + H#J.", "query": "H#P", "answer": 2, "cot": "H#J = 5 -> H#J = 5\nG#O = 5 -> G#O = 5\nH#K = G#O + H#J\n = 5 + 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => H#K = 3\nH#P = 6 / H#K\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => H#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#J": 1, "L#M": 1, "G#O": 1, "I#I": 1, "G#L": 2, "H#K": 2, "K#N": 2, "H#P": 3}, "id": "mod7_arith_d3_0246"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := J#L - G#N. J#L := G#N / 5 + F#M. G#N := 3. I#I := F#M + G#N + 6. F#M := 4. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := J#L - G#N. J#L := G#N / 5 + F#M. G#N := 3. I#I := F#M + G#N + 6. F#M := 4.", "query": "H#J", "answer": 3, "cot": "G#N = 3 -> G#N = 3\nF#M = 4 -> F#M = 4\nJ#L = G#N / 5 + F#M\n = 3 / 5 + 4\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n 2 + 4 = (2 + 4) mod 7 = 6\n => J#L = 6\nH#J = J#L - G#N\n = 6 - 3\n 6 - 3 = (6 - 3) mod 7 = 3\n => H#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 5, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#N": 1, "F#M": 1, "I#I": 2, "J#L": 2, "H#J": 3}, "id": "mod7_arith_d3_0247"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#N := F#J / L#O. I#K := H#M. H#N := 3 / F#J / L#O. F#J := 5. L#O := 4. H#M := F#J - L#O. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#N := F#J / L#O. I#K := H#M. H#N := 3 / F#J / L#O. F#J := 5. L#O := 4. H#M := F#J - L#O.", "query": "I#K", "answer": 1, "cot": "F#J = 5 -> F#J = 5\nL#O = 4 -> L#O = 4\nH#M = F#J - L#O\n = 5 - 4\n 5 - 4 = (5 - 4) mod 7 = 1\n => H#M = 1\nI#K = H#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 6, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#J": 1, "L#O": 1, "H#N": 2, "L#N": 2, "H#M": 2, "I#K": 3}, "id": "mod7_arith_d3_0248"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := 4. G#O := 3 + G#K. G#I := 6. G#K := 1. H#K := 6. J#L := 5 + I#N. I#N := L#J * H#K. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := 4. G#O := 3 + G#K. G#I := 6. G#K := 1. H#K := 6. J#L := 5 + I#N. I#N := L#J * H#K.", "query": "J#L", "answer": 1, "cot": "L#J = 4 -> L#J = 4\nH#K = 6 -> H#K = 6\nI#N = L#J * H#K\n = 4 * 6\n 4 * 6 = (4 × 6) mod 7 = 3\n => I#N = 3\nJ#L = 5 + I#N\n = 5 + 3\n 5 + 3 = (5 + 3) mod 7 = 1\n => J#L = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 3, "achieved_depth": 3, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#J": 1, "H#K": 1, "G#K": 1, "G#I": 1, "I#N": 2, "G#O": 2, "J#L": 3}, "id": "mod7_arith_d3_0249"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := 5. F#L := 5 - J#M. J#N := 5 - I#J + I#P. J#M := I#J - I#P. K#P := 6 - G#L. F#P := J#N / 2 / 5. I#J := 2. F#I := K#P - I#P / G#L. G#L := 1. I#K := 6 - 3 / F#L. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := 5. F#L := 5 - J#M. J#N := 5 - I#J + I#P. J#M := I#J - I#P. K#P := 6 - G#L. F#P := J#N / 2 / 5. I#J := 2. F#I := K#P - I#P / G#L. G#L := 1. I#K := 6 - 3 / F#L.", "query": "I#K", "answer": 3, "cot": "I#J = 2 -> I#J = 2\nI#P = 5 -> I#P = 5\nJ#M = I#J - I#P\n = 2 - 5\n 2 - 5 = (2 - 5) mod 7 = 4\n => J#M = 4\nF#L = 5 - J#M\n = 5 - 4\n 5 - 4 = (5 - 4) mod 7 = 1\n => F#L = 1\nI#K = 6 - 3 / F#L\n = 6 - 3 / 1\n 6 - 3 = (6 - 3) mod 7 = 3\n 3 / 1 = (3 × 1⁻¹) mod 7 = 3\n => I#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#J": 1, "I#P": 1, "G#L": 1, "K#P": 2, "J#N": 2, "J#M": 2, "F#P": 3, "F#L": 3, "F#I": 3, "I#K": 4}, "id": "mod7_arith_d4_0000"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#K := H#O * J#I + I#P. G#I := J#I + G#J. I#P := H#O - G#I. F#K := 3. J#I := 3. H#O := 1. E#P := G#I. E#M := J#I + H#O. G#J := 4. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#K := H#O * J#I + I#P. G#I := J#I + G#J. I#P := H#O - G#I. F#K := 3. J#I := 3. H#O := 1. E#P := G#I. E#M := J#I + H#O. G#J := 4.", "query": "I#K", "answer": 4, "cot": "J#I = 3 -> J#I = 3\nH#O = 1 -> H#O = 1\nG#J = 4 -> G#J = 4\nG#I = J#I + G#J\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => G#I = 0\nI#P = H#O - G#I\n = 1 - 0\n 1 - 0 = (1 - 0) mod 7 = 1\n => I#P = 1\nI#K = H#O * J#I + I#P\n = 1 * 3 + 1\n 1 * 3 = (1 × 3) mod 7 = 3\n 3 + 1 = (3 + 1) mod 7 = 4\n => I#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#I": 1, "F#K": 1, "H#O": 1, "G#J": 1, "E#M": 2, "G#I": 2, "E#P": 3, "I#P": 3, "I#K": 4}, "id": "mod7_arith_d4_0001"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#K := 1. J#M := J#K - 4 * G#J. G#J := 5. H#O := J#M + 3. J#L := I#I + 6 / J#M. I#I := J#K + G#J. L#J := J#M * J#K. E#J := 6 - J#L * I#I. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#K := 1. J#M := J#K - 4 * G#J. G#J := 5. H#O := J#M + 3. J#L := I#I + 6 / J#M. I#I := J#K + G#J. L#J := J#M * J#K. E#J := 6 - J#L * I#I.", "query": "E#J", "answer": 3, "cot": "G#J = 5 -> G#J = 5\nJ#K = 1 -> J#K = 1\nI#I = J#K + G#J\n = 1 + 5\n 1 + 5 = (1 + 5) mod 7 = 6\n => I#I = 6\nJ#M = J#K - 4 * G#J\n = 1 - 4 * 5\n 1 - 4 = (1 - 4) mod 7 = 4\n 4 * 5 = (4 × 5) mod 7 = 6\n => J#M = 6\nJ#L = I#I + 6 / J#M\n = 6 + 6 / 6\n 6 + 6 = (6 + 6) mod 7 = 5\n 5 / 6 = (5 × 6⁻¹) mod 7 = 2\n => J#L = 2\nE#J = 6 - J#L * I#I\n = 6 - 2 * 6\n 6 - 2 = (6 - 2) mod 7 = 4\n 4 * 6 = (4 × 6) mod 7 = 3\n => E#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#J": 1, "J#K": 1, "I#I": 2, "J#M": 2, "J#L": 3, "H#O": 3, "L#J": 3, "E#J": 4}, "id": "mod7_arith_d4_0002"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := L#O - G#K. H#J := L#O + K#I. E#I := L#O. G#K := F#M. L#O := 2. J#L := G#K. F#J := L#O / F#M. F#M := 2. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := L#O - G#K. H#J := L#O + K#I. E#I := L#O. G#K := F#M. L#O := 2. J#L := G#K. F#J := L#O / F#M. F#M := 2.", "query": "H#J", "answer": 2, "cot": "F#M = 2 -> F#M = 2\nL#O = 2 -> L#O = 2\nG#K = F#M = 2\nK#I = L#O - G#K\n = 2 - 2\n 2 - 2 = (2 - 2) mod 7 = 0\n => K#I = 0\nH#J = L#O + K#I\n = 2 + 0\n 2 + 0 = (2 + 0) mod 7 = 2\n => H#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#M": 1, "L#O": 1, "F#J": 2, "E#I": 2, "G#K": 2, "K#I": 3, "J#L": 3, "H#J": 4}, "id": "mod7_arith_d4_0003"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#M := G#J - 4. F#M := L#M + L#O. H#P := 1. L#M := H#P / K#K. K#I := L#O * H#P / L#M. K#L := H#P. L#O := 5. F#N := 6. K#K := 6. G#J := K#K + L#M. E#M := L#O * 5 - K#K. I#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#M := G#J - 4. F#M := L#M + L#O. H#P := 1. L#M := H#P / K#K. K#I := L#O * H#P / L#M. K#L := H#P. L#O := 5. F#N := 6. K#K := 6. G#J := K#K + L#M. E#M := L#O * 5 - K#K.", "query": "I#M", "answer": 1, "cot": "K#K = 6 -> K#K = 6\nH#P = 1 -> H#P = 1\nL#M = H#P / K#K\n = 1 / 6\n 1 / 6 = (1 × 6⁻¹) mod 7 = 6\n => L#M = 6\nG#J = K#K + L#M\n = 6 + 6\n 6 + 6 = (6 + 6) mod 7 = 5\n => G#J = 5\nI#M = G#J - 4\n = 5 - 4\n 5 - 4 = (5 - 4) mod 7 = 1\n => I#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#N": 1, "K#K": 1, "H#P": 1, "L#O": 1, "L#M": 2, "E#M": 2, "K#L": 2, "G#J": 3, "F#M": 3, "K#I": 3, "I#M": 4}, "id": "mod7_arith_d4_0004"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := 6. K#J := E#N + J#J. J#M := 1. E#N := G#P / E#J. L#M := 5. I#O := K#J. E#J := 4. L#N := L#M * J#M / 6. J#J := G#P - L#M. G#J := L#N - E#J - G#P. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := 6. K#J := E#N + J#J. J#M := 1. E#N := G#P / E#J. L#M := 5. I#O := K#J. E#J := 4. L#N := L#M * J#M / 6. J#J := G#P - L#M. G#J := L#N - E#J - G#P.", "query": "I#O", "answer": 6, "cot": "E#J = 4 -> E#J = 4\nL#M = 5 -> L#M = 5\nG#P = 6 -> G#P = 6\nJ#J = G#P - L#M\n = 6 - 5\n 6 - 5 = (6 - 5) mod 7 = 1\n => J#J = 1\nE#N = G#P / E#J\n = 6 / 4\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n => E#N = 5\nK#J = E#N + J#J\n = 5 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n => K#J = 6\nI#O = K#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"E#J": 1, "L#M": 1, "G#P": 1, "J#M": 1, "J#J": 2, "E#N": 2, "L#N": 2, "K#J": 3, "G#J": 3, "I#O": 4}, "id": "mod7_arith_d4_0005"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := 4. F#M := I#N - H#K / J#J. J#J := H#K / I#N. H#J := H#K + I#N. I#O := J#J. I#N := 3. H#L := 4 / F#M. H#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := 4. F#M := I#N - H#K / J#J. J#J := H#K / I#N. H#J := H#K + I#N. I#O := J#J. I#N := 3. H#L := 4 / F#M.", "query": "H#L", "answer": 4, "cot": "H#K = 4 -> H#K = 4\nI#N = 3 -> I#N = 3\nJ#J = H#K / I#N\n = 4 / 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => J#J = 6\nF#M = I#N - H#K / J#J\n = 3 - 4 / 6\n 3 - 4 = (3 - 4) mod 7 = 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => F#M = 1\nH#L = 4 / F#M\n = 4 / 1\n 4 / 1 = (4 × 1⁻¹) mod 7 = 4\n => H#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#K": 1, "I#N": 1, "J#J": 2, "H#J": 2, "I#O": 3, "F#M": 3, "H#L": 4}, "id": "mod7_arith_d4_0006"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#O := 2. F#K := E#J / 6. E#J := E#O. I#J := 4. K#O := H#I + I#J / E#J. H#I := E#O. I#N := K#O + F#K. I#K := 3 / E#J / I#J. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#O := 2. F#K := E#J / 6. E#J := E#O. I#J := 4. K#O := H#I + I#J / E#J. H#I := E#O. I#N := K#O + F#K. I#K := 3 / E#J / I#J.", "query": "I#N", "answer": 1, "cot": "E#O = 2 -> E#O = 2\nI#J = 4 -> I#J = 4\nH#I = E#O = 2\nE#J = E#O = 2\nK#O = H#I + I#J / E#J\n = 2 + 4 / 2\n 2 + 4 = (2 + 4) mod 7 = 6\n 6 / 2 = (6 × 2⁻¹) mod 7 = 3\n => K#O = 3\nF#K = E#J / 6\n = 2 / 6\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => F#K = 5\nI#N = K#O + F#K\n = 3 + 5\n 3 + 5 = (3 + 5) mod 7 = 1\n => I#N = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 7, "num_distractors": 0, "var_layer": {"E#O": 1, "I#J": 1, "H#I": 2, "E#J": 2, "K#O": 3, "F#K": 3, "I#K": 3, "I#N": 4}, "id": "mod7_arith_d4_0007"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := K#N - 6 / L#M. F#O := I#M. G#L := 2. K#N := 4 / G#L + 2. J#L := K#N. L#M := G#J / I#M * G#L. J#K := 1. K#O := G#O. G#J := 2. I#M := 6. E#M := G#L * K#N. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := K#N - 6 / L#M. F#O := I#M. G#L := 2. K#N := 4 / G#L + 2. J#L := K#N. L#M := G#J / I#M * G#L. J#K := 1. K#O := G#O. G#J := 2. I#M := 6. E#M := G#L * K#N.", "query": "K#O", "answer": 4, "cot": "G#J = 2 -> G#J = 2\nI#M = 6 -> I#M = 6\nG#L = 2 -> G#L = 2\nL#M = G#J / I#M * G#L\n = 2 / 6 * 2\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n 5 * 2 = (5 × 2) mod 7 = 3\n => L#M = 3\nK#N = 4 / G#L + 2\n = 4 / 2 + 2\n 4 / 2 = (4 × 2⁻¹) mod 7 = 2\n 2 + 2 = (2 + 2) mod 7 = 4\n => K#N = 4\nG#O = K#N - 6 / L#M\n = 4 - 6 / 3\n 4 - 6 = (4 - 6) mod 7 = 5\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n => G#O = 4\nK#O = G#O = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 7, "num_distractors": 0, "var_layer": {"G#J": 1, "I#M": 1, "J#K": 1, "G#L": 1, "F#O": 2, "L#M": 2, "K#N": 2, "G#O": 3, "E#M": 3, "J#L": 3, "K#O": 4}, "id": "mod7_arith_d4_0008"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#N := I#O + J#J. K#N := J#K. J#K := 3. G#M := 4. J#P := J#M / 4. J#J := 4. L#K := K#N * I#O. I#O := 5. J#M := L#N. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#N := I#O + J#J. K#N := J#K. J#K := 3. G#M := 4. J#P := J#M / 4. J#J := 4. L#K := K#N * I#O. I#O := 5. J#M := L#N.", "query": "J#P", "answer": 4, "cot": "I#O = 5 -> I#O = 5\nJ#J = 4 -> J#J = 4\nL#N = I#O + J#J\n = 5 + 4\n 5 + 4 = (5 + 4) mod 7 = 2\n => L#N = 2\nJ#M = L#N = 2\nJ#P = J#M / 4\n = 2 / 4\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n => J#P = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#M": 1, "I#O": 1, "J#J": 1, "J#K": 1, "K#N": 2, "L#N": 2, "J#M": 3, "L#K": 3, "J#P": 4}, "id": "mod7_arith_d4_0009"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#O := E#J * 3. K#L := 3. F#N := E#J. E#J := K#L * I#M. I#M := 2. F#P := I#M. G#I := 6. H#P := K#O - F#N. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#O := E#J * 3. K#L := 3. F#N := E#J. E#J := K#L * I#M. I#M := 2. F#P := I#M. G#I := 6. H#P := K#O - F#N.", "query": "H#P", "answer": 5, "cot": "K#L = 3 -> K#L = 3\nI#M = 2 -> I#M = 2\nE#J = K#L * I#M\n = 3 * 2\n 3 * 2 = (3 × 2) mod 7 = 6\n => E#J = 6\nF#N = E#J = 6\nK#O = E#J * 3\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => K#O = 4\nH#P = K#O - F#N\n = 4 - 6\n 4 - 6 = (4 - 6) mod 7 = 5\n => H#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#I": 1, "K#L": 1, "I#M": 1, "F#P": 2, "E#J": 2, "F#N": 3, "K#O": 3, "H#P": 4}, "id": "mod7_arith_d4_0010"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 6 - I#K. I#K := K#P * 5. H#K := J#M + L#M - J#N. G#O := 3. L#M := G#O - K#P. K#P := 6. J#M := 2 * L#M - I#K. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 6 - I#K. I#K := K#P * 5. H#K := J#M + L#M - J#N. G#O := 3. L#M := G#O - K#P. K#P := 6. J#M := 2 * L#M - I#K.", "query": "H#K", "answer": 6, "cot": "G#O = 3 -> G#O = 3\nK#P = 6 -> K#P = 6\nL#M = G#O - K#P\n = 3 - 6\n 3 - 6 = (3 - 6) mod 7 = 4\n => L#M = 4\nI#K = K#P * 5\n = 6 * 5\n 6 * 5 = (6 × 5) mod 7 = 2\n => I#K = 2\nJ#M = 2 * L#M - I#K\n = 2 * 4 - 2\n 2 * 4 = (2 × 4) mod 7 = 1\n 1 - 2 = (1 - 2) mod 7 = 6\n => J#M = 6\nJ#N = 6 - I#K\n = 6 - 2\n 6 - 2 = (6 - 2) mod 7 = 4\n => J#N = 4\nH#K = J#M + L#M - J#N\n = 6 + 4 - 4\n 6 + 4 = (6 + 4) mod 7 = 3\n 3 - 4 = (3 - 4) mod 7 = 6\n => H#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 7, "num_distractors": 0, "var_layer": {"G#O": 1, "K#P": 1, "L#M": 2, "I#K": 2, "J#M": 3, "J#N": 3, "H#K": 4}, "id": "mod7_arith_d4_0011"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := 4. F#I := K#I + F#K. L#N := 3. F#K := 3. K#I := K#J. H#N := G#P / F#K. G#P := K#J. J#O := H#N. I#K := F#K - L#N * 3. I#L := G#P * 4 / F#K. J#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := 4. F#I := K#I + F#K. L#N := 3. F#K := 3. K#I := K#J. H#N := G#P / F#K. G#P := K#J. J#O := H#N. I#K := F#K - L#N * 3. I#L := G#P * 4 / F#K.", "query": "J#O", "answer": 6, "cot": "K#J = 4 -> K#J = 4\nF#K = 3 -> F#K = 3\nG#P = K#J = 4\nH#N = G#P / F#K\n = 4 / 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => H#N = 6\nJ#O = H#N = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#J": 1, "F#K": 1, "L#N": 1, "K#I": 2, "G#P": 2, "I#K": 2, "I#L": 3, "H#N": 3, "F#I": 3, "J#O": 4}, "id": "mod7_arith_d4_0012"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := H#K. H#K := 2. K#L := H#K / K#N. I#I := K#L. K#N := H#K * J#K. J#K := 3. E#N := K#N * E#K. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := H#K. H#K := 2. K#L := H#K / K#N. I#I := K#L. K#N := H#K * J#K. J#K := 3. E#N := K#N * E#K.", "query": "I#I", "answer": 5, "cot": "J#K = 3 -> J#K = 3\nH#K = 2 -> H#K = 2\nK#N = H#K * J#K\n = 2 * 3\n 2 * 3 = (2 × 3) mod 7 = 6\n => K#N = 6\nK#L = H#K / K#N\n = 2 / 6\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => K#L = 5\nI#I = K#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#K": 1, "H#K": 1, "E#K": 2, "K#N": 2, "K#L": 3, "E#N": 3, "I#I": 4}, "id": "mod7_arith_d4_0013"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#O := F#P - G#L / 5. F#P := 3. I#L := G#K - F#P. H#P := I#L / G#L. G#K := F#P. H#N := K#O / F#P. G#L := 5. J#L := F#P + K#O + G#L. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#O := F#P - G#L / 5. F#P := 3. I#L := G#K - F#P. H#P := I#L / G#L. G#K := F#P. H#N := K#O / F#P. G#L := 5. J#L := F#P + K#O + G#L.", "query": "H#P", "answer": 0, "cot": "F#P = 3 -> F#P = 3\nG#L = 5 -> G#L = 5\nG#K = F#P = 3\nI#L = G#K - F#P\n = 3 - 3\n 3 - 3 = (3 - 3) mod 7 = 0\n => I#L = 0\nH#P = I#L / G#L\n = 0 / 5\n 0 / 5 = (0 × 5⁻¹) mod 7 = 0\n => H#P = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#P": 1, "G#L": 1, "G#K": 2, "K#O": 2, "J#L": 3, "H#N": 3, "I#L": 3, "H#P": 4}, "id": "mod7_arith_d4_0014"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#P := L#O * K#J / I#L. L#O := I#N * L#K. K#L := I#N. I#N := L#K - F#P. F#P := 1. L#K := 1. K#J := F#P * 6. I#L := H#M * K#J / 6. H#M := F#P. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#P := L#O * K#J / I#L. L#O := I#N * L#K. K#L := I#N. I#N := L#K - F#P. F#P := 1. L#K := 1. K#J := F#P * 6. I#L := H#M * K#J / 6. H#M := F#P.", "query": "H#P", "answer": 0, "cot": "L#K = 1 -> L#K = 1\nF#P = 1 -> F#P = 1\nI#N = L#K - F#P\n = 1 - 1\n 1 - 1 = (1 - 1) mod 7 = 0\n => I#N = 0\nH#M = F#P = 1\nK#J = F#P * 6\n = 1 * 6\n 1 * 6 = (1 × 6) mod 7 = 6\n => K#J = 6\nL#O = I#N * L#K\n = 0 * 1\n 0 * 1 = (0 × 1) mod 7 = 0\n => L#O = 0\nI#L = H#M * K#J / 6\n = 1 * 6 / 6\n 1 * 6 = (1 × 6) mod 7 = 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => I#L = 1\nH#P = L#O * K#J / I#L\n = 0 * 6 / 1\n 0 * 6 = (0 × 6) mod 7 = 0\n 0 / 1 = (0 × 1⁻¹) mod 7 = 0\n => H#P = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 8, "num_distractors": 0, "var_layer": {"L#K": 1, "F#P": 1, "I#N": 2, "H#M": 2, "K#J": 2, "L#O": 3, "I#L": 3, "K#L": 3, "H#P": 4}, "id": "mod7_arith_d4_0015"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := 5. E#M := 3 * G#N - F#O. J#O := H#P. H#P := F#O / G#N. G#J := H#P - E#M. F#O := 2. L#M := 6. L#K := F#O - E#M - G#J. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := 5. E#M := 3 * G#N - F#O. J#O := H#P. H#P := F#O / G#N. G#J := H#P - E#M. F#O := 2. L#M := 6. L#K := F#O - E#M - G#J.", "query": "L#K", "answer": 3, "cot": "G#N = 5 -> G#N = 5\nF#O = 2 -> F#O = 2\nE#M = 3 * G#N - F#O\n = 3 * 5 - 2\n 3 * 5 = (3 × 5) mod 7 = 1\n 1 - 2 = (1 - 2) mod 7 = 6\n => E#M = 6\nH#P = F#O / G#N\n = 2 / 5\n 2 / 5 = (2 × 5⁻¹) mod 7 = 6\n => H#P = 6\nG#J = H#P - E#M\n = 6 - 6\n 6 - 6 = (6 - 6) mod 7 = 0\n => G#J = 0\nL#K = F#O - E#M - G#J\n = 2 - 6 - 0\n 2 - 6 = (2 - 6) mod 7 = 3\n 3 - 0 = (3 - 0) mod 7 = 3\n => L#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#N": 1, "F#O": 1, "L#M": 1, "E#M": 2, "H#P": 2, "G#J": 3, "J#O": 3, "L#K": 4}, "id": "mod7_arith_d4_0016"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#I := 1. H#I := G#J. I#M := J#I * K#L. G#M := J#M - J#I - G#J. K#J := 4. G#J := K#J * J#I * 3. F#K := 3 + J#I - K#J. K#L := G#J * 6. J#M := J#I + K#J. I#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#I := 1. H#I := G#J. I#M := J#I * K#L. G#M := J#M - J#I - G#J. K#J := 4. G#J := K#J * J#I * 3. F#K := 3 + J#I - K#J. K#L := G#J * 6. J#M := J#I + K#J.", "query": "I#M", "answer": 2, "cot": "K#J = 4 -> K#J = 4\nJ#I = 1 -> J#I = 1\nG#J = K#J * J#I * 3\n = 4 * 1 * 3\n 4 * 1 = (4 × 1) mod 7 = 4\n 4 * 3 = (4 × 3) mod 7 = 5\n => G#J = 5\nK#L = G#J * 6\n = 5 * 6\n 5 * 6 = (5 × 6) mod 7 = 2\n => K#L = 2\nI#M = J#I * K#L\n = 1 * 2\n 1 * 2 = (1 × 2) mod 7 = 2\n => I#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#J": 1, "J#I": 1, "J#M": 2, "G#J": 2, "F#K": 2, "K#L": 3, "G#M": 3, "H#I": 3, "I#M": 4}, "id": "mod7_arith_d4_0017"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#P := G#O * G#J. G#J := 4 - F#K + G#O. G#M := F#K + G#O. K#N := 5 + G#M * E#K. F#K := 6. E#K := G#J + G#O - 5. L#M := G#M. G#O := 6. K#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#P := G#O * G#J. G#J := 4 - F#K + G#O. G#M := F#K + G#O. K#N := 5 + G#M * E#K. F#K := 6. E#K := G#J + G#O - 5. L#M := G#M. G#O := 6.", "query": "K#N", "answer": 1, "cot": "F#K = 6 -> F#K = 6\nG#O = 6 -> G#O = 6\nG#M = F#K + G#O\n = 6 + 6\n 6 + 6 = (6 + 6) mod 7 = 5\n => G#M = 5\nG#J = 4 - F#K + G#O\n = 4 - 6 + 6\n 4 - 6 = (4 - 6) mod 7 = 5\n 5 + 6 = (5 + 6) mod 7 = 4\n => G#J = 4\nE#K = G#J + G#O - 5\n = 4 + 6 - 5\n 4 + 6 = (4 + 6) mod 7 = 3\n 3 - 5 = (3 - 5) mod 7 = 5\n => E#K = 5\nK#N = 5 + G#M * E#K\n = 5 + 5 * 5\n 5 + 5 = (5 + 5) mod 7 = 3\n 3 * 5 = (3 × 5) mod 7 = 1\n => K#N = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#K": 1, "G#O": 1, "G#M": 2, "G#J": 2, "K#P": 3, "L#M": 3, "E#K": 3, "K#N": 4}, "id": "mod7_arith_d4_0018"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := 6. G#N := I#N. E#O := E#L * J#I * I#N. G#K := J#I + L#M. F#M := 6. I#N := F#M + E#L / L#M. I#I := E#O. L#K := F#M + E#L. J#I := L#M. L#M := 1. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := 6. G#N := I#N. E#O := E#L * J#I * I#N. G#K := J#I + L#M. F#M := 6. I#N := F#M + E#L / L#M. I#I := E#O. L#K := F#M + E#L. J#I := L#M. L#M := 1.", "query": "I#I", "answer": 2, "cot": "L#M = 1 -> L#M = 1\nF#M = 6 -> F#M = 6\nE#L = 6 -> E#L = 6\nI#N = F#M + E#L / L#M\n = 6 + 6 / 1\n 6 + 6 = (6 + 6) mod 7 = 5\n 5 / 1 = (5 × 1⁻¹) mod 7 = 5\n => I#N = 5\nJ#I = L#M = 1\nE#O = E#L * J#I * I#N\n = 6 * 1 * 5\n 6 * 1 = (6 × 1) mod 7 = 6\n 6 * 5 = (6 × 5) mod 7 = 2\n => E#O = 2\nI#I = E#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"L#M": 1, "F#M": 1, "E#L": 1, "L#K": 2, "I#N": 2, "J#I": 2, "E#O": 3, "G#N": 3, "G#K": 3, "I#I": 4}, "id": "mod7_arith_d4_0019"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := I#N. K#K := L#N. F#K := H#P - E#J. G#K := H#L. H#P := L#N - E#J. L#N := 3. I#N := H#L. E#J := 4. H#L := L#N - E#J. J#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := I#N. K#K := L#N. F#K := H#P - E#J. G#K := H#L. H#P := L#N - E#J. L#N := 3. I#N := H#L. E#J := 4. H#L := L#N - E#J.", "query": "J#N", "answer": 6, "cot": "L#N = 3 -> L#N = 3\nE#J = 4 -> E#J = 4\nH#L = L#N - E#J\n = 3 - 4\n 3 - 4 = (3 - 4) mod 7 = 6\n => H#L = 6\nI#N = H#L = 6\nJ#N = I#N = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#N": 1, "E#J": 1, "H#P": 2, "H#L": 2, "K#K": 2, "G#K": 3, "I#N": 3, "F#K": 3, "J#N": 4}, "id": "mod7_arith_d4_0020"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#M := 3 * I#N. E#O := L#N / I#N. L#N := 3. G#L := 6. I#N := G#L. L#O := I#M. G#K := I#N * L#N * 4. J#J := 6 + G#L - L#N. L#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#M := 3 * I#N. E#O := L#N / I#N. L#N := 3. G#L := 6. I#N := G#L. L#O := I#M. G#K := I#N * L#N * 4. J#J := 6 + G#L - L#N.", "query": "L#O", "answer": 4, "cot": "G#L = 6 -> G#L = 6\nI#N = G#L = 6\nI#M = 3 * I#N\n = 3 * 6\n 3 * 6 = (3 × 6) mod 7 = 4\n => I#M = 4\nL#O = I#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#N": 1, "G#L": 1, "J#J": 2, "I#N": 2, "E#O": 3, "G#K": 3, "I#M": 3, "L#O": 4}, "id": "mod7_arith_d4_0021"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#I := L#O + K#J. H#K := F#I / L#O. K#P := K#J / L#O. L#N := K#P * F#I + L#O. L#O := 4. F#K := L#N - H#K. K#J := 5. G#I := L#O + K#P. F#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#I := L#O + K#J. H#K := F#I / L#O. K#P := K#J / L#O. L#N := K#P * F#I + L#O. L#O := 4. F#K := L#N - H#K. K#J := 5. G#I := L#O + K#P.", "query": "F#K", "answer": 6, "cot": "L#O = 4 -> L#O = 4\nK#J = 5 -> K#J = 5\nF#I = L#O + K#J\n = 4 + 5\n 4 + 5 = (4 + 5) mod 7 = 2\n => F#I = 2\nK#P = K#J / L#O\n = 5 / 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => K#P = 3\nL#N = K#P * F#I + L#O\n = 3 * 2 + 4\n 3 * 2 = (3 × 2) mod 7 = 6\n 6 + 4 = (6 + 4) mod 7 = 3\n => L#N = 3\nH#K = F#I / L#O\n = 2 / 4\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n => H#K = 4\nF#K = L#N - H#K\n = 3 - 4\n 3 - 4 = (3 - 4) mod 7 = 6\n => F#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 7, "num_distractors": 0, "var_layer": {"L#O": 1, "K#J": 1, "F#I": 2, "K#P": 2, "L#N": 3, "G#I": 3, "H#K": 3, "F#K": 4}, "id": "mod7_arith_d4_0022"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := H#P. E#I := K#O. J#L := J#O. J#O := K#O. K#O := 6. E#N := 4. L#P := L#M. H#P := L#P + E#I. E#O := E#I. L#M := 2. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := H#P. E#I := K#O. J#L := J#O. J#O := K#O. K#O := 6. E#N := 4. L#P := L#M. H#P := L#P + E#I. E#O := E#I. L#M := 2.", "query": "I#J", "answer": 1, "cot": "L#M = 2 -> L#M = 2\nK#O = 6 -> K#O = 6\nL#P = L#M = 2\nE#I = K#O = 6\nH#P = L#P + E#I\n = 2 + 6\n 2 + 6 = (2 + 6) mod 7 = 1\n => H#P = 1\nI#J = H#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"L#M": 1, "K#O": 1, "E#N": 1, "L#P": 2, "J#O": 2, "E#I": 2, "E#O": 3, "J#L": 3, "H#P": 3, "I#J": 4}, "id": "mod7_arith_d4_0023"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := 2 - H#I. K#L := I#I * 3. E#L := K#O - J#M. K#K := H#I. I#I := K#O / J#M. K#P := H#I + K#L. K#O := 5. J#M := 5. H#I := 4 / J#M * K#O. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := 2 - H#I. K#L := I#I * 3. E#L := K#O - J#M. K#K := H#I. I#I := K#O / J#M. K#P := H#I + K#L. K#O := 5. J#M := 5. H#I := 4 / J#M * K#O.", "query": "K#P", "answer": 0, "cot": "J#M = 5 -> J#M = 5\nK#O = 5 -> K#O = 5\nI#I = K#O / J#M\n = 5 / 5\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n => I#I = 1\nH#I = 4 / J#M * K#O\n = 4 / 5 * 5\n 4 / 5 = (4 × 5⁻¹) mod 7 = 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => H#I = 4\nK#L = I#I * 3\n = 1 * 3\n 1 * 3 = (1 × 3) mod 7 = 3\n => K#L = 3\nK#P = H#I + K#L\n = 4 + 3\n 4 + 3 = (4 + 3) mod 7 = 0\n => K#P = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#M": 1, "K#O": 1, "I#I": 2, "E#L": 2, "H#I": 2, "K#K": 3, "K#L": 3, "K#J": 3, "K#P": 4}, "id": "mod7_arith_d4_0024"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := E#O + I#L * J#N. J#N := 2. I#L := J#N + F#N - L#N. H#L := G#N * 3. H#I := I#L / F#N - I#P. L#N := 4. F#N := 1. I#P := J#N - L#N. G#N := I#L * 4 * E#O. E#O := 1. H#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := E#O + I#L * J#N. J#N := 2. I#L := J#N + F#N - L#N. H#L := G#N * 3. H#I := I#L / F#N - I#P. L#N := 4. F#N := 1. I#P := J#N - L#N. G#N := I#L * 4 * E#O. E#O := 1.", "query": "H#L", "answer": 2, "cot": "F#N = 1 -> F#N = 1\nJ#N = 2 -> J#N = 2\nL#N = 4 -> L#N = 4\nE#O = 1 -> E#O = 1\nI#L = J#N + F#N - L#N\n = 2 + 1 - 4\n 2 + 1 = (2 + 1) mod 7 = 3\n 3 - 4 = (3 - 4) mod 7 = 6\n => I#L = 6\nG#N = I#L * 4 * E#O\n = 6 * 4 * 1\n 6 * 4 = (6 × 4) mod 7 = 3\n 3 * 1 = (3 × 1) mod 7 = 3\n => G#N = 3\nH#L = G#N * 3\n = 3 * 3\n 3 * 3 = (3 × 3) mod 7 = 2\n => H#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"F#N": 1, "J#N": 1, "L#N": 1, "E#O": 1, "I#L": 2, "I#P": 2, "G#N": 3, "I#J": 3, "H#I": 3, "H#L": 4}, "id": "mod7_arith_d4_0025"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#J := E#O / 2. K#J := 6. E#O := K#O * G#M / H#K. G#O := G#M. H#K := 6. G#M := H#K - 4. E#K := 5. K#O := 5 / 3 - H#K. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#J := E#O / 2. K#J := 6. E#O := K#O * G#M / H#K. G#O := G#M. H#K := 6. G#M := H#K - 4. E#K := 5. K#O := 5 / 3 - H#K.", "query": "E#J", "answer": 2, "cot": "H#K = 6 -> H#K = 6\nG#M = H#K - 4\n = 6 - 4\n 6 - 4 = (6 - 4) mod 7 = 2\n => G#M = 2\nK#O = 5 / 3 - H#K\n = 5 / 3 - 6\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n 4 - 6 = (4 - 6) mod 7 = 5\n => K#O = 5\nE#O = K#O * G#M / H#K\n = 5 * 2 / 6\n 5 * 2 = (5 × 2) mod 7 = 3\n 3 / 6 = (3 × 6⁻¹) mod 7 = 4\n => E#O = 4\nE#J = E#O / 2\n = 4 / 2\n 4 / 2 = (4 × 2⁻¹) mod 7 = 2\n => E#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"E#K": 1, "K#J": 1, "H#K": 1, "G#M": 2, "K#O": 2, "E#O": 3, "G#O": 3, "E#J": 4}, "id": "mod7_arith_d4_0026"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := 1. G#L := L#L. H#K := E#J. E#J := 2. L#L := J#K. F#L := F#P + H#K + J#K. J#K := F#P * E#J. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := 1. G#L := L#L. H#K := E#J. E#J := 2. L#L := J#K. F#L := F#P + H#K + J#K. J#K := F#P * E#J.", "query": "G#L", "answer": 2, "cot": "E#J = 2 -> E#J = 2\nF#P = 1 -> F#P = 1\nJ#K = F#P * E#J\n = 1 * 2\n 1 * 2 = (1 × 2) mod 7 = 2\n => J#K = 2\nL#L = J#K = 2\nG#L = L#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"E#J": 1, "F#P": 1, "J#K": 2, "H#K": 2, "F#L": 3, "L#L": 3, "G#L": 4}, "id": "mod7_arith_d4_0027"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := K#N - 4 - G#J. H#M := I#P / H#J. G#J := H#I + I#P - H#M. K#N := H#J * H#M. I#P := 2. H#I := H#J - I#P. H#J := 4. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := K#N - 4 - G#J. H#M := I#P / H#J. G#J := H#I + I#P - H#M. K#N := H#J * H#M. I#P := 2. H#I := H#J - I#P. H#J := 4.", "query": "L#I", "answer": 5, "cot": "H#J = 4 -> H#J = 4\nI#P = 2 -> I#P = 2\nH#I = H#J - I#P\n = 4 - 2\n 4 - 2 = (4 - 2) mod 7 = 2\n => H#I = 2\nH#M = I#P / H#J\n = 2 / 4\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n => H#M = 4\nK#N = H#J * H#M\n = 4 * 4\n 4 * 4 = (4 × 4) mod 7 = 2\n => K#N = 2\nG#J = H#I + I#P - H#M\n = 2 + 2 - 4\n 2 + 2 = (2 + 2) mod 7 = 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => G#J = 0\nL#I = K#N - 4 - G#J\n = 2 - 4 - 0\n 2 - 4 = (2 - 4) mod 7 = 5\n 5 - 0 = (5 - 0) mod 7 = 5\n => L#I = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 7, "num_distractors": 0, "var_layer": {"H#J": 1, "I#P": 1, "H#I": 2, "H#M": 2, "K#N": 3, "G#J": 3, "L#I": 4}, "id": "mod7_arith_d4_0028"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := E#P * G#I. E#P := E#J. L#M := I#I. H#N := I#O. E#J := 1. F#O := F#L / H#N + E#P. F#L := 6. I#O := 6. G#I := E#J - F#L. G#M := G#I + F#L. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := E#P * G#I. E#P := E#J. L#M := I#I. H#N := I#O. E#J := 1. F#O := F#L / H#N + E#P. F#L := 6. I#O := 6. G#I := E#J - F#L. G#M := G#I + F#L.", "query": "L#M", "answer": 2, "cot": "E#J = 1 -> E#J = 1\nF#L = 6 -> F#L = 6\nE#P = E#J = 1\nG#I = E#J - F#L\n = 1 - 6\n 1 - 6 = (1 - 6) mod 7 = 2\n => G#I = 2\nI#I = E#P * G#I\n = 1 * 2\n 1 * 2 = (1 × 2) mod 7 = 2\n => I#I = 2\nL#M = I#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#J": 1, "F#L": 1, "I#O": 1, "H#N": 2, "E#P": 2, "G#I": 2, "G#M": 3, "F#O": 3, "I#I": 3, "L#M": 4}, "id": "mod7_arith_d4_0029"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#P := E#K + 2. F#O := 5. L#L := F#O + G#K. E#N := G#N * E#K. E#L := E#N * J#P. E#K := F#O + G#K. G#K := 5. K#N := L#L - G#N. G#N := 2 - F#O / G#K. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#P := E#K + 2. F#O := 5. L#L := F#O + G#K. E#N := G#N * E#K. E#L := E#N * J#P. E#K := F#O + G#K. G#K := 5. K#N := L#L - G#N. G#N := 2 - F#O / G#K.", "query": "E#L", "answer": 5, "cot": "F#O = 5 -> F#O = 5\nG#K = 5 -> G#K = 5\nG#N = 2 - F#O / G#K\n = 2 - 5 / 5\n 2 - 5 = (2 - 5) mod 7 = 4\n 4 / 5 = (4 × 5⁻¹) mod 7 = 5\n => G#N = 5\nE#K = F#O + G#K\n = 5 + 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => E#K = 3\nJ#P = E#K + 2\n = 3 + 2\n 3 + 2 = (3 + 2) mod 7 = 5\n => J#P = 5\nE#N = G#N * E#K\n = 5 * 3\n 5 * 3 = (5 × 3) mod 7 = 1\n => E#N = 1\nE#L = E#N * J#P\n = 1 * 5\n 1 * 5 = (1 × 5) mod 7 = 5\n => E#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"F#O": 1, "G#K": 1, "G#N": 2, "L#L": 2, "E#K": 2, "K#N": 3, "J#P": 3, "E#N": 3, "E#L": 4}, "id": "mod7_arith_d4_0030"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := 2. F#N := 2. G#O := F#L. E#M := H#I. I#P := F#N * F#L. I#M := 3 + G#O. J#J := 2 * F#N. H#I := I#P + F#L / 6. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := 2. F#N := 2. G#O := F#L. E#M := H#I. I#P := F#N * F#L. I#M := 3 + G#O. J#J := 2 * F#N. H#I := I#P + F#L / 6.", "query": "E#M", "answer": 1, "cot": "F#N = 2 -> F#N = 2\nF#L = 2 -> F#L = 2\nI#P = F#N * F#L\n = 2 * 2\n 2 * 2 = (2 × 2) mod 7 = 4\n => I#P = 4\nH#I = I#P + F#L / 6\n = 4 + 2 / 6\n 4 + 2 = (4 + 2) mod 7 = 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => H#I = 1\nE#M = H#I = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#N": 1, "F#L": 1, "J#J": 2, "I#P": 2, "G#O": 2, "H#I": 3, "I#M": 3, "E#M": 4}, "id": "mod7_arith_d4_0031"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#O := I#M / I#J. E#K := J#M * I#I. I#I := G#N. J#M := 4. I#J := J#J + I#I / 6. E#L := J#J * G#N. I#M := J#M - G#N. G#N := 4. J#J := 1. F#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#O := I#M / I#J. E#K := J#M * I#I. I#I := G#N. J#M := 4. I#J := J#J + I#I / 6. E#L := J#J * G#N. I#M := J#M - G#N. G#N := 4. J#J := 1.", "query": "F#O", "answer": 0, "cot": "J#J = 1 -> J#J = 1\nJ#M = 4 -> J#M = 4\nG#N = 4 -> G#N = 4\nI#I = G#N = 4\nI#M = J#M - G#N\n = 4 - 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => I#M = 0\nI#J = J#J + I#I / 6\n = 1 + 4 / 6\n 1 + 4 = (1 + 4) mod 7 = 5\n 5 / 6 = (5 × 6⁻¹) mod 7 = 2\n => I#J = 2\nF#O = I#M / I#J\n = 0 / 2\n 0 / 2 = (0 × 2⁻¹) mod 7 = 0\n => F#O = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"J#J": 1, "J#M": 1, "G#N": 1, "I#I": 2, "E#L": 2, "I#M": 2, "E#K": 3, "I#J": 3, "F#O": 4}, "id": "mod7_arith_d4_0032"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := K#I - L#N. L#L := K#I - L#M. K#N := 6. K#L := I#J + E#N. L#N := 2. I#J := E#N * K#I. G#K := 3 * I#N. K#I := 3. L#M := 3. I#N := L#N / K#I. K#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := K#I - L#N. L#L := K#I - L#M. K#N := 6. K#L := I#J + E#N. L#N := 2. I#J := E#N * K#I. G#K := 3 * I#N. K#I := 3. L#M := 3. I#N := L#N / K#I.", "query": "K#L", "answer": 4, "cot": "K#I = 3 -> K#I = 3\nL#N = 2 -> L#N = 2\nE#N = K#I - L#N\n = 3 - 2\n 3 - 2 = (3 - 2) mod 7 = 1\n => E#N = 1\nI#J = E#N * K#I\n = 1 * 3\n 1 * 3 = (1 × 3) mod 7 = 3\n => I#J = 3\nK#L = I#J + E#N\n = 3 + 1\n 3 + 1 = (3 + 1) mod 7 = 4\n => K#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#M": 1, "K#N": 1, "K#I": 1, "L#N": 1, "I#N": 2, "L#L": 2, "E#N": 2, "G#K": 3, "I#J": 3, "K#L": 4}, "id": "mod7_arith_d4_0033"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := 5. L#M := 6. F#O := 2. G#I := E#K - 4. L#L := 6 * G#I - K#L. E#K := 6. K#L := F#O + E#K * L#M. I#O := L#J / K#I. L#J := L#M. E#J := 6 / L#L - K#I. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := 5. L#M := 6. F#O := 2. G#I := E#K - 4. L#L := 6 * G#I - K#L. E#K := 6. K#L := F#O + E#K * L#M. I#O := L#J / K#I. L#J := L#M. E#J := 6 / L#L - K#I.", "query": "E#J", "answer": 3, "cot": "K#I = 5 -> K#I = 5\nF#O = 2 -> F#O = 2\nL#M = 6 -> L#M = 6\nE#K = 6 -> E#K = 6\nK#L = F#O + E#K * L#M\n = 2 + 6 * 6\n 2 + 6 = (2 + 6) mod 7 = 1\n 1 * 6 = (1 × 6) mod 7 = 6\n => K#L = 6\nG#I = E#K - 4\n = 6 - 4\n 6 - 4 = (6 - 4) mod 7 = 2\n => G#I = 2\nL#L = 6 * G#I - K#L\n = 6 * 2 - 6\n 6 * 2 = (6 × 2) mod 7 = 5\n 5 - 6 = (5 - 6) mod 7 = 6\n => L#L = 6\nE#J = 6 / L#L - K#I\n = 6 / 6 - 5\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n 1 - 5 = (1 - 5) mod 7 = 3\n => E#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 8, "num_distractors": 0, "var_layer": {"K#I": 1, "F#O": 1, "L#M": 1, "E#K": 1, "L#J": 2, "K#L": 2, "G#I": 2, "I#O": 3, "L#L": 3, "E#J": 4}, "id": "mod7_arith_d4_0034"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#K := J#N. J#P := 4. E#N := 4 - G#J + J#N. G#J := 2. H#P := J#J - G#J. J#J := G#K. E#K := J#N + G#J. J#N := 5. K#O := E#K * E#N. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#K := J#N. J#P := 4. E#N := 4 - G#J + J#N. G#J := 2. H#P := J#J - G#J. J#J := G#K. E#K := J#N + G#J. J#N := 5. K#O := E#K * E#N.", "query": "H#P", "answer": 3, "cot": "G#J = 2 -> G#J = 2\nJ#N = 5 -> J#N = 5\nG#K = J#N = 5\nJ#J = G#K = 5\nH#P = J#J - G#J\n = 5 - 2\n 5 - 2 = (5 - 2) mod 7 = 3\n => H#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#J": 1, "J#N": 1, "J#P": 1, "E#N": 2, "E#K": 2, "G#K": 2, "K#O": 3, "J#J": 3, "H#P": 4}, "id": "mod7_arith_d4_0035"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := 1. J#J := 4. J#I := J#J - I#P / H#P. I#I := G#I + L#O. I#O := L#O * I#I. L#N := J#I. H#P := 2. L#O := H#P / J#J * 2. G#I := H#P / 6. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := 1. J#J := 4. J#I := J#J - I#P / H#P. I#I := G#I + L#O. I#O := L#O * I#I. L#N := J#I. H#P := 2. L#O := H#P / J#J * 2. G#I := H#P / 6.", "query": "I#O", "answer": 6, "cot": "J#J = 4 -> J#J = 4\nH#P = 2 -> H#P = 2\nG#I = H#P / 6\n = 2 / 6\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => G#I = 5\nL#O = H#P / J#J * 2\n = 2 / 4 * 2\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n 4 * 2 = (4 × 2) mod 7 = 1\n => L#O = 1\nI#I = G#I + L#O\n = 5 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n => I#I = 6\nI#O = L#O * I#I\n = 1 * 6\n 1 * 6 = (1 × 6) mod 7 = 6\n => I#O = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#J": 1, "H#P": 1, "I#P": 1, "G#I": 2, "L#O": 2, "J#I": 2, "I#I": 3, "L#N": 3, "I#O": 4}, "id": "mod7_arith_d4_0036"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := H#L. L#P := E#L - I#M * H#O. F#O := J#O + E#L - F#K. J#O := 3. H#L := F#K * J#O. F#K := 6. I#M := F#O + H#L. E#L := 3. L#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := H#L. L#P := E#L - I#M * H#O. F#O := J#O + E#L - F#K. J#O := 3. H#L := F#K * J#O. F#K := 6. I#M := F#O + H#L. E#L := 3.", "query": "L#P", "answer": 3, "cot": "J#O = 3 -> J#O = 3\nF#K = 6 -> F#K = 6\nE#L = 3 -> E#L = 3\nF#O = J#O + E#L - F#K\n = 3 + 3 - 6\n 3 + 3 = (3 + 3) mod 7 = 6\n 6 - 6 = (6 - 6) mod 7 = 0\n => F#O = 0\nH#L = F#K * J#O\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => H#L = 4\nH#O = H#L = 4\nI#M = F#O + H#L\n = 0 + 4\n 0 + 4 = (0 + 4) mod 7 = 4\n => I#M = 4\nL#P = E#L - I#M * H#O\n = 3 - 4 * 4\n 3 - 4 = (3 - 4) mod 7 = 6\n 6 * 4 = (6 × 4) mod 7 = 3\n => L#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 8, "num_distractors": 0, "var_layer": {"J#O": 1, "F#K": 1, "E#L": 1, "F#O": 2, "H#L": 2, "H#O": 3, "I#M": 3, "L#P": 4}, "id": "mod7_arith_d4_0037"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#M := I#J / K#J. I#J := 5. L#L := 2 / K#J * E#I. K#J := 4. I#P := 6. E#I := I#J + J#M. E#K := J#M - F#O - I#J. L#I := 6 * 3 - I#J. F#O := K#J - I#J. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#M := I#J / K#J. I#J := 5. L#L := 2 / K#J * E#I. K#J := 4. I#P := 6. E#I := I#J + J#M. E#K := J#M - F#O - I#J. L#I := 6 * 3 - I#J. F#O := K#J - I#J.", "query": "L#L", "answer": 4, "cot": "K#J = 4 -> K#J = 4\nI#J = 5 -> I#J = 5\nJ#M = I#J / K#J\n = 5 / 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => J#M = 3\nE#I = I#J + J#M\n = 5 + 3\n 5 + 3 = (5 + 3) mod 7 = 1\n => E#I = 1\nL#L = 2 / K#J * E#I\n = 2 / 4 * 1\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n 4 * 1 = (4 × 1) mod 7 = 4\n => L#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#P": 1, "K#J": 1, "I#J": 1, "F#O": 2, "J#M": 2, "L#I": 2, "E#I": 3, "E#K": 3, "L#L": 4}, "id": "mod7_arith_d4_0038"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := K#O - E#N. E#N := K#O - K#N. I#M := K#N / E#N. L#I := 6 + K#N + K#O. K#P := E#N. J#N := K#P * 4 * I#M. K#N := 5. K#O := 4. J#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := K#O - E#N. E#N := K#O - K#N. I#M := K#N / E#N. L#I := 6 + K#N + K#O. K#P := E#N. J#N := K#P * 4 * I#M. K#N := 5. K#O := 4.", "query": "J#N", "answer": 6, "cot": "K#O = 4 -> K#O = 4\nK#N = 5 -> K#N = 5\nE#N = K#O - K#N\n = 4 - 5\n 4 - 5 = (4 - 5) mod 7 = 6\n => E#N = 6\nI#M = K#N / E#N\n = 5 / 6\n 5 / 6 = (5 × 6⁻¹) mod 7 = 2\n => I#M = 2\nK#P = E#N = 6\nJ#N = K#P * 4 * I#M\n = 6 * 4 * 2\n 6 * 4 = (6 × 4) mod 7 = 3\n 3 * 2 = (3 × 2) mod 7 = 6\n => J#N = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#O": 1, "K#N": 1, "L#I": 2, "E#N": 2, "I#M": 3, "K#P": 3, "F#K": 3, "J#N": 4}, "id": "mod7_arith_d4_0039"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#J := L#O / G#P + E#K. G#K := 6 * G#P + F#P. F#P := 2. I#N := I#M * G#K. E#K := F#P. I#M := 5 + L#O. G#P := 1. L#O := 4 / F#P. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#J := L#O / G#P + E#K. G#K := 6 * G#P + F#P. F#P := 2. I#N := I#M * G#K. E#K := F#P. I#M := 5 + L#O. G#P := 1. L#O := 4 / F#P.", "query": "I#N", "answer": 0, "cot": "G#P = 1 -> G#P = 1\nF#P = 2 -> F#P = 2\nG#K = 6 * G#P + F#P\n = 6 * 1 + 2\n 6 * 1 = (6 × 1) mod 7 = 6\n 6 + 2 = (6 + 2) mod 7 = 1\n => G#K = 1\nL#O = 4 / F#P\n = 4 / 2\n 4 / 2 = (4 × 2⁻¹) mod 7 = 2\n => L#O = 2\nI#M = 5 + L#O\n = 5 + 2\n 5 + 2 = (5 + 2) mod 7 = 0\n => I#M = 0\nI#N = I#M * G#K\n = 0 * 1\n 0 * 1 = (0 × 1) mod 7 = 0\n => I#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#P": 1, "F#P": 1, "G#K": 2, "E#K": 2, "L#O": 2, "I#M": 3, "E#J": 3, "I#N": 4}, "id": "mod7_arith_d4_0040"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := L#I / F#J - 5. I#K := F#M * K#J / J#N. F#M := K#J / F#J. K#K := 5 * L#I * J#J. K#J := 4. L#L := 4. F#O := L#I * 6. F#J := 6. J#J := 2. L#I := K#J. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := L#I / F#J - 5. I#K := F#M * K#J / J#N. F#M := K#J / F#J. K#K := 5 * L#I * J#J. K#J := 4. L#L := 4. F#O := L#I * 6. F#J := 6. J#J := 2. L#I := K#J.", "query": "I#K", "answer": 1, "cot": "F#J = 6 -> F#J = 6\nK#J = 4 -> K#J = 4\nL#I = K#J = 4\nF#M = K#J / F#J\n = 4 / 6\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n => F#M = 3\nJ#N = L#I / F#J - 5\n = 4 / 6 - 5\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n 3 - 5 = (3 - 5) mod 7 = 5\n => J#N = 5\nI#K = F#M * K#J / J#N\n = 3 * 4 / 5\n 3 * 4 = (3 × 4) mod 7 = 5\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n => I#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"L#L": 1, "F#J": 1, "K#J": 1, "J#J": 1, "L#I": 2, "F#M": 2, "F#O": 3, "J#N": 3, "K#K": 3, "I#K": 4}, "id": "mod7_arith_d4_0041"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := 3. L#I := H#O / H#N. E#M := 6. K#M := 1. H#O := J#L - G#O. J#L := E#N - K#M. H#N := G#O - E#N. G#O := K#M + 6. E#N := 6. J#N := G#O + F#L. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := 3. L#I := H#O / H#N. E#M := 6. K#M := 1. H#O := J#L - G#O. J#L := E#N - K#M. H#N := G#O - E#N. G#O := K#M + 6. E#N := 6. J#N := G#O + F#L.", "query": "L#I", "answer": 5, "cot": "K#M = 1 -> K#M = 1\nE#N = 6 -> E#N = 6\nJ#L = E#N - K#M\n = 6 - 1\n 6 - 1 = (6 - 1) mod 7 = 5\n => J#L = 5\nG#O = K#M + 6\n = 1 + 6\n 1 + 6 = (1 + 6) mod 7 = 0\n => G#O = 0\nH#O = J#L - G#O\n = 5 - 0\n 5 - 0 = (5 - 0) mod 7 = 5\n => H#O = 5\nH#N = G#O - E#N\n = 0 - 6\n 0 - 6 = (0 - 6) mod 7 = 1\n => H#N = 1\nL#I = H#O / H#N\n = 5 / 1\n 5 / 1 = (5 × 1⁻¹) mod 7 = 5\n => L#I = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"K#M": 1, "F#L": 1, "E#N": 1, "E#M": 1, "J#L": 2, "G#O": 2, "H#O": 3, "H#N": 3, "J#N": 3, "L#I": 4}, "id": "mod7_arith_d4_0042"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#K := 1. H#I := H#P. G#L := 5. L#N := 6 + J#O * J#M. J#M := G#K - G#L. H#P := G#L. J#O := 1. I#O := J#M / G#K + L#N. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#K := 1. H#I := H#P. G#L := 5. L#N := 6 + J#O * J#M. J#M := G#K - G#L. H#P := G#L. J#O := 1. I#O := J#M / G#K + L#N.", "query": "I#O", "answer": 3, "cot": "G#K = 1 -> G#K = 1\nG#L = 5 -> G#L = 5\nJ#O = 1 -> J#O = 1\nJ#M = G#K - G#L\n = 1 - 5\n 1 - 5 = (1 - 5) mod 7 = 3\n => J#M = 3\nL#N = 6 + J#O * J#M\n = 6 + 1 * 3\n 6 + 1 = (6 + 1) mod 7 = 0\n 0 * 3 = (0 × 3) mod 7 = 0\n => L#N = 0\nI#O = J#M / G#K + L#N\n = 3 / 1 + 0\n 3 / 1 = (3 × 1⁻¹) mod 7 = 3\n 3 + 0 = (3 + 0) mod 7 = 3\n => I#O = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#K": 1, "G#L": 1, "J#O": 1, "J#M": 2, "H#P": 2, "H#I": 3, "L#N": 3, "I#O": 4}, "id": "mod7_arith_d4_0043"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#P := 4 / 3 / I#P. L#P := 5. J#N := K#O + L#P. F#M := 3 - J#P / I#J. K#O := 4. I#P := J#P + K#O. J#P := L#P. I#J := L#P - K#O. K#M := J#N. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#P := 4 / 3 / I#P. L#P := 5. J#N := K#O + L#P. F#M := 3 - J#P / I#J. K#O := 4. I#P := J#P + K#O. J#P := L#P. I#J := L#P - K#O. K#M := J#N.", "query": "H#P", "answer": 3, "cot": "L#P = 5 -> L#P = 5\nK#O = 4 -> K#O = 4\nJ#P = L#P = 5\nI#P = J#P + K#O\n = 5 + 4\n 5 + 4 = (5 + 4) mod 7 = 2\n => I#P = 2\nH#P = 4 / 3 / I#P\n = 4 / 3 / 2\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n 6 / 2 = (6 × 2⁻¹) mod 7 = 3\n => H#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#P": 1, "K#O": 1, "I#J": 2, "J#N": 2, "J#P": 2, "F#M": 3, "K#M": 3, "I#P": 3, "H#P": 4}, "id": "mod7_arith_d4_0044"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := L#N. I#P := 3. H#N := 6. K#M := 1. E#J := L#N. L#N := H#N / I#P. H#P := H#I - E#J. H#I := L#N - 5 / J#N. J#N := H#N. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := L#N. I#P := 3. H#N := 6. K#M := 1. E#J := L#N. L#N := H#N / I#P. H#P := H#I - E#J. H#I := L#N - 5 / J#N. J#N := H#N.", "query": "H#P", "answer": 1, "cot": "I#P = 3 -> I#P = 3\nH#N = 6 -> H#N = 6\nJ#N = H#N = 6\nL#N = H#N / I#P\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => L#N = 2\nE#J = L#N = 2\nH#I = L#N - 5 / J#N\n = 2 - 5 / 6\n 2 - 5 = (2 - 5) mod 7 = 4\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n => H#I = 3\nH#P = H#I - E#J\n = 3 - 2\n 3 - 2 = (3 - 2) mod 7 = 1\n => H#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"K#M": 1, "I#P": 1, "H#N": 1, "J#N": 2, "L#N": 2, "E#J": 3, "F#N": 3, "H#I": 3, "H#P": 4}, "id": "mod7_arith_d4_0045"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#P := I#N - I#M + K#P. G#K := L#P * H#P. L#P := 6. I#N := L#P. L#K := G#K - 3. K#P := 1. E#I := G#N - I#M - L#P. H#L := 3 - I#N - G#N. H#P := L#P * G#N. G#N := 4. I#M := 3. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#P := I#N - I#M + K#P. G#K := L#P * H#P. L#P := 6. I#N := L#P. L#K := G#K - 3. K#P := 1. E#I := G#N - I#M - L#P. H#L := 3 - I#N - G#N. H#P := L#P * G#N. G#N := 4. I#M := 3.", "query": "L#K", "answer": 1, "cot": "G#N = 4 -> G#N = 4\nL#P = 6 -> L#P = 6\nH#P = L#P * G#N\n = 6 * 4\n 6 * 4 = (6 × 4) mod 7 = 3\n => H#P = 3\nG#K = L#P * H#P\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => G#K = 4\nL#K = G#K - 3\n = 4 - 3\n 4 - 3 = (4 - 3) mod 7 = 1\n => L#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#M": 1, "K#P": 1, "G#N": 1, "L#P": 1, "H#P": 2, "I#N": 2, "E#I": 2, "E#P": 3, "H#L": 3, "G#K": 3, "L#K": 4}, "id": "mod7_arith_d4_0046"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := L#P - H#I. E#P := H#I * F#K. H#I := 1. F#K := L#P * 6. L#P := 3. F#I := 2 + K#J. K#J := F#K - 2. F#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := L#P - H#I. E#P := H#I * F#K. H#I := 1. F#K := L#P * 6. L#P := 3. F#I := 2 + K#J. K#J := F#K - 2.", "query": "F#I", "answer": 4, "cot": "L#P = 3 -> L#P = 3\nF#K = L#P * 6\n = 3 * 6\n 3 * 6 = (3 × 6) mod 7 = 4\n => F#K = 4\nK#J = F#K - 2\n = 4 - 2\n 4 - 2 = (4 - 2) mod 7 = 2\n => K#J = 2\nF#I = 2 + K#J\n = 2 + 2\n 2 + 2 = (2 + 2) mod 7 = 4\n => F#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#P": 1, "H#I": 1, "F#N": 2, "F#K": 2, "E#P": 3, "K#J": 3, "F#I": 4}, "id": "mod7_arith_d4_0047"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := K#J - K#P / H#J. L#N := 5. F#M := K#P - K#J. K#J := H#J - L#N + L#J. K#P := 2 - L#N. L#J := 2. F#J := F#P / K#J * L#J. H#J := 2. K#M := K#J / H#J + 4. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := K#J - K#P / H#J. L#N := 5. F#M := K#P - K#J. K#J := H#J - L#N + L#J. K#P := 2 - L#N. L#J := 2. F#J := F#P / K#J * L#J. H#J := 2. K#M := K#J / H#J + 4.", "query": "F#J", "answer": 5, "cot": "H#J = 2 -> H#J = 2\nL#N = 5 -> L#N = 5\nL#J = 2 -> L#J = 2\nK#J = H#J - L#N + L#J\n = 2 - 5 + 2\n 2 - 5 = (2 - 5) mod 7 = 4\n 4 + 2 = (4 + 2) mod 7 = 6\n => K#J = 6\nK#P = 2 - L#N\n = 2 - 5\n 2 - 5 = (2 - 5) mod 7 = 4\n => K#P = 4\nF#P = K#J - K#P / H#J\n = 6 - 4 / 2\n 6 - 4 = (6 - 4) mod 7 = 2\n 2 / 2 = (2 × 2⁻¹) mod 7 = 1\n => F#P = 1\nF#J = F#P / K#J * L#J\n = 1 / 6 * 2\n 1 / 6 = (1 × 6⁻¹) mod 7 = 6\n 6 * 2 = (6 × 2) mod 7 = 5\n => F#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"H#J": 1, "L#N": 1, "L#J": 1, "K#J": 2, "K#P": 2, "F#P": 3, "K#M": 3, "F#M": 3, "F#J": 4}, "id": "mod7_arith_d4_0048"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#N := E#M * G#L. E#L := H#J - G#L + G#K. L#J := 1. E#M := 2 / E#P - H#J. G#K := 1. G#L := L#J * E#P + G#K. E#P := 6. F#N := G#K / 5 + L#J. K#I := 2. H#J := K#I - 4. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#N := E#M * G#L. E#L := H#J - G#L + G#K. L#J := 1. E#M := 2 / E#P - H#J. G#K := 1. G#L := L#J * E#P + G#K. E#P := 6. F#N := G#K / 5 + L#J. K#I := 2. H#J := K#I - 4.", "query": "I#N", "answer": 0, "cot": "G#K = 1 -> G#K = 1\nL#J = 1 -> L#J = 1\nE#P = 6 -> E#P = 6\nK#I = 2 -> K#I = 2\nG#L = L#J * E#P + G#K\n = 1 * 6 + 1\n 1 * 6 = (1 × 6) mod 7 = 6\n 6 + 1 = (6 + 1) mod 7 = 0\n => G#L = 0\nH#J = K#I - 4\n = 2 - 4\n 2 - 4 = (2 - 4) mod 7 = 5\n => H#J = 5\nE#M = 2 / E#P - H#J\n = 2 / 6 - 5\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n 5 - 5 = (5 - 5) mod 7 = 0\n => E#M = 0\nI#N = E#M * G#L\n = 0 * 0\n 0 * 0 = (0 × 0) mod 7 = 0\n => I#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 8, "num_distractors": 0, "var_layer": {"G#K": 1, "L#J": 1, "E#P": 1, "K#I": 1, "F#N": 2, "G#L": 2, "H#J": 2, "E#L": 3, "E#M": 3, "I#N": 4}, "id": "mod7_arith_d4_0049"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#O := J#M. K#N := F#K + E#M. E#M := 4. J#M := G#J. J#J := 3. G#J := J#J + E#M. F#K := 5. E#P := E#M / K#N. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#O := J#M. K#N := F#K + E#M. E#M := 4. J#M := G#J. J#J := 3. G#J := J#J + E#M. F#K := 5. E#P := E#M / K#N.", "query": "K#O", "answer": 0, "cot": "J#J = 3 -> J#J = 3\nE#M = 4 -> E#M = 4\nG#J = J#J + E#M\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => G#J = 0\nJ#M = G#J = 0\nK#O = J#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#J": 1, "E#M": 1, "F#K": 1, "G#J": 2, "K#N": 2, "J#M": 3, "E#P": 3, "K#O": 4}, "id": "mod7_arith_d4_0050"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := J#P * I#J. K#K := 4. H#I := 6. K#L := 2. H#N := H#I * 2 - K#O. I#J := 3 / K#K. I#N := 3. J#P := K#O. K#O := H#I / 3. J#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := J#P * I#J. K#K := 4. H#I := 6. K#L := 2. H#N := H#I * 2 - K#O. I#J := 3 / K#K. I#N := 3. J#P := K#O. K#O := H#I / 3.", "query": "J#O", "answer": 5, "cot": "K#K = 4 -> K#K = 4\nH#I = 6 -> H#I = 6\nI#J = 3 / K#K\n = 3 / 4\n 3 / 4 = (3 × 4⁻¹) mod 7 = 6\n => I#J = 6\nK#O = H#I / 3\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => K#O = 2\nJ#P = K#O = 2\nJ#O = J#P * I#J\n = 2 * 6\n 2 * 6 = (2 × 6) mod 7 = 5\n => J#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#K": 1, "H#I": 1, "K#L": 1, "I#N": 1, "I#J": 2, "K#O": 2, "J#P": 3, "H#N": 3, "J#O": 4}, "id": "mod7_arith_d4_0051"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := 5. E#M := G#J. L#P := F#J + 2 / 4. G#J := L#M. F#J := 4. L#M := 3. G#N := F#J / G#J * L#M. E#J := G#N. G#P := L#P. E#I := 6. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := 5. E#M := G#J. L#P := F#J + 2 / 4. G#J := L#M. F#J := 4. L#M := 3. G#N := F#J / G#J * L#M. E#J := G#N. G#P := L#P. E#I := 6.", "query": "E#J", "answer": 4, "cot": "F#J = 4 -> F#J = 4\nL#M = 3 -> L#M = 3\nG#J = L#M = 3\nG#N = F#J / G#J * L#M\n = 4 / 3 * 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n 6 * 3 = (6 × 3) mod 7 = 4\n => G#N = 4\nE#J = G#N = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#J": 1, "G#O": 1, "L#M": 1, "E#I": 1, "L#P": 2, "G#J": 2, "G#P": 3, "E#M": 3, "G#N": 3, "E#J": 4}, "id": "mod7_arith_d4_0052"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := I#O * J#M. J#O := J#M. H#J := I#O - F#N * 2. F#I := F#N - J#I. K#J := I#I. I#I := I#O. J#I := I#I / J#M. J#M := 4. I#O := 3. F#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := I#O * J#M. J#O := J#M. H#J := I#O - F#N * 2. F#I := F#N - J#I. K#J := I#I. I#I := I#O. J#I := I#I / J#M. J#M := 4. I#O := 3.", "query": "F#I", "answer": 6, "cot": "J#M = 4 -> J#M = 4\nI#O = 3 -> I#O = 3\nI#I = I#O = 3\nF#N = I#O * J#M\n = 3 * 4\n 3 * 4 = (3 × 4) mod 7 = 5\n => F#N = 5\nJ#I = I#I / J#M\n = 3 / 4\n 3 / 4 = (3 × 4⁻¹) mod 7 = 6\n => J#I = 6\nF#I = F#N - J#I\n = 5 - 6\n 5 - 6 = (5 - 6) mod 7 = 6\n => F#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#M": 1, "I#O": 1, "I#I": 2, "F#N": 2, "J#O": 2, "K#J": 3, "J#I": 3, "H#J": 3, "F#I": 4}, "id": "mod7_arith_d4_0053"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#J := I#I + J#L. I#I := J#L / 2 * I#L. J#L := 3. L#N := E#J - L#O - I#L. F#J := 3. F#P := F#O. F#O := J#L + F#J + I#L. L#O := 5 * I#I. I#L := 1. L#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#J := I#I + J#L. I#I := J#L / 2 * I#L. J#L := 3. L#N := E#J - L#O - I#L. F#J := 3. F#P := F#O. F#O := J#L + F#J + I#L. L#O := 5 * I#I. I#L := 1.", "query": "L#N", "answer": 3, "cot": "J#L = 3 -> J#L = 3\nI#L = 1 -> I#L = 1\nI#I = J#L / 2 * I#L\n = 3 / 2 * 1\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n 5 * 1 = (5 × 1) mod 7 = 5\n => I#I = 5\nL#O = 5 * I#I\n = 5 * 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => L#O = 4\nE#J = I#I + J#L\n = 5 + 3\n 5 + 3 = (5 + 3) mod 7 = 1\n => E#J = 1\nL#N = E#J - L#O - I#L\n = 1 - 4 - 1\n 1 - 4 = (1 - 4) mod 7 = 4\n 4 - 1 = (4 - 1) mod 7 = 3\n => L#N = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#L": 1, "I#L": 1, "F#J": 1, "F#O": 2, "I#I": 2, "F#P": 3, "L#O": 3, "E#J": 3, "L#N": 4}, "id": "mod7_arith_d4_0054"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := H#K * H#P + 4. G#N := H#P. H#P := 1. J#J := G#N * H#I. H#I := H#K * H#P. G#J := E#K - 6. K#K := J#J * H#I. H#K := 2. J#P := G#N + E#K. K#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := H#K * H#P + 4. G#N := H#P. H#P := 1. J#J := G#N * H#I. H#I := H#K * H#P. G#J := E#K - 6. K#K := J#J * H#I. H#K := 2. J#P := G#N + E#K.", "query": "K#K", "answer": 4, "cot": "H#P = 1 -> H#P = 1\nH#K = 2 -> H#K = 2\nH#I = H#K * H#P\n = 2 * 1\n 2 * 1 = (2 × 1) mod 7 = 2\n => H#I = 2\nG#N = H#P = 1\nJ#J = G#N * H#I\n = 1 * 2\n 1 * 2 = (1 × 2) mod 7 = 2\n => J#J = 2\nK#K = J#J * H#I\n = 2 * 2\n 2 * 2 = (2 × 2) mod 7 = 4\n => K#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"H#P": 1, "H#K": 1, "H#I": 2, "G#N": 2, "E#K": 2, "J#J": 3, "J#P": 3, "G#J": 3, "K#K": 4}, "id": "mod7_arith_d4_0055"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := K#I / L#K. F#I := I#M. K#I := 6. H#I := I#M * 6 - K#I. L#K := 4. K#P := 5 / I#M * K#O. K#O := K#I * I#M + F#J. I#M := L#K / K#I. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := K#I / L#K. F#I := I#M. K#I := 6. H#I := I#M * 6 - K#I. L#K := 4. K#P := 5 / I#M * K#O. K#O := K#I * I#M + F#J. I#M := L#K / K#I.", "query": "K#P", "answer": 1, "cot": "K#I = 6 -> K#I = 6\nL#K = 4 -> L#K = 4\nI#M = L#K / K#I\n = 4 / 6\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n => I#M = 3\nF#J = K#I / L#K\n = 6 / 4\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n => F#J = 5\nK#O = K#I * I#M + F#J\n = 6 * 3 + 5\n 6 * 3 = (6 × 3) mod 7 = 4\n 4 + 5 = (4 + 5) mod 7 = 2\n => K#O = 2\nK#P = 5 / I#M * K#O\n = 5 / 3 * 2\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n 4 * 2 = (4 × 2) mod 7 = 1\n => K#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#I": 1, "L#K": 1, "I#M": 2, "F#J": 2, "F#I": 3, "H#I": 3, "K#O": 3, "K#P": 4}, "id": "mod7_arith_d4_0056"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := E#P - 5. E#P := G#L + K#L. F#N := 1. L#M := 2 - E#N. K#O := 1. G#L := 3. I#I := 1. K#L := I#I / 4. G#N := 5 + 5 + E#N. L#L := F#N + G#L * I#I. E#N := F#N. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := E#P - 5. E#P := G#L + K#L. F#N := 1. L#M := 2 - E#N. K#O := 1. G#L := 3. I#I := 1. K#L := I#I / 4. G#N := 5 + 5 + E#N. L#L := F#N + G#L * I#I. E#N := F#N.", "query": "F#J", "answer": 0, "cot": "G#L = 3 -> G#L = 3\nI#I = 1 -> I#I = 1\nK#L = I#I / 4\n = 1 / 4\n 1 / 4 = (1 × 4⁻¹) mod 7 = 2\n => K#L = 2\nE#P = G#L + K#L\n = 3 + 2\n 3 + 2 = (3 + 2) mod 7 = 5\n => E#P = 5\nF#J = E#P - 5\n = 5 - 5\n 5 - 5 = (5 - 5) mod 7 = 0\n => F#J = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#N": 1, "K#O": 1, "G#L": 1, "I#I": 1, "E#N": 2, "K#L": 2, "L#L": 2, "L#M": 3, "G#N": 3, "E#P": 3, "F#J": 4}, "id": "mod7_arith_d4_0057"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#O := 4. G#M := 1. I#P := 2 + J#O. L#M := L#K * 2 / E#J. E#J := J#O + 4. E#M := E#O - L#K. G#P := 2. E#P := G#P / 4. J#O := E#O * G#M. L#K := 4 + G#P. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#O := 4. G#M := 1. I#P := 2 + J#O. L#M := L#K * 2 / E#J. E#J := J#O + 4. E#M := E#O - L#K. G#P := 2. E#P := G#P / 4. J#O := E#O * G#M. L#K := 4 + G#P.", "query": "L#M", "answer": 5, "cot": "G#P = 2 -> G#P = 2\nE#O = 4 -> E#O = 4\nG#M = 1 -> G#M = 1\nL#K = 4 + G#P\n = 4 + 2\n 4 + 2 = (4 + 2) mod 7 = 6\n => L#K = 6\nJ#O = E#O * G#M\n = 4 * 1\n 4 * 1 = (4 × 1) mod 7 = 4\n => J#O = 4\nE#J = J#O + 4\n = 4 + 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => E#J = 1\nL#M = L#K * 2 / E#J\n = 6 * 2 / 1\n 6 * 2 = (6 × 2) mod 7 = 5\n 5 / 1 = (5 × 1⁻¹) mod 7 = 5\n => L#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"G#P": 1, "E#O": 1, "G#M": 1, "L#K": 2, "E#P": 2, "J#O": 2, "I#P": 3, "E#M": 3, "E#J": 3, "L#M": 4}, "id": "mod7_arith_d4_0058"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#K := K#I * E#N * 4. K#I := 6. K#O := J#J * K#N * 6. E#O := E#N * K#I. K#K := K#I - H#I. E#N := 5 / K#I. K#N := 5. H#I := K#O. L#I := K#I - J#J * K#N. J#J := 3. K#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#K := K#I * E#N * 4. K#I := 6. K#O := J#J * K#N * 6. E#O := E#N * K#I. K#K := K#I - H#I. E#N := 5 / K#I. K#N := 5. H#I := K#O. L#I := K#I - J#J * K#N. J#J := 3.", "query": "K#K", "answer": 0, "cot": "K#I = 6 -> K#I = 6\nJ#J = 3 -> J#J = 3\nK#N = 5 -> K#N = 5\nK#O = J#J * K#N * 6\n = 3 * 5 * 6\n 3 * 5 = (3 × 5) mod 7 = 1\n 1 * 6 = (1 × 6) mod 7 = 6\n => K#O = 6\nH#I = K#O = 6\nK#K = K#I - H#I\n = 6 - 6\n 6 - 6 = (6 - 6) mod 7 = 0\n => K#K = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#I": 1, "J#J": 1, "K#N": 1, "E#N": 2, "L#I": 2, "K#O": 2, "L#K": 3, "E#O": 3, "H#I": 3, "K#K": 4}, "id": "mod7_arith_d4_0059"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := 5 + J#P. K#O := F#O. J#J := J#P / F#P. K#K := 1. J#P := 4. H#I := 4. F#J := G#O + F#P + F#O. F#P := 3. F#O := 6 / F#P. G#L := J#P * F#J. L#M := H#I * F#O. G#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := 5 + J#P. K#O := F#O. J#J := J#P / F#P. K#K := 1. J#P := 4. H#I := 4. F#J := G#O + F#P + F#O. F#P := 3. F#O := 6 / F#P. G#L := J#P * F#J. L#M := H#I * F#O.", "query": "G#L", "answer": 0, "cot": "F#P = 3 -> F#P = 3\nJ#P = 4 -> J#P = 4\nF#O = 6 / F#P\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => F#O = 2\nG#O = 5 + J#P\n = 5 + 4\n 5 + 4 = (5 + 4) mod 7 = 2\n => G#O = 2\nF#J = G#O + F#P + F#O\n = 2 + 3 + 2\n 2 + 3 = (2 + 3) mod 7 = 5\n 5 + 2 = (5 + 2) mod 7 = 0\n => F#J = 0\nG#L = J#P * F#J\n = 4 * 0\n 4 * 0 = (4 × 0) mod 7 = 0\n => G#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#P": 1, "J#P": 1, "H#I": 1, "K#K": 1, "F#O": 2, "J#J": 2, "G#O": 2, "L#M": 3, "K#O": 3, "F#J": 3, "G#L": 4}, "id": "mod7_arith_d4_0060"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := F#K - I#N. F#K := E#M / F#N. F#N := 1. G#L := I#N - 3. I#N := E#M + F#N. E#M := 5. J#L := K#J. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := F#K - I#N. F#K := E#M / F#N. F#N := 1. G#L := I#N - 3. I#N := E#M + F#N. E#M := 5. J#L := K#J.", "query": "J#L", "answer": 6, "cot": "F#N = 1 -> F#N = 1\nE#M = 5 -> E#M = 5\nF#K = E#M / F#N\n = 5 / 1\n 5 / 1 = (5 × 1⁻¹) mod 7 = 5\n => F#K = 5\nI#N = E#M + F#N\n = 5 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n => I#N = 6\nK#J = F#K - I#N\n = 5 - 6\n 5 - 6 = (5 - 6) mod 7 = 6\n => K#J = 6\nJ#L = K#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#N": 1, "E#M": 1, "F#K": 2, "I#N": 2, "G#L": 3, "K#J": 3, "J#L": 4}, "id": "mod7_arith_d4_0061"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := J#K. J#O := 3 * K#N. K#N := 2. F#L := G#M * J#K. G#J := J#O. G#M := F#P * J#O. J#K := 4. H#I := 6 + J#O. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := J#K. J#O := 3 * K#N. K#N := 2. F#L := G#M * J#K. G#J := J#O. G#M := F#P * J#O. J#K := 4. H#I := 6 + J#O.", "query": "F#L", "answer": 5, "cot": "J#K = 4 -> J#K = 4\nK#N = 2 -> K#N = 2\nF#P = J#K = 4\nJ#O = 3 * K#N\n = 3 * 2\n 3 * 2 = (3 × 2) mod 7 = 6\n => J#O = 6\nG#M = F#P * J#O\n = 4 * 6\n 4 * 6 = (4 × 6) mod 7 = 3\n => G#M = 3\nF#L = G#M * J#K\n = 3 * 4\n 3 * 4 = (3 × 4) mod 7 = 5\n => F#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#K": 1, "K#N": 1, "F#P": 2, "J#O": 2, "G#J": 3, "G#M": 3, "H#I": 3, "F#L": 4}, "id": "mod7_arith_d4_0062"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 2. K#P := H#N. L#N := 5. H#N := 2. G#N := 5 * L#N. I#P := H#N - K#P. F#K := 3. E#I := G#N. I#I := E#I + 4. E#N := K#P. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 2. K#P := H#N. L#N := 5. H#N := 2. G#N := 5 * L#N. I#P := H#N - K#P. F#K := 3. E#I := G#N. I#I := E#I + 4. E#N := K#P.", "query": "I#I", "answer": 1, "cot": "L#N = 5 -> L#N = 5\nG#N = 5 * L#N\n = 5 * 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => G#N = 4\nE#I = G#N = 4\nI#I = E#I + 4\n = 4 + 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => I#I = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#K": 1, "J#N": 1, "L#N": 1, "H#N": 1, "K#P": 2, "G#N": 2, "I#P": 3, "E#N": 3, "E#I": 3, "I#I": 4}, "id": "mod7_arith_d4_0063"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := 3. F#L := 3 / I#I. L#O := G#M / I#I. I#I := G#P / G#M. G#M := 3. I#L := G#M + H#J. I#K := L#O. H#J := 4 + 5 - G#P. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := 3. F#L := 3 / I#I. L#O := G#M / I#I. I#I := G#P / G#M. G#M := 3. I#L := G#M + H#J. I#K := L#O. H#J := 4 + 5 - G#P.", "query": "I#K", "answer": 3, "cot": "G#P = 3 -> G#P = 3\nG#M = 3 -> G#M = 3\nI#I = G#P / G#M\n = 3 / 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => I#I = 1\nL#O = G#M / I#I\n = 3 / 1\n 3 / 1 = (3 × 1⁻¹) mod 7 = 3\n => L#O = 3\nI#K = L#O = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#P": 1, "G#M": 1, "H#J": 2, "I#I": 2, "I#L": 3, "F#L": 3, "L#O": 3, "I#K": 4}, "id": "mod7_arith_d4_0064"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#L := H#M * E#I. E#I := 1. J#P := J#L / E#I. L#O := H#M * E#J - E#I. J#M := H#M + J#L. H#M := 4. F#N := J#M / H#N - 5. H#N := L#O. E#J := 1. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#L := H#M * E#I. E#I := 1. J#P := J#L / E#I. L#O := H#M * E#J - E#I. J#M := H#M + J#L. H#M := 4. F#N := J#M / H#N - 5. H#N := L#O. E#J := 1.", "query": "F#N", "answer": 0, "cot": "E#I = 1 -> E#I = 1\nH#M = 4 -> H#M = 4\nE#J = 1 -> E#J = 1\nJ#L = H#M * E#I\n = 4 * 1\n 4 * 1 = (4 × 1) mod 7 = 4\n => J#L = 4\nL#O = H#M * E#J - E#I\n = 4 * 1 - 1\n 4 * 1 = (4 × 1) mod 7 = 4\n 4 - 1 = (4 - 1) mod 7 = 3\n => L#O = 3\nJ#M = H#M + J#L\n = 4 + 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => J#M = 1\nH#N = L#O = 3\nF#N = J#M / H#N - 5\n = 1 / 3 - 5\n 1 / 3 = (1 × 3⁻¹) mod 7 = 5\n 5 - 5 = (5 - 5) mod 7 = 0\n => F#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 8, "num_distractors": 0, "var_layer": {"E#I": 1, "H#M": 1, "E#J": 1, "J#L": 2, "L#O": 2, "J#P": 3, "J#M": 3, "H#N": 3, "F#N": 4}, "id": "mod7_arith_d4_0065"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#O := G#P * G#K. K#O := 1. E#L := 2. G#K := K#P * 5. G#P := K#P. K#P := 5. H#O := K#O + I#O. H#I := G#P. L#I := 1. H#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#O := G#P * G#K. K#O := 1. E#L := 2. G#K := K#P * 5. G#P := K#P. K#P := 5. H#O := K#O + I#O. H#I := G#P. L#I := 1.", "query": "H#O", "answer": 0, "cot": "K#O = 1 -> K#O = 1\nK#P = 5 -> K#P = 5\nG#P = K#P = 5\nG#K = K#P * 5\n = 5 * 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => G#K = 4\nI#O = G#P * G#K\n = 5 * 4\n 5 * 4 = (5 × 4) mod 7 = 6\n => I#O = 6\nH#O = K#O + I#O\n = 1 + 6\n 1 + 6 = (1 + 6) mod 7 = 0\n => H#O = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#O": 1, "E#L": 1, "L#I": 1, "K#P": 1, "G#P": 2, "G#K": 2, "I#O": 3, "H#I": 3, "H#O": 4}, "id": "mod7_arith_d4_0066"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := 6. K#L := I#O. E#I := 4. E#N := H#P - 2 - L#J. J#J := 4. K#J := L#J + E#I. F#P := 3. F#I := I#N / J#J. H#P := K#J + I#O / L#J. I#N := E#I. I#O := J#J. E#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := 6. K#L := I#O. E#I := 4. E#N := H#P - 2 - L#J. J#J := 4. K#J := L#J + E#I. F#P := 3. F#I := I#N / J#J. H#P := K#J + I#O / L#J. I#N := E#I. I#O := J#J.", "query": "E#N", "answer": 6, "cot": "J#J = 4 -> J#J = 4\nE#I = 4 -> E#I = 4\nL#J = 6 -> L#J = 6\nI#O = J#J = 4\nK#J = L#J + E#I\n = 6 + 4\n 6 + 4 = (6 + 4) mod 7 = 3\n => K#J = 3\nH#P = K#J + I#O / L#J\n = 3 + 4 / 6\n 3 + 4 = (3 + 4) mod 7 = 0\n 0 / 6 = (0 × 6⁻¹) mod 7 = 0\n => H#P = 0\nE#N = H#P - 2 - L#J\n = 0 - 2 - 6\n 0 - 2 = (0 - 2) mod 7 = 5\n 5 - 6 = (5 - 6) mod 7 = 6\n => E#N = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 7, "num_distractors": 0, "var_layer": {"J#J": 1, "E#I": 1, "F#P": 1, "L#J": 1, "I#N": 2, "I#O": 2, "K#J": 2, "K#L": 3, "H#P": 3, "F#I": 3, "E#N": 4}, "id": "mod7_arith_d4_0067"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#K := 3. G#N := L#P * L#K. I#O := G#N / 3. L#L := L#P. F#P := 2. L#P := L#N / F#P. L#N := 5. H#I := 1. K#M := J#N - L#P. J#N := H#I * L#N. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#K := 3. G#N := L#P * L#K. I#O := G#N / 3. L#L := L#P. F#P := 2. L#P := L#N / F#P. L#N := 5. H#I := 1. K#M := J#N - L#P. J#N := H#I * L#N.", "query": "I#O", "answer": 6, "cot": "L#N = 5 -> L#N = 5\nL#K = 3 -> L#K = 3\nF#P = 2 -> F#P = 2\nL#P = L#N / F#P\n = 5 / 2\n 5 / 2 = (5 × 2⁻¹) mod 7 = 6\n => L#P = 6\nG#N = L#P * L#K\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => G#N = 4\nI#O = G#N / 3\n = 4 / 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => I#O = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"H#I": 1, "L#N": 1, "L#K": 1, "F#P": 1, "J#N": 2, "L#P": 2, "G#N": 3, "K#M": 3, "L#L": 3, "I#O": 4}, "id": "mod7_arith_d4_0068"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#L := 6. G#I := K#J. J#J := L#M / I#L. G#P := I#L * 2. L#M := I#L. K#K := G#L. K#J := 2. I#L := K#J * 5. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#L := 6. G#I := K#J. J#J := L#M / I#L. G#P := I#L * 2. L#M := I#L. K#K := G#L. K#J := 2. I#L := K#J * 5.", "query": "J#J", "answer": 1, "cot": "K#J = 2 -> K#J = 2\nI#L = K#J * 5\n = 2 * 5\n 2 * 5 = (2 × 5) mod 7 = 3\n => I#L = 3\nL#M = I#L = 3\nJ#J = L#M / I#L\n = 3 / 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => J#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#J": 1, "G#L": 1, "K#K": 2, "G#I": 2, "I#L": 2, "L#M": 3, "G#P": 3, "J#J": 4}, "id": "mod7_arith_d4_0069"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#L := I#N. I#N := 1. F#I := L#L. J#N := F#M * H#O. F#M := 4. E#M := J#N * F#M. J#I := J#N. E#O := E#M * 6. H#O := 6. E#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#L := I#N. I#N := 1. F#I := L#L. J#N := F#M * H#O. F#M := 4. E#M := J#N * F#M. J#I := J#N. E#O := E#M * 6. H#O := 6.", "query": "E#O", "answer": 2, "cot": "F#M = 4 -> F#M = 4\nH#O = 6 -> H#O = 6\nJ#N = F#M * H#O\n = 4 * 6\n 4 * 6 = (4 × 6) mod 7 = 3\n => J#N = 3\nE#M = J#N * F#M\n = 3 * 4\n 3 * 4 = (3 × 4) mod 7 = 5\n => E#M = 5\nE#O = E#M * 6\n = 5 * 6\n 5 * 6 = (5 × 6) mod 7 = 2\n => E#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#M": 1, "H#O": 1, "I#N": 1, "J#N": 2, "L#L": 2, "F#I": 3, "J#I": 3, "E#M": 3, "E#O": 4}, "id": "mod7_arith_d4_0070"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#I := I#P * L#K. K#L := K#O / L#K. F#O := L#K / I#P. L#O := G#I - F#O * E#K. L#K := 2. K#O := G#I. I#P := 2. E#K := 4 / L#K. K#K := E#K. K#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#I := I#P * L#K. K#L := K#O / L#K. F#O := L#K / I#P. L#O := G#I - F#O * E#K. L#K := 2. K#O := G#I. I#P := 2. E#K := 4 / L#K. K#K := E#K.", "query": "K#L", "answer": 2, "cot": "I#P = 2 -> I#P = 2\nL#K = 2 -> L#K = 2\nG#I = I#P * L#K\n = 2 * 2\n 2 * 2 = (2 × 2) mod 7 = 4\n => G#I = 4\nK#O = G#I = 4\nK#L = K#O / L#K\n = 4 / 2\n 4 / 2 = (4 × 2⁻¹) mod 7 = 2\n => K#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#P": 1, "L#K": 1, "E#K": 2, "G#I": 2, "F#O": 2, "K#O": 3, "L#O": 3, "K#K": 3, "K#L": 4}, "id": "mod7_arith_d4_0071"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#N := F#M / J#M / F#I. J#M := K#L - F#I. H#L := 5. F#M := J#K. K#L := 6. G#P := 2. I#N := 1. F#L := F#I * J#K. J#K := H#L. F#I := 2 * I#N. L#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#N := F#M / J#M / F#I. J#M := K#L - F#I. H#L := 5. F#M := J#K. K#L := 6. G#P := 2. I#N := 1. F#L := F#I * J#K. J#K := H#L. F#I := 2 * I#N.", "query": "L#N", "answer": 5, "cot": "I#N = 1 -> I#N = 1\nK#L = 6 -> K#L = 6\nH#L = 5 -> H#L = 5\nF#I = 2 * I#N\n = 2 * 1\n 2 * 1 = (2 × 1) mod 7 = 2\n => F#I = 2\nJ#K = H#L = 5\nF#M = J#K = 5\nJ#M = K#L - F#I\n = 6 - 2\n 6 - 2 = (6 - 2) mod 7 = 4\n => J#M = 4\nL#N = F#M / J#M / F#I\n = 5 / 4 / 2\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => L#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 8, "num_distractors": 0, "var_layer": {"I#N": 1, "K#L": 1, "G#P": 1, "H#L": 1, "F#I": 2, "J#K": 2, "F#M": 3, "J#M": 3, "F#L": 3, "L#N": 4}, "id": "mod7_arith_d4_0072"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := J#L + E#J. F#N := 5. E#J := 4. H#M := E#J + 2. E#K := 6 + E#L - H#M. E#L := K#K. H#N := E#K. E#N := 5. J#L := E#J. K#K := 6. H#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := J#L + E#J. F#N := 5. E#J := 4. H#M := E#J + 2. E#K := 6 + E#L - H#M. E#L := K#K. H#N := E#K. E#N := 5. J#L := E#J. K#K := 6.", "query": "H#N", "answer": 6, "cot": "K#K = 6 -> K#K = 6\nE#J = 4 -> E#J = 4\nE#L = K#K = 6\nH#M = E#J + 2\n = 4 + 2\n 4 + 2 = (4 + 2) mod 7 = 6\n => H#M = 6\nE#K = 6 + E#L - H#M\n = 6 + 6 - 6\n 6 + 6 = (6 + 6) mod 7 = 5\n 5 - 6 = (5 - 6) mod 7 = 6\n => E#K = 6\nH#N = E#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#N": 1, "F#N": 1, "K#K": 1, "E#J": 1, "E#L": 2, "H#M": 2, "J#L": 2, "E#K": 3, "L#P": 3, "H#N": 4}, "id": "mod7_arith_d4_0073"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := F#K / G#L. F#K := E#I - G#N. L#M := 6. G#N := 6. G#L := G#N. L#L := E#O / 3 - 3. E#O := G#L * F#K. I#O := 1. E#P := E#I + 6 / G#L. E#I := 2. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := F#K / G#L. F#K := E#I - G#N. L#M := 6. G#N := 6. G#L := G#N. L#L := E#O / 3 - 3. E#O := G#L * F#K. I#O := 1. E#P := E#I + 6 / G#L. E#I := 2.", "query": "L#L", "answer": 3, "cot": "E#I = 2 -> E#I = 2\nG#N = 6 -> G#N = 6\nG#L = G#N = 6\nF#K = E#I - G#N\n = 2 - 6\n 2 - 6 = (2 - 6) mod 7 = 3\n => F#K = 3\nE#O = G#L * F#K\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => E#O = 4\nL#L = E#O / 3 - 3\n = 4 / 3 - 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n 6 - 3 = (6 - 3) mod 7 = 3\n => L#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#I": 1, "I#O": 1, "L#M": 1, "G#N": 1, "G#L": 2, "F#K": 2, "I#I": 3, "E#P": 3, "E#O": 3, "L#L": 4}, "id": "mod7_arith_d4_0074"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := G#J - K#P. E#N := G#J / F#M - G#O. E#K := F#M / K#P. K#P := G#J - G#O. G#N := L#O * G#O - 6. F#M := 3. G#J := 1. L#O := 2. E#J := J#N. F#P := F#M * E#N. G#O := 5. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := G#J - K#P. E#N := G#J / F#M - G#O. E#K := F#M / K#P. K#P := G#J - G#O. G#N := L#O * G#O - 6. F#M := 3. G#J := 1. L#O := 2. E#J := J#N. F#P := F#M * E#N. G#O := 5.", "query": "E#J", "answer": 5, "cot": "G#J = 1 -> G#J = 1\nG#O = 5 -> G#O = 5\nK#P = G#J - G#O\n = 1 - 5\n 1 - 5 = (1 - 5) mod 7 = 3\n => K#P = 3\nJ#N = G#J - K#P\n = 1 - 3\n 1 - 3 = (1 - 3) mod 7 = 5\n => J#N = 5\nE#J = J#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#O": 1, "G#J": 1, "G#O": 1, "F#M": 1, "E#N": 2, "K#P": 2, "G#N": 2, "J#N": 3, "F#P": 3, "E#K": 3, "E#J": 4}, "id": "mod7_arith_d4_0075"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#N := 3 * L#P - F#M. I#L := F#M. L#P := K#I. F#M := 4. E#N := L#P * I#L. K#I := 4. H#P := K#N + L#P. J#M := 2. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#N := 3 * L#P - F#M. I#L := F#M. L#P := K#I. F#M := 4. E#N := L#P * I#L. K#I := 4. H#P := K#N + L#P. J#M := 2.", "query": "H#P", "answer": 5, "cot": "K#I = 4 -> K#I = 4\nF#M = 4 -> F#M = 4\nL#P = K#I = 4\nK#N = 3 * L#P - F#M\n = 3 * 4 - 4\n 3 * 4 = (3 × 4) mod 7 = 5\n 5 - 4 = (5 - 4) mod 7 = 1\n => K#N = 1\nH#P = K#N + L#P\n = 1 + 4\n 1 + 4 = (1 + 4) mod 7 = 5\n => H#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#I": 1, "F#M": 1, "J#M": 1, "I#L": 2, "L#P": 2, "E#N": 3, "K#N": 3, "H#P": 4}, "id": "mod7_arith_d4_0076"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := 6. H#L := 6. I#K := I#M. E#L := H#L. L#P := 6 + E#K. J#L := 1. I#M := F#M. F#M := E#K - H#L. G#L := 4 + F#M. H#I := 5. G#I := E#K - F#M. I#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := 6. H#L := 6. I#K := I#M. E#L := H#L. L#P := 6 + E#K. J#L := 1. I#M := F#M. F#M := E#K - H#L. G#L := 4 + F#M. H#I := 5. G#I := E#K - F#M.", "query": "I#K", "answer": 0, "cot": "E#K = 6 -> E#K = 6\nH#L = 6 -> H#L = 6\nF#M = E#K - H#L\n = 6 - 6\n 6 - 6 = (6 - 6) mod 7 = 0\n => F#M = 0\nI#M = F#M = 0\nI#K = I#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#L": 1, "H#I": 1, "E#K": 1, "H#L": 1, "E#L": 2, "F#M": 2, "L#P": 2, "I#M": 3, "G#I": 3, "G#L": 3, "I#K": 4}, "id": "mod7_arith_d4_0077"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := 6 / I#K - 5. F#M := F#L * H#I + F#J. F#L := 1. L#I := F#N. F#N := I#K - J#I - F#L. I#L := 4 * 3 / F#M. I#K := H#I * 2 * F#L. J#I := F#J / H#I - F#L. H#I := 4. F#J := 6. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := 6 / I#K - 5. F#M := F#L * H#I + F#J. F#L := 1. L#I := F#N. F#N := I#K - J#I - F#L. I#L := 4 * 3 / F#M. I#K := H#I * 2 * F#L. J#I := F#J / H#I - F#L. H#I := 4. F#J := 6.", "query": "L#I", "answer": 3, "cot": "F#L = 1 -> F#L = 1\nH#I = 4 -> H#I = 4\nF#J = 6 -> F#J = 6\nJ#I = F#J / H#I - F#L\n = 6 / 4 - 1\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n 5 - 1 = (5 - 1) mod 7 = 4\n => J#I = 4\nI#K = H#I * 2 * F#L\n = 4 * 2 * 1\n 4 * 2 = (4 × 2) mod 7 = 1\n 1 * 1 = (1 × 1) mod 7 = 1\n => I#K = 1\nF#N = I#K - J#I - F#L\n = 1 - 4 - 1\n 1 - 4 = (1 - 4) mod 7 = 4\n 4 - 1 = (4 - 1) mod 7 = 3\n => F#N = 3\nL#I = F#N = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"F#L": 1, "H#I": 1, "F#J": 1, "J#I": 2, "I#K": 2, "F#M": 2, "I#L": 3, "F#N": 3, "L#M": 3, "L#I": 4}, "id": "mod7_arith_d4_0078"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := 3. H#K := J#J + I#O. H#J := I#O. K#P := J#J. K#M := 2. I#O := J#J. E#J := H#K - 3. J#J := 3. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := 3. H#K := J#J + I#O. H#J := I#O. K#P := J#J. K#M := 2. I#O := J#J. E#J := H#K - 3. J#J := 3.", "query": "E#J", "answer": 3, "cot": "J#J = 3 -> J#J = 3\nI#O = J#J = 3\nH#K = J#J + I#O\n = 3 + 3\n 3 + 3 = (3 + 3) mod 7 = 6\n => H#K = 6\nE#J = H#K - 3\n = 6 - 3\n 6 - 3 = (6 - 3) mod 7 = 3\n => E#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#J": 1, "K#M": 1, "G#O": 1, "K#P": 2, "I#O": 2, "H#J": 3, "H#K": 3, "E#J": 4}, "id": "mod7_arith_d4_0079"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := 4. L#I := 5. E#K := L#I / L#K * K#I. G#L := G#J - L#K. G#J := L#K * 5 + L#I. J#P := L#I. L#M := K#L + 4. K#L := 6 * L#I / G#J. L#K := 4. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := 4. L#I := 5. E#K := L#I / L#K * K#I. G#L := G#J - L#K. G#J := L#K * 5 + L#I. J#P := L#I. L#M := K#L + 4. K#L := 6 * L#I / G#J. L#K := 4.", "query": "L#M", "answer": 1, "cot": "L#I = 5 -> L#I = 5\nL#K = 4 -> L#K = 4\nG#J = L#K * 5 + L#I\n = 4 * 5 + 5\n 4 * 5 = (4 × 5) mod 7 = 6\n 6 + 5 = (6 + 5) mod 7 = 4\n => G#J = 4\nK#L = 6 * L#I / G#J\n = 6 * 5 / 4\n 6 * 5 = (6 × 5) mod 7 = 2\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n => K#L = 4\nL#M = K#L + 4\n = 4 + 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => L#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#I": 1, "L#I": 1, "L#K": 1, "E#K": 2, "G#J": 2, "J#P": 2, "G#L": 3, "K#L": 3, "L#M": 4}, "id": "mod7_arith_d4_0080"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#J := G#O. L#L := 2. J#O := 5. J#I := I#L. I#L := L#L / J#O. E#K := 5 + E#J. L#I := 2. F#P := L#I - I#L. G#O := J#O. K#P := L#I + 4 * J#O. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#J := G#O. L#L := 2. J#O := 5. J#I := I#L. I#L := L#L / J#O. E#K := 5 + E#J. L#I := 2. F#P := L#I - I#L. G#O := J#O. K#P := L#I + 4 * J#O.", "query": "E#K", "answer": 3, "cot": "J#O = 5 -> J#O = 5\nG#O = J#O = 5\nE#J = G#O = 5\nE#K = 5 + E#J\n = 5 + 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => E#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#O": 1, "L#I": 1, "L#L": 1, "K#P": 2, "I#L": 2, "G#O": 2, "F#P": 3, "J#I": 3, "E#J": 3, "E#K": 4}, "id": "mod7_arith_d4_0081"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#K := H#N * J#J - H#M. K#I := I#K. J#P := 5. H#O := L#P. H#M := 3. K#O := K#I / H#O. L#P := H#N * 5. J#J := 1. L#N := 4 / H#M. H#N := 2. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#K := H#N * J#J - H#M. K#I := I#K. J#P := 5. H#O := L#P. H#M := 3. K#O := K#I / H#O. L#P := H#N * 5. J#J := 1. L#N := 4 / H#M. H#N := 2.", "query": "K#O", "answer": 2, "cot": "J#J = 1 -> J#J = 1\nH#M = 3 -> H#M = 3\nH#N = 2 -> H#N = 2\nL#P = H#N * 5\n = 2 * 5\n 2 * 5 = (2 × 5) mod 7 = 3\n => L#P = 3\nI#K = H#N * J#J - H#M\n = 2 * 1 - 3\n 2 * 1 = (2 × 1) mod 7 = 2\n 2 - 3 = (2 - 3) mod 7 = 6\n => I#K = 6\nH#O = L#P = 3\nK#I = I#K = 6\nK#O = K#I / H#O\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => K#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 8, "num_distractors": 0, "var_layer": {"J#J": 1, "H#M": 1, "H#N": 1, "J#P": 1, "L#P": 2, "I#K": 2, "L#N": 2, "H#O": 3, "K#I": 3, "K#O": 4}, "id": "mod7_arith_d4_0082"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := H#K * H#O. F#K := E#I. H#K := 1. J#P := 5 + 6 - F#K. G#N := E#I / H#O. E#I := H#K. H#O := 1. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := H#K * H#O. F#K := E#I. H#K := 1. J#P := 5 + 6 - F#K. G#N := E#I / H#O. E#I := H#K. H#O := 1.", "query": "J#P", "answer": 3, "cot": "H#K = 1 -> H#K = 1\nE#I = H#K = 1\nF#K = E#I = 1\nJ#P = 5 + 6 - F#K\n = 5 + 6 - 1\n 5 + 6 = (5 + 6) mod 7 = 4\n 4 - 1 = (4 - 1) mod 7 = 3\n => J#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#O": 1, "H#K": 1, "E#I": 2, "G#P": 2, "F#K": 3, "G#N": 3, "J#P": 4}, "id": "mod7_arith_d4_0083"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := E#I. K#K := 4. F#K := 1. E#K := F#K / I#I. I#I := 6. K#P := I#I * K#K. E#I := I#I + F#K + K#K. L#P := F#K - E#K. J#L := E#I / I#J. K#M := 1. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := E#I. K#K := 4. F#K := 1. E#K := F#K / I#I. I#I := 6. K#P := I#I * K#K. E#I := I#I + F#K + K#K. L#P := F#K - E#K. J#L := E#I / I#J. K#M := 1.", "query": "J#L", "answer": 1, "cot": "I#I = 6 -> I#I = 6\nK#K = 4 -> K#K = 4\nF#K = 1 -> F#K = 1\nE#I = I#I + F#K + K#K\n = 6 + 1 + 4\n 6 + 1 = (6 + 1) mod 7 = 0\n 0 + 4 = (0 + 4) mod 7 = 4\n => E#I = 4\nI#J = E#I = 4\nJ#L = E#I / I#J\n = 4 / 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => J#L = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"I#I": 1, "K#K": 1, "K#M": 1, "F#K": 1, "E#I": 2, "E#K": 2, "K#P": 2, "L#P": 3, "I#J": 3, "J#L": 4}, "id": "mod7_arith_d4_0084"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := 3. I#M := L#P - L#K. G#I := F#K - H#O. L#P := L#K. F#N := I#M - L#P - 3. L#K := 4. G#K := 4. H#O := F#K. L#I := F#L. F#L := G#K. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := 3. I#M := L#P - L#K. G#I := F#K - H#O. L#P := L#K. F#N := I#M - L#P - 3. L#K := 4. G#K := 4. H#O := F#K. L#I := F#L. F#L := G#K.", "query": "F#N", "answer": 0, "cot": "L#K = 4 -> L#K = 4\nL#P = L#K = 4\nI#M = L#P - L#K\n = 4 - 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => I#M = 0\nF#N = I#M - L#P - 3\n = 0 - 4 - 3\n 0 - 4 = (0 - 4) mod 7 = 3\n 3 - 3 = (3 - 3) mod 7 = 0\n => F#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#K": 1, "G#K": 1, "F#K": 1, "H#O": 2, "L#P": 2, "F#L": 2, "G#I": 3, "I#M": 3, "L#I": 3, "F#N": 4}, "id": "mod7_arith_d4_0085"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := L#O + J#N. L#O := 3 * E#N. H#P := E#N. E#N := 1. I#L := L#O - L#I. H#J := J#N + H#P. J#N := 5. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := L#O + J#N. L#O := 3 * E#N. H#P := E#N. E#N := 1. I#L := L#O - L#I. H#J := J#N + H#P. J#N := 5.", "query": "I#L", "answer": 2, "cot": "E#N = 1 -> E#N = 1\nJ#N = 5 -> J#N = 5\nL#O = 3 * E#N\n = 3 * 1\n 3 * 1 = (3 × 1) mod 7 = 3\n => L#O = 3\nL#I = L#O + J#N\n = 3 + 5\n 3 + 5 = (3 + 5) mod 7 = 1\n => L#I = 1\nI#L = L#O - L#I\n = 3 - 1\n 3 - 1 = (3 - 1) mod 7 = 2\n => I#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"E#N": 1, "J#N": 1, "L#O": 2, "H#P": 2, "H#J": 3, "L#I": 3, "I#L": 4}, "id": "mod7_arith_d4_0086"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := K#I - G#J / J#K. J#K := K#I. G#J := L#M. L#M := 3. E#N := 2 * L#P * G#J. K#I := 1. L#P := L#M / K#I + J#K. E#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := K#I - G#J / J#K. J#K := K#I. G#J := L#M. L#M := 3. E#N := 2 * L#P * G#J. K#I := 1. L#P := L#M / K#I + J#K.", "query": "E#N", "answer": 3, "cot": "L#M = 3 -> L#M = 3\nK#I = 1 -> K#I = 1\nG#J = L#M = 3\nJ#K = K#I = 1\nL#P = L#M / K#I + J#K\n = 3 / 1 + 1\n 3 / 1 = (3 × 1⁻¹) mod 7 = 3\n 3 + 1 = (3 + 1) mod 7 = 4\n => L#P = 4\nE#N = 2 * L#P * G#J\n = 2 * 4 * 3\n 2 * 4 = (2 × 4) mod 7 = 1\n 1 * 3 = (1 × 3) mod 7 = 3\n => E#N = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"L#M": 1, "K#I": 1, "G#J": 2, "J#K": 2, "L#P": 3, "G#N": 3, "E#N": 4}, "id": "mod7_arith_d4_0087"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := L#I. K#P := E#O * F#P. L#I := E#N. I#J := 4. E#N := 4. E#O := E#N / I#J * I#O. E#M := 6. I#K := L#I / 3. I#O := I#J. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := L#I. K#P := E#O * F#P. L#I := E#N. I#J := 4. E#N := 4. E#O := E#N / I#J * I#O. E#M := 6. I#K := L#I / 3. I#O := I#J.", "query": "K#P", "answer": 2, "cot": "E#N = 4 -> E#N = 4\nI#J = 4 -> I#J = 4\nI#O = I#J = 4\nL#I = E#N = 4\nE#O = E#N / I#J * I#O\n = 4 / 4 * 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n 1 * 4 = (1 × 4) mod 7 = 4\n => E#O = 4\nF#P = L#I = 4\nK#P = E#O * F#P\n = 4 * 4\n 4 * 4 = (4 × 4) mod 7 = 2\n => K#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"E#N": 1, "E#M": 1, "I#J": 1, "I#O": 2, "L#I": 2, "E#O": 3, "F#P": 3, "I#K": 3, "K#P": 4}, "id": "mod7_arith_d4_0088"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := 3 / E#P. I#P := H#O - F#P. L#P := 1. H#O := L#P - F#I. F#P := 4. G#O := L#M. E#P := F#P. F#I := 5. G#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := 3 / E#P. I#P := H#O - F#P. L#P := 1. H#O := L#P - F#I. F#P := 4. G#O := L#M. E#P := F#P. F#I := 5.", "query": "G#O", "answer": 6, "cot": "F#P = 4 -> F#P = 4\nE#P = F#P = 4\nL#M = 3 / E#P\n = 3 / 4\n 3 / 4 = (3 × 4⁻¹) mod 7 = 6\n => L#M = 6\nG#O = L#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#P": 1, "F#P": 1, "F#I": 1, "H#O": 2, "E#P": 2, "I#P": 3, "L#M": 3, "G#O": 4}, "id": "mod7_arith_d4_0089"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#J := E#I / H#M - K#K. F#J := E#I * K#O + K#K. L#K := F#J * 5. K#O := 1. I#K := J#J. E#I := 3. F#I := I#K. H#M := 5. K#K := 6. I#L := F#J + E#I. F#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#J := E#I / H#M - K#K. F#J := E#I * K#O + K#K. L#K := F#J * 5. K#O := 1. I#K := J#J. E#I := 3. F#I := I#K. H#M := 5. K#K := 6. I#L := F#J + E#I.", "query": "F#I", "answer": 3, "cot": "K#K = 6 -> K#K = 6\nE#I = 3 -> E#I = 3\nH#M = 5 -> H#M = 5\nJ#J = E#I / H#M - K#K\n = 3 / 5 - 6\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n 2 - 6 = (2 - 6) mod 7 = 3\n => J#J = 3\nI#K = J#J = 3\nF#I = I#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#K": 1, "E#I": 1, "H#M": 1, "K#O": 1, "J#J": 2, "F#J": 2, "L#K": 3, "I#K": 3, "I#L": 3, "F#I": 4}, "id": "mod7_arith_d4_0090"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := 4. K#I := F#J + L#J / 2. J#N := G#K / E#K - I#N. I#N := 5. H#M := 4 + K#I * I#N. E#K := L#O + F#J. G#K := L#O. I#J := H#M. L#O := 5. F#P := 5 + L#O * E#K. L#J := 6. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := 4. K#I := F#J + L#J / 2. J#N := G#K / E#K - I#N. I#N := 5. H#M := 4 + K#I * I#N. E#K := L#O + F#J. G#K := L#O. I#J := H#M. L#O := 5. F#P := 5 + L#O * E#K. L#J := 6.", "query": "I#J", "answer": 3, "cot": "L#J = 6 -> L#J = 6\nF#J = 4 -> F#J = 4\nI#N = 5 -> I#N = 5\nK#I = F#J + L#J / 2\n = 4 + 6 / 2\n 4 + 6 = (4 + 6) mod 7 = 3\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => K#I = 5\nH#M = 4 + K#I * I#N\n = 4 + 5 * 5\n 4 + 5 = (4 + 5) mod 7 = 2\n 2 * 5 = (2 × 5) mod 7 = 3\n => H#M = 3\nI#J = H#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 6, "num_distractors": 0, "var_layer": {"L#J": 1, "F#J": 1, "I#N": 1, "L#O": 1, "E#K": 2, "K#I": 2, "G#K": 2, "F#P": 3, "H#M": 3, "J#N": 3, "I#J": 4}, "id": "mod7_arith_d4_0091"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#P := G#I. K#L := 4. J#J := K#L. G#I := K#I. I#L := K#I * J#J. E#J := 6. F#P := 6. K#I := F#P / K#L - E#J. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#P := G#I. K#L := 4. J#J := K#L. G#I := K#I. I#L := K#I * J#J. E#J := 6. F#P := 6. K#I := F#P / K#L - E#J.", "query": "H#P", "answer": 6, "cot": "F#P = 6 -> F#P = 6\nE#J = 6 -> E#J = 6\nK#L = 4 -> K#L = 4\nK#I = F#P / K#L - E#J\n = 6 / 4 - 6\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n 5 - 6 = (5 - 6) mod 7 = 6\n => K#I = 6\nG#I = K#I = 6\nH#P = G#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#P": 1, "E#J": 1, "K#L": 1, "K#I": 2, "J#J": 2, "G#I": 3, "I#L": 3, "H#P": 4}, "id": "mod7_arith_d4_0092"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := L#M + I#J. J#J := L#N. K#O := J#J / L#N * L#J. I#J := F#K - K#K. K#K := 4. K#I := 3 / L#M. L#M := F#K. L#N := 4 - F#K. F#K := 5. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := L#M + I#J. J#J := L#N. K#O := J#J / L#N * L#J. I#J := F#K - K#K. K#K := 4. K#I := 3 / L#M. L#M := F#K. L#N := 4 - F#K. F#K := 5.", "query": "K#O", "answer": 6, "cot": "F#K = 5 -> F#K = 5\nK#K = 4 -> K#K = 4\nL#N = 4 - F#K\n = 4 - 5\n 4 - 5 = (4 - 5) mod 7 = 6\n => L#N = 6\nI#J = F#K - K#K\n = 5 - 4\n 5 - 4 = (5 - 4) mod 7 = 1\n => I#J = 1\nL#M = F#K = 5\nJ#J = L#N = 6\nL#J = L#M + I#J\n = 5 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n => L#J = 6\nK#O = J#J / L#N * L#J\n = 6 / 6 * 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n 1 * 6 = (1 × 6) mod 7 = 6\n => K#O = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 8, "num_distractors": 0, "var_layer": {"F#K": 1, "K#K": 1, "L#N": 2, "I#J": 2, "L#M": 2, "J#J": 3, "L#J": 3, "K#I": 3, "K#O": 4}, "id": "mod7_arith_d4_0093"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := 2. E#N := 3 * K#N. J#L := 1. L#M := K#M + J#L + K#N. K#M := 4. H#M := L#M. K#P := K#N + E#P. E#P := K#M + E#N. K#N := 3. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := 2. E#N := 3 * K#N. J#L := 1. L#M := K#M + J#L + K#N. K#M := 4. H#M := L#M. K#P := K#N + E#P. E#P := K#M + E#N. K#N := 3.", "query": "K#P", "answer": 2, "cot": "K#N = 3 -> K#N = 3\nK#M = 4 -> K#M = 4\nE#N = 3 * K#N\n = 3 * 3\n 3 * 3 = (3 × 3) mod 7 = 2\n => E#N = 2\nE#P = K#M + E#N\n = 4 + 2\n 4 + 2 = (4 + 2) mod 7 = 6\n => E#P = 6\nK#P = K#N + E#P\n = 3 + 6\n 3 + 6 = (3 + 6) mod 7 = 2\n => K#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#L": 1, "K#N": 1, "K#M": 1, "H#K": 1, "L#M": 2, "E#N": 2, "H#M": 3, "E#P": 3, "K#P": 4}, "id": "mod7_arith_d4_0094"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := H#J - J#I + K#N. G#M := I#I * 5 - K#N. K#N := 6. L#J := K#N / 6. H#J := K#N / I#I. J#I := K#N * G#M. K#L := H#J * I#I. I#I := 6. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := H#J - J#I + K#N. G#M := I#I * 5 - K#N. K#N := 6. L#J := K#N / 6. H#J := K#N / I#I. J#I := K#N * G#M. K#L := H#J * I#I. I#I := 6.", "query": "F#J", "answer": 3, "cot": "K#N = 6 -> K#N = 6\nI#I = 6 -> I#I = 6\nH#J = K#N / I#I\n = 6 / 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => H#J = 1\nG#M = I#I * 5 - K#N\n = 6 * 5 - 6\n 6 * 5 = (6 × 5) mod 7 = 2\n 2 - 6 = (2 - 6) mod 7 = 3\n => G#M = 3\nJ#I = K#N * G#M\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => J#I = 4\nF#J = H#J - J#I + K#N\n = 1 - 4 + 6\n 1 - 4 = (1 - 4) mod 7 = 4\n 4 + 6 = (4 + 6) mod 7 = 3\n => F#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#N": 1, "I#I": 1, "H#J": 2, "L#J": 2, "G#M": 2, "K#L": 3, "J#I": 3, "F#J": 4}, "id": "mod7_arith_d4_0095"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#L := 3. K#L := 3. H#N := K#L * L#P. J#I := G#J - 5. G#J := L#P + K#L. F#M := H#L - K#L. F#P := K#L - F#M. E#I := 3 + F#P. L#P := 1. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#L := 3. K#L := 3. H#N := K#L * L#P. J#I := G#J - 5. G#J := L#P + K#L. F#M := H#L - K#L. F#P := K#L - F#M. E#I := 3 + F#P. L#P := 1.", "query": "E#I", "answer": 6, "cot": "H#L = 3 -> H#L = 3\nK#L = 3 -> K#L = 3\nF#M = H#L - K#L\n = 3 - 3\n 3 - 3 = (3 - 3) mod 7 = 0\n => F#M = 0\nF#P = K#L - F#M\n = 3 - 0\n 3 - 0 = (3 - 0) mod 7 = 3\n => F#P = 3\nE#I = 3 + F#P\n = 3 + 3\n 3 + 3 = (3 + 3) mod 7 = 6\n => E#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#L": 1, "K#L": 1, "L#P": 1, "G#J": 2, "H#N": 2, "F#M": 2, "F#P": 3, "J#I": 3, "E#I": 4}, "id": "mod7_arith_d4_0096"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := G#O - J#O. G#O := 4. J#P := J#O * G#O / 3. L#J := E#K / J#P * G#O. H#K := H#L. J#O := 3. H#L := J#P + E#K. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := G#O - J#O. G#O := 4. J#P := J#O * G#O / 3. L#J := E#K / J#P * G#O. H#K := H#L. J#O := 3. H#L := J#P + E#K.", "query": "H#K", "answer": 5, "cot": "G#O = 4 -> G#O = 4\nJ#O = 3 -> J#O = 3\nE#K = G#O - J#O\n = 4 - 3\n 4 - 3 = (4 - 3) mod 7 = 1\n => E#K = 1\nJ#P = J#O * G#O / 3\n = 3 * 4 / 3\n 3 * 4 = (3 × 4) mod 7 = 5\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n => J#P = 4\nH#L = J#P + E#K\n = 4 + 1\n 4 + 1 = (4 + 1) mod 7 = 5\n => H#L = 5\nH#K = H#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#O": 1, "J#O": 1, "E#K": 2, "J#P": 2, "H#L": 3, "L#J": 3, "H#K": 4}, "id": "mod7_arith_d4_0097"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := 3 / F#O + G#K. I#I := G#P. G#K := 4. L#J := 2. K#J := L#J * G#K / 4. L#P := 4 / I#J. F#O := 5. F#K := 3 * F#O. G#P := I#J - 2. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := 3 / F#O + G#K. I#I := G#P. G#K := 4. L#J := 2. K#J := L#J * G#K / 4. L#P := 4 / I#J. F#O := 5. F#K := 3 * F#O. G#P := I#J - 2.", "query": "I#I", "answer": 4, "cot": "F#O = 5 -> F#O = 5\nG#K = 4 -> G#K = 4\nI#J = 3 / F#O + G#K\n = 3 / 5 + 4\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n 2 + 4 = (2 + 4) mod 7 = 6\n => I#J = 6\nG#P = I#J - 2\n = 6 - 2\n 6 - 2 = (6 - 2) mod 7 = 4\n => G#P = 4\nI#I = G#P = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#J": 1, "F#O": 1, "G#K": 1, "I#J": 2, "F#K": 2, "K#J": 2, "G#P": 3, "L#P": 3, "I#I": 4}, "id": "mod7_arith_d4_0098"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#L := H#O * 4. H#N := J#L + L#I. L#O := 2. E#K := 2. H#L := J#L. K#K := L#O / 3. H#O := 2. L#I := 5 - J#L. E#I := 2. H#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#L := H#O * 4. H#N := J#L + L#I. L#O := 2. E#K := 2. H#L := J#L. K#K := L#O / 3. H#O := 2. L#I := 5 - J#L. E#I := 2.", "query": "H#N", "answer": 5, "cot": "H#O = 2 -> H#O = 2\nJ#L = H#O * 4\n = 2 * 4\n 2 * 4 = (2 × 4) mod 7 = 1\n => J#L = 1\nL#I = 5 - J#L\n = 5 - 1\n 5 - 1 = (5 - 1) mod 7 = 4\n => L#I = 4\nH#N = J#L + L#I\n = 1 + 4\n 1 + 4 = (1 + 4) mod 7 = 5\n => H#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#O": 1, "L#O": 1, "E#K": 1, "E#I": 1, "K#K": 2, "J#L": 2, "H#L": 3, "L#I": 3, "H#N": 4}, "id": "mod7_arith_d4_0099"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#O := E#N * 5. H#P := 3. J#O := 4 * H#P - E#N. I#J := 5. I#I := E#N. G#P := E#N - E#O. L#P := I#J. E#N := H#P + 6. E#L := H#P / 5. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#O := E#N * 5. H#P := 3. J#O := 4 * H#P - E#N. I#J := 5. I#I := E#N. G#P := E#N - E#O. L#P := I#J. E#N := H#P + 6. E#L := H#P / 5.", "query": "G#P", "answer": 6, "cot": "H#P = 3 -> H#P = 3\nE#N = H#P + 6\n = 3 + 6\n 3 + 6 = (3 + 6) mod 7 = 2\n => E#N = 2\nE#O = E#N * 5\n = 2 * 5\n 2 * 5 = (2 × 5) mod 7 = 3\n => E#O = 3\nG#P = E#N - E#O\n = 2 - 3\n 2 - 3 = (2 - 3) mod 7 = 6\n => G#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#P": 1, "I#J": 1, "E#L": 2, "E#N": 2, "L#P": 2, "J#O": 3, "I#I": 3, "E#O": 3, "G#P": 4}, "id": "mod7_arith_d4_0100"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := 5. J#L := L#P - J#K. K#J := 4. J#K := I#O. I#O := H#J * 4. L#P := K#J. E#J := F#J - K#J. J#J := I#O / 4. F#J := 2. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := 5. J#L := L#P - J#K. K#J := 4. J#K := I#O. I#O := H#J * 4. L#P := K#J. E#J := F#J - K#J. J#J := I#O / 4. F#J := 2.", "query": "J#L", "answer": 5, "cot": "H#J = 5 -> H#J = 5\nK#J = 4 -> K#J = 4\nL#P = K#J = 4\nI#O = H#J * 4\n = 5 * 4\n 5 * 4 = (5 × 4) mod 7 = 6\n => I#O = 6\nJ#K = I#O = 6\nJ#L = L#P - J#K\n = 4 - 6\n 4 - 6 = (4 - 6) mod 7 = 5\n => J#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#J": 1, "H#J": 1, "K#J": 1, "E#J": 2, "L#P": 2, "I#O": 2, "J#K": 3, "J#J": 3, "J#L": 4}, "id": "mod7_arith_d4_0101"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := I#O * 5. I#O := H#P * L#M. F#I := I#O * L#N. I#P := F#I + L#N * L#P. L#N := L#M + H#P. H#P := 2. L#M := 1. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := I#O * 5. I#O := H#P * L#M. F#I := I#O * L#N. I#P := F#I + L#N * L#P. L#N := L#M + H#P. H#P := 2. L#M := 1.", "query": "I#P", "answer": 6, "cot": "H#P = 2 -> H#P = 2\nL#M = 1 -> L#M = 1\nI#O = H#P * L#M\n = 2 * 1\n 2 * 1 = (2 × 1) mod 7 = 2\n => I#O = 2\nL#N = L#M + H#P\n = 1 + 2\n 1 + 2 = (1 + 2) mod 7 = 3\n => L#N = 3\nF#I = I#O * L#N\n = 2 * 3\n 2 * 3 = (2 × 3) mod 7 = 6\n => F#I = 6\nL#P = I#O * 5\n = 2 * 5\n 2 * 5 = (2 × 5) mod 7 = 3\n => L#P = 3\nI#P = F#I + L#N * L#P\n = 6 + 3 * 3\n 6 + 3 = (6 + 3) mod 7 = 2\n 2 * 3 = (2 × 3) mod 7 = 6\n => I#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 7, "num_distractors": 0, "var_layer": {"H#P": 1, "L#M": 1, "I#O": 2, "L#N": 2, "F#I": 3, "L#P": 3, "I#P": 4}, "id": "mod7_arith_d4_0102"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#I := 1. L#M := I#M. G#N := 5. I#M := 4. F#P := G#N * J#I. F#N := F#K * 6. F#K := L#M. K#M := 6 + L#M - I#M. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#I := 1. L#M := I#M. G#N := 5. I#M := 4. F#P := G#N * J#I. F#N := F#K * 6. F#K := L#M. K#M := 6 + L#M - I#M.", "query": "F#N", "answer": 3, "cot": "I#M = 4 -> I#M = 4\nL#M = I#M = 4\nF#K = L#M = 4\nF#N = F#K * 6\n = 4 * 6\n 4 * 6 = (4 × 6) mod 7 = 3\n => F#N = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#M": 1, "G#N": 1, "J#I": 1, "F#P": 2, "L#M": 2, "F#K": 3, "K#M": 3, "F#N": 4}, "id": "mod7_arith_d4_0103"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := G#P * H#O. E#O := 3. I#M := K#M. G#P := K#I * K#M / E#O. J#K := G#P - I#M. I#I := 6 * E#L. K#M := 2. K#I := 2. H#O := K#M - E#O. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := G#P * H#O. E#O := 3. I#M := K#M. G#P := K#I * K#M / E#O. J#K := G#P - I#M. I#I := 6 * E#L. K#M := 2. K#I := 2. H#O := K#M - E#O.", "query": "I#I", "answer": 6, "cot": "K#I = 2 -> K#I = 2\nK#M = 2 -> K#M = 2\nE#O = 3 -> E#O = 3\nH#O = K#M - E#O\n = 2 - 3\n 2 - 3 = (2 - 3) mod 7 = 6\n => H#O = 6\nG#P = K#I * K#M / E#O\n = 2 * 2 / 3\n 2 * 2 = (2 × 2) mod 7 = 4\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => G#P = 6\nE#L = G#P * H#O\n = 6 * 6\n 6 * 6 = (6 × 6) mod 7 = 1\n => E#L = 1\nI#I = 6 * E#L\n = 6 * 1\n 6 * 1 = (6 × 1) mod 7 = 6\n => I#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"K#I": 1, "K#M": 1, "E#O": 1, "H#O": 2, "I#M": 2, "G#P": 2, "J#K": 3, "E#L": 3, "I#I": 4}, "id": "mod7_arith_d4_0104"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := E#J. J#P := 3. I#M := J#P * E#J. H#K := 6. H#J := H#M / 2 + K#L. H#M := E#J. G#O := J#P / H#K. E#J := J#P. H#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := E#J. J#P := 3. I#M := J#P * E#J. H#K := 6. H#J := H#M / 2 + K#L. H#M := E#J. G#O := J#P / H#K. E#J := J#P.", "query": "H#J", "answer": 1, "cot": "J#P = 3 -> J#P = 3\nE#J = J#P = 3\nH#M = E#J = 3\nK#L = E#J = 3\nH#J = H#M / 2 + K#L\n = 3 / 2 + 3\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n 5 + 3 = (5 + 3) mod 7 = 1\n => H#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#K": 1, "J#P": 1, "E#J": 2, "G#O": 2, "H#M": 3, "K#L": 3, "I#M": 3, "H#J": 4}, "id": "mod7_arith_d4_0105"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := 3. L#M := 6 * G#I + I#J. K#I := L#M. K#J := E#K - 3. G#I := 3 - 6 + E#K. H#L := I#J. I#J := 6. I#K := H#L. L#I := 1. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := 3. L#M := 6 * G#I + I#J. K#I := L#M. K#J := E#K - 3. G#I := 3 - 6 + E#K. H#L := I#J. I#J := 6. I#K := H#L. L#I := 1.", "query": "K#I", "answer": 6, "cot": "E#K = 3 -> E#K = 3\nI#J = 6 -> I#J = 6\nG#I = 3 - 6 + E#K\n = 3 - 6 + 3\n 3 - 6 = (3 - 6) mod 7 = 4\n 4 + 3 = (4 + 3) mod 7 = 0\n => G#I = 0\nL#M = 6 * G#I + I#J\n = 6 * 0 + 6\n 6 * 0 = (6 × 0) mod 7 = 0\n 0 + 6 = (0 + 6) mod 7 = 6\n => L#M = 6\nK#I = L#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#I": 1, "E#K": 1, "I#J": 1, "G#I": 2, "K#J": 2, "H#L": 2, "I#K": 3, "L#M": 3, "K#I": 4}, "id": "mod7_arith_d4_0106"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#P := 5 * E#M. K#I := E#M - E#J. J#J := E#I. L#K := 4. H#O := K#I / K#P. E#I := 5. E#M := 5. H#K := K#P / 6. E#J := E#M / E#I. J#L := 2. H#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#P := 5 * E#M. K#I := E#M - E#J. J#J := E#I. L#K := 4. H#O := K#I / K#P. E#I := 5. E#M := 5. H#K := K#P / 6. E#J := E#M / E#I. J#L := 2.", "query": "H#O", "answer": 1, "cot": "E#M = 5 -> E#M = 5\nE#I = 5 -> E#I = 5\nE#J = E#M / E#I\n = 5 / 5\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n => E#J = 1\nK#P = 5 * E#M\n = 5 * 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => K#P = 4\nK#I = E#M - E#J\n = 5 - 1\n 5 - 1 = (5 - 1) mod 7 = 4\n => K#I = 4\nH#O = K#I / K#P\n = 4 / 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => H#O = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#M": 1, "L#K": 1, "J#L": 1, "E#I": 1, "E#J": 2, "J#J": 2, "K#P": 2, "K#I": 3, "H#K": 3, "H#O": 4}, "id": "mod7_arith_d4_0107"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := 4 / K#I. F#P := E#J * I#P. E#J := 6. L#N := 4. K#K := J#L. E#K := I#M / K#I. H#M := E#K * 6. I#P := 4. K#I := 4 + I#M. I#M := 2. J#L := E#J * I#P - L#N. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := 4 / K#I. F#P := E#J * I#P. E#J := 6. L#N := 4. K#K := J#L. E#K := I#M / K#I. H#M := E#K * 6. I#P := 4. K#I := 4 + I#M. I#M := 2. J#L := E#J * I#P - L#N.", "query": "H#M", "answer": 2, "cot": "I#M = 2 -> I#M = 2\nK#I = 4 + I#M\n = 4 + 2\n 4 + 2 = (4 + 2) mod 7 = 6\n => K#I = 6\nE#K = I#M / K#I\n = 2 / 6\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => E#K = 5\nH#M = E#K * 6\n = 5 * 6\n 5 * 6 = (5 × 6) mod 7 = 2\n => H#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#N": 1, "E#J": 1, "I#P": 1, "I#M": 1, "K#I": 2, "J#L": 2, "F#P": 2, "E#K": 3, "K#K": 3, "K#J": 3, "H#M": 4}, "id": "mod7_arith_d4_0108"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := K#M * H#L. E#M := K#O - H#L + G#O. H#L := G#O - K#M. I#L := 3 - G#O * H#L. L#N := 4 - K#M * G#O. G#O := 2. K#M := 4. K#O := L#N / H#L. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := K#M * H#L. E#M := K#O - H#L + G#O. H#L := G#O - K#M. I#L := 3 - G#O * H#L. L#N := 4 - K#M * G#O. G#O := 2. K#M := 4. K#O := L#N / H#L.", "query": "E#M", "answer": 4, "cot": "K#M = 4 -> K#M = 4\nG#O = 2 -> G#O = 2\nL#N = 4 - K#M * G#O\n = 4 - 4 * 2\n 4 - 4 = (4 - 4) mod 7 = 0\n 0 * 2 = (0 × 2) mod 7 = 0\n => L#N = 0\nH#L = G#O - K#M\n = 2 - 4\n 2 - 4 = (2 - 4) mod 7 = 5\n => H#L = 5\nK#O = L#N / H#L\n = 0 / 5\n 0 / 5 = (0 × 5⁻¹) mod 7 = 0\n => K#O = 0\nE#M = K#O - H#L + G#O\n = 0 - 5 + 2\n 0 - 5 = (0 - 5) mod 7 = 2\n 2 + 2 = (2 + 2) mod 7 = 4\n => E#M = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#M": 1, "G#O": 1, "L#N": 2, "H#L": 2, "K#O": 3, "I#L": 3, "L#M": 3, "E#M": 4}, "id": "mod7_arith_d4_0109"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#N := 5 / H#K. H#L := 5. F#L := 1. K#K := 6. J#O := F#L - 5 - H#L. E#J := I#M. H#K := 5 + H#L - F#L. I#M := 6 / H#K * 2. G#K := 6. I#L := H#K * H#L. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#N := 5 / H#K. H#L := 5. F#L := 1. K#K := 6. J#O := F#L - 5 - H#L. E#J := I#M. H#K := 5 + H#L - F#L. I#M := 6 / H#K * 2. G#K := 6. I#L := H#K * H#L.", "query": "E#J", "answer": 6, "cot": "F#L = 1 -> F#L = 1\nH#L = 5 -> H#L = 5\nH#K = 5 + H#L - F#L\n = 5 + 5 - 1\n 5 + 5 = (5 + 5) mod 7 = 3\n 3 - 1 = (3 - 1) mod 7 = 2\n => H#K = 2\nI#M = 6 / H#K * 2\n = 6 / 2 * 2\n 6 / 2 = (6 × 2⁻¹) mod 7 = 3\n 3 * 2 = (3 × 2) mod 7 = 6\n => I#M = 6\nE#J = I#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#L": 1, "H#L": 1, "K#K": 1, "G#K": 1, "H#K": 2, "J#O": 2, "I#N": 3, "I#M": 3, "I#L": 3, "E#J": 4}, "id": "mod7_arith_d4_0110"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#M := L#M. F#M := 4. L#M := J#J / H#K. J#O := H#K * J#N. H#K := 1. J#N := J#J / F#M * H#K. J#J := 5. E#P := J#N + J#O / F#M. E#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#M := L#M. F#M := 4. L#M := J#J / H#K. J#O := H#K * J#N. H#K := 1. J#N := J#J / F#M * H#K. J#J := 5. E#P := J#N + J#O / F#M.", "query": "E#P", "answer": 5, "cot": "H#K = 1 -> H#K = 1\nF#M = 4 -> F#M = 4\nJ#J = 5 -> J#J = 5\nJ#N = J#J / F#M * H#K\n = 5 / 4 * 1\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n 3 * 1 = (3 × 1) mod 7 = 3\n => J#N = 3\nJ#O = H#K * J#N\n = 1 * 3\n 1 * 3 = (1 × 3) mod 7 = 3\n => J#O = 3\nE#P = J#N + J#O / F#M\n = 3 + 3 / 4\n 3 + 3 = (3 + 3) mod 7 = 6\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n => E#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"H#K": 1, "F#M": 1, "J#J": 1, "J#N": 2, "L#M": 2, "G#M": 3, "J#O": 3, "E#P": 4}, "id": "mod7_arith_d4_0111"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := F#I + H#M - K#J. H#M := G#M. G#N := G#M + I#P. H#L := K#J. G#M := 2. L#N := F#I + G#M. F#I := 2. K#J := F#I + G#M. G#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := F#I + H#M - K#J. H#M := G#M. G#N := G#M + I#P. H#L := K#J. G#M := 2. L#N := F#I + G#M. F#I := 2. K#J := F#I + G#M.", "query": "G#N", "answer": 2, "cot": "F#I = 2 -> F#I = 2\nG#M = 2 -> G#M = 2\nK#J = F#I + G#M\n = 2 + 2\n 2 + 2 = (2 + 2) mod 7 = 4\n => K#J = 4\nH#M = G#M = 2\nI#P = F#I + H#M - K#J\n = 2 + 2 - 4\n 2 + 2 = (2 + 2) mod 7 = 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => I#P = 0\nG#N = G#M + I#P\n = 2 + 0\n 2 + 0 = (2 + 0) mod 7 = 2\n => G#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#I": 1, "G#M": 1, "K#J": 2, "L#N": 2, "H#M": 2, "H#L": 3, "I#P": 3, "G#N": 4}, "id": "mod7_arith_d4_0112"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := H#N. L#I := 2 / G#O + G#J. G#O := 6. H#M := G#O * E#P. E#P := 6. G#J := I#P / E#P - G#O. H#N := G#J. I#P := 4. J#J := 3 - E#P + G#O. G#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := H#N. L#I := 2 / G#O + G#J. G#O := 6. H#M := G#O * E#P. E#P := 6. G#J := I#P / E#P - G#O. H#N := G#J. I#P := 4. J#J := 3 - E#P + G#O.", "query": "G#N", "answer": 4, "cot": "E#P = 6 -> E#P = 6\nI#P = 4 -> I#P = 4\nG#O = 6 -> G#O = 6\nG#J = I#P / E#P - G#O\n = 4 / 6 - 6\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n 3 - 6 = (3 - 6) mod 7 = 4\n => G#J = 4\nH#N = G#J = 4\nG#N = H#N = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#P": 1, "I#P": 1, "G#O": 1, "H#M": 2, "G#J": 2, "J#J": 2, "L#I": 3, "H#N": 3, "G#N": 4}, "id": "mod7_arith_d4_0113"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := 5. K#P := G#N * I#N. I#L := G#N - K#P * 3. J#M := 4. G#N := 3. J#L := E#L / G#N. K#K := K#M + 6 / J#M. I#N := 6. K#M := K#P * 6. K#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := 5. K#P := G#N * I#N. I#L := G#N - K#P * 3. J#M := 4. G#N := 3. J#L := E#L / G#N. K#K := K#M + 6 / J#M. I#N := 6. K#M := K#P * 6.", "query": "K#K", "answer": 4, "cot": "J#M = 4 -> J#M = 4\nG#N = 3 -> G#N = 3\nI#N = 6 -> I#N = 6\nK#P = G#N * I#N\n = 3 * 6\n 3 * 6 = (3 × 6) mod 7 = 4\n => K#P = 4\nK#M = K#P * 6\n = 4 * 6\n 4 * 6 = (4 × 6) mod 7 = 3\n => K#M = 3\nK#K = K#M + 6 / J#M\n = 3 + 6 / 4\n 3 + 6 = (3 + 6) mod 7 = 2\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n => K#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#L": 1, "J#M": 1, "G#N": 1, "I#N": 1, "K#P": 2, "J#L": 2, "K#M": 3, "I#L": 3, "K#K": 4}, "id": "mod7_arith_d4_0114"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#O := G#K * H#O - K#J. I#L := G#J + L#I + H#O. G#K := L#I. G#J := 5. L#L := L#I / L#O / H#O. L#I := 3. H#O := 6. I#P := G#K * I#L. K#J := L#I + G#J. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#O := G#K * H#O - K#J. I#L := G#J + L#I + H#O. G#K := L#I. G#J := 5. L#L := L#I / L#O / H#O. L#I := 3. H#O := 6. I#P := G#K * I#L. K#J := L#I + G#J.", "query": "L#L", "answer": 6, "cot": "H#O = 6 -> H#O = 6\nG#J = 5 -> G#J = 5\nL#I = 3 -> L#I = 3\nK#J = L#I + G#J\n = 3 + 5\n 3 + 5 = (3 + 5) mod 7 = 1\n => K#J = 1\nG#K = L#I = 3\nL#O = G#K * H#O - K#J\n = 3 * 6 - 1\n 3 * 6 = (3 × 6) mod 7 = 4\n 4 - 1 = (4 - 1) mod 7 = 3\n => L#O = 3\nL#L = L#I / L#O / H#O\n = 3 / 3 / 6\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n 1 / 6 = (1 × 6⁻¹) mod 7 = 6\n => L#L = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"H#O": 1, "G#J": 1, "L#I": 1, "K#J": 2, "G#K": 2, "I#L": 2, "I#P": 3, "L#O": 3, "L#L": 4}, "id": "mod7_arith_d4_0115"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := 4. K#N := F#P + E#L / L#M. L#M := L#P / 2 * 4. L#J := E#L. F#N := 3. E#L := J#M. J#M := 5. I#P := L#M / J#M. E#M := L#J / F#P - I#P. L#P := 5. E#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := 4. K#N := F#P + E#L / L#M. L#M := L#P / 2 * 4. L#J := E#L. F#N := 3. E#L := J#M. J#M := 5. I#P := L#M / J#M. E#M := L#J / F#P - I#P. L#P := 5.", "query": "E#M", "answer": 1, "cot": "L#P = 5 -> L#P = 5\nJ#M = 5 -> J#M = 5\nF#P = 4 -> F#P = 4\nL#M = L#P / 2 * 4\n = 5 / 2 * 4\n 5 / 2 = (5 × 2⁻¹) mod 7 = 6\n 6 * 4 = (6 × 4) mod 7 = 3\n => L#M = 3\nE#L = J#M = 5\nL#J = E#L = 5\nI#P = L#M / J#M\n = 3 / 5\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n => I#P = 2\nE#M = L#J / F#P - I#P\n = 5 / 4 - 2\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n 3 - 2 = (3 - 2) mod 7 = 1\n => E#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 8, "num_distractors": 0, "var_layer": {"L#P": 1, "J#M": 1, "F#N": 1, "F#P": 1, "L#M": 2, "E#L": 2, "L#J": 3, "I#P": 3, "K#N": 3, "E#M": 4}, "id": "mod7_arith_d4_0116"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#M := 1. L#K := K#J + 4. H#M := L#K. L#J := G#L. K#J := G#J. G#L := G#J. G#J := 5. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#M := 1. L#K := K#J + 4. H#M := L#K. L#J := G#L. K#J := G#J. G#L := G#J. G#J := 5.", "query": "H#M", "answer": 2, "cot": "G#J = 5 -> G#J = 5\nK#J = G#J = 5\nL#K = K#J + 4\n = 5 + 4\n 5 + 4 = (5 + 4) mod 7 = 2\n => L#K = 2\nH#M = L#K = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#J": 1, "F#M": 1, "K#J": 2, "G#L": 2, "L#K": 3, "L#J": 3, "H#M": 4}, "id": "mod7_arith_d4_0117"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := H#L + J#P * 5. F#O := H#O * 4. L#O := J#P * H#O. H#L := 5. K#L := L#I. J#P := 5. L#I := H#L. I#I := F#O. H#O := 2 * J#P. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := H#L + J#P * 5. F#O := H#O * 4. L#O := J#P * H#O. H#L := 5. K#L := L#I. J#P := 5. L#I := H#L. I#I := F#O. H#O := 2 * J#P.", "query": "I#I", "answer": 5, "cot": "J#P = 5 -> J#P = 5\nH#O = 2 * J#P\n = 2 * 5\n 2 * 5 = (2 × 5) mod 7 = 3\n => H#O = 3\nF#O = H#O * 4\n = 3 * 4\n 3 * 4 = (3 × 4) mod 7 = 5\n => F#O = 5\nI#I = F#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#L": 1, "J#P": 1, "K#M": 2, "L#I": 2, "H#O": 2, "L#O": 3, "K#L": 3, "F#O": 3, "I#I": 4}, "id": "mod7_arith_d4_0118"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#L := 3. E#J := 2. E#M := K#M + H#L. F#K := L#M. E#O := K#M / 6. K#M := 2. I#K := I#M / E#M. L#M := 2 / E#M / I#J. I#M := K#M / 4. I#J := 3. F#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#L := 3. E#J := 2. E#M := K#M + H#L. F#K := L#M. E#O := K#M / 6. K#M := 2. I#K := I#M / E#M. L#M := 2 / E#M / I#J. I#M := K#M / 4. I#J := 3.", "query": "F#K", "answer": 2, "cot": "K#M = 2 -> K#M = 2\nI#J = 3 -> I#J = 3\nH#L = 3 -> H#L = 3\nE#M = K#M + H#L\n = 2 + 3\n 2 + 3 = (2 + 3) mod 7 = 5\n => E#M = 5\nL#M = 2 / E#M / I#J\n = 2 / 5 / 3\n 2 / 5 = (2 × 5⁻¹) mod 7 = 6\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => L#M = 2\nF#K = L#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#M": 1, "I#J": 1, "E#J": 1, "H#L": 1, "E#M": 2, "I#M": 2, "E#O": 2, "I#K": 3, "L#M": 3, "F#K": 4}, "id": "mod7_arith_d4_0119"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#K := L#I. L#I := 5. E#P := 4. F#I := L#I * 4 * E#P. I#N := K#K / F#I. I#J := 6 + G#K / L#I. I#M := K#K + F#I. G#P := E#P. G#K := G#P - K#K - F#I. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#K := L#I. L#I := 5. E#P := 4. F#I := L#I * 4 * E#P. I#N := K#K / F#I. I#J := 6 + G#K / L#I. I#M := K#K + F#I. G#P := E#P. G#K := G#P - K#K - F#I.", "query": "I#J", "answer": 6, "cot": "L#I = 5 -> L#I = 5\nE#P = 4 -> E#P = 4\nF#I = L#I * 4 * E#P\n = 5 * 4 * 4\n 5 * 4 = (5 × 4) mod 7 = 6\n 6 * 4 = (6 × 4) mod 7 = 3\n => F#I = 3\nK#K = L#I = 5\nG#P = E#P = 4\nG#K = G#P - K#K - F#I\n = 4 - 5 - 3\n 4 - 5 = (4 - 5) mod 7 = 6\n 6 - 3 = (6 - 3) mod 7 = 3\n => G#K = 3\nI#J = 6 + G#K / L#I\n = 6 + 3 / 5\n 6 + 3 = (6 + 3) mod 7 = 2\n 2 / 5 = (2 × 5⁻¹) mod 7 = 6\n => I#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"L#I": 1, "E#P": 1, "F#I": 2, "K#K": 2, "G#P": 2, "I#M": 3, "G#K": 3, "I#N": 3, "I#J": 4}, "id": "mod7_arith_d4_0120"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#L := K#O + K#N. L#P := J#K / K#O. I#I := G#L + G#J. J#K := K#N. I#M := K#O. K#N := 3. G#J := K#O - J#K. G#P := J#K + I#M - K#N. K#O := 1. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#L := K#O + K#N. L#P := J#K / K#O. I#I := G#L + G#J. J#K := K#N. I#M := K#O. K#N := 3. G#J := K#O - J#K. G#P := J#K + I#M - K#N. K#O := 1.", "query": "I#I", "answer": 2, "cot": "K#N = 3 -> K#N = 3\nK#O = 1 -> K#O = 1\nJ#K = K#N = 3\nG#L = K#O + K#N\n = 1 + 3\n 1 + 3 = (1 + 3) mod 7 = 4\n => G#L = 4\nG#J = K#O - J#K\n = 1 - 3\n 1 - 3 = (1 - 3) mod 7 = 5\n => G#J = 5\nI#I = G#L + G#J\n = 4 + 5\n 4 + 5 = (4 + 5) mod 7 = 2\n => I#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#N": 1, "K#O": 1, "I#M": 2, "J#K": 2, "G#L": 2, "L#P": 3, "G#J": 3, "G#P": 3, "I#I": 4}, "id": "mod7_arith_d4_0121"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := F#J + E#J. F#J := 4. H#I := F#J + G#O. F#K := G#O - F#J. E#J := 3. L#I := 6 / 4 * G#O. I#M := G#O / H#I * E#J. I#P := F#J / E#J. I#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := F#J + E#J. F#J := 4. H#I := F#J + G#O. F#K := G#O - F#J. E#J := 3. L#I := 6 / 4 * G#O. I#M := G#O / H#I * E#J. I#P := F#J / E#J.", "query": "I#M", "answer": 0, "cot": "F#J = 4 -> F#J = 4\nE#J = 3 -> E#J = 3\nG#O = F#J + E#J\n = 4 + 3\n 4 + 3 = (4 + 3) mod 7 = 0\n => G#O = 0\nH#I = F#J + G#O\n = 4 + 0\n 4 + 0 = (4 + 0) mod 7 = 4\n => H#I = 4\nI#M = G#O / H#I * E#J\n = 0 / 4 * 3\n 0 / 4 = (0 × 4⁻¹) mod 7 = 0\n 0 * 3 = (0 × 3) mod 7 = 0\n => I#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#J": 1, "E#J": 1, "G#O": 2, "I#P": 2, "L#I": 3, "F#K": 3, "H#I": 3, "I#M": 4}, "id": "mod7_arith_d4_0122"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := G#K. J#P := H#N + F#O. K#I := G#K / E#L. K#J := 3. E#L := 4. F#O := 2 + J#O. G#K := 5. H#N := G#K - K#I. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := G#K. J#P := H#N + F#O. K#I := G#K / E#L. K#J := 3. E#L := 4. F#O := 2 + J#O. G#K := 5. H#N := G#K - K#I.", "query": "J#P", "answer": 2, "cot": "E#L = 4 -> E#L = 4\nG#K = 5 -> G#K = 5\nK#I = G#K / E#L\n = 5 / 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => K#I = 3\nJ#O = G#K = 5\nH#N = G#K - K#I\n = 5 - 3\n 5 - 3 = (5 - 3) mod 7 = 2\n => H#N = 2\nF#O = 2 + J#O\n = 2 + 5\n 2 + 5 = (2 + 5) mod 7 = 0\n => F#O = 0\nJ#P = H#N + F#O\n = 2 + 0\n 2 + 0 = (2 + 0) mod 7 = 2\n => J#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 7, "num_distractors": 0, "var_layer": {"E#L": 1, "G#K": 1, "K#J": 1, "K#I": 2, "J#O": 2, "H#N": 3, "F#O": 3, "J#P": 4}, "id": "mod7_arith_d4_0123"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := 3. E#N := I#P. I#J := K#I + I#N * I#L. K#I := 1. I#L := 5. I#N := 6. E#K := I#L / G#O - K#I. I#P := E#K / G#O. G#L := G#O * I#J / I#N. E#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := 3. E#N := I#P. I#J := K#I + I#N * I#L. K#I := 1. I#L := 5. I#N := 6. E#K := I#L / G#O - K#I. I#P := E#K / G#O. G#L := G#O * I#J / I#N.", "query": "E#N", "answer": 1, "cot": "G#O = 3 -> G#O = 3\nK#I = 1 -> K#I = 1\nI#L = 5 -> I#L = 5\nE#K = I#L / G#O - K#I\n = 5 / 3 - 1\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n 4 - 1 = (4 - 1) mod 7 = 3\n => E#K = 3\nI#P = E#K / G#O\n = 3 / 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => I#P = 1\nE#N = I#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"I#N": 1, "G#O": 1, "K#I": 1, "I#L": 1, "E#K": 2, "I#J": 2, "G#L": 3, "I#P": 3, "E#N": 4}, "id": "mod7_arith_d4_0124"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := 3. J#J := 2. J#M := G#P / L#J + F#O. F#I := 6. L#M := F#O + J#M. L#J := J#J. F#O := G#P / J#J. H#N := L#J + F#O. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := 3. J#J := 2. J#M := G#P / L#J + F#O. F#I := 6. L#M := F#O + J#M. L#J := J#J. F#O := G#P / J#J. H#N := L#J + F#O.", "query": "L#M", "answer": 1, "cot": "J#J = 2 -> J#J = 2\nG#P = 3 -> G#P = 3\nL#J = J#J = 2\nF#O = G#P / J#J\n = 3 / 2\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => F#O = 5\nJ#M = G#P / L#J + F#O\n = 3 / 2 + 5\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => J#M = 3\nL#M = F#O + J#M\n = 5 + 3\n 5 + 3 = (5 + 3) mod 7 = 1\n => L#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#J": 1, "F#I": 1, "G#P": 1, "L#J": 2, "F#O": 2, "J#M": 3, "H#N": 3, "L#M": 4}, "id": "mod7_arith_d4_0125"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := H#O. J#N := K#N + 5. K#N := 5. H#O := K#O. H#M := J#N - 6. E#M := 2. H#L := H#O + 6. F#N := J#N + 3 - H#L. K#O := 3. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := H#O. J#N := K#N + 5. K#N := 5. H#O := K#O. H#M := J#N - 6. E#M := 2. H#L := H#O + 6. F#N := J#N + 3 - H#L. K#O := 3.", "query": "F#N", "answer": 4, "cot": "K#N = 5 -> K#N = 5\nK#O = 3 -> K#O = 3\nJ#N = K#N + 5\n = 5 + 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => J#N = 3\nH#O = K#O = 3\nH#L = H#O + 6\n = 3 + 6\n 3 + 6 = (3 + 6) mod 7 = 2\n => H#L = 2\nF#N = J#N + 3 - H#L\n = 3 + 3 - 2\n 3 + 3 = (3 + 3) mod 7 = 6\n 6 - 2 = (6 - 2) mod 7 = 4\n => F#N = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#N": 1, "E#M": 1, "K#O": 1, "J#N": 2, "H#O": 2, "L#M": 3, "H#L": 3, "H#M": 3, "F#N": 4}, "id": "mod7_arith_d4_0126"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#I := G#L - 2. G#L := K#M. I#O := 6. G#M := G#L. K#M := 2. L#N := I#O. F#L := 5. I#L := 5 + F#I + G#M. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#I := G#L - 2. G#L := K#M. I#O := 6. G#M := G#L. K#M := 2. L#N := I#O. F#L := 5. I#L := 5 + F#I + G#M.", "query": "I#L", "answer": 0, "cot": "K#M = 2 -> K#M = 2\nG#L = K#M = 2\nG#M = G#L = 2\nF#I = G#L - 2\n = 2 - 2\n 2 - 2 = (2 - 2) mod 7 = 0\n => F#I = 0\nI#L = 5 + F#I + G#M\n = 5 + 0 + 2\n 5 + 0 = (5 + 0) mod 7 = 5\n 5 + 2 = (5 + 2) mod 7 = 0\n => I#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#M": 1, "I#O": 1, "F#L": 1, "G#L": 2, "L#N": 2, "G#M": 3, "F#I": 3, "I#L": 4}, "id": "mod7_arith_d4_0127"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := I#M - E#O. I#J := L#J. I#M := L#J. E#K := 2 + I#M. K#I := 6. L#J := 5. E#O := 4 * K#I. J#M := L#J - E#K. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := I#M - E#O. I#J := L#J. I#M := L#J. E#K := 2 + I#M. K#I := 6. L#J := 5. E#O := 4 * K#I. J#M := L#J - E#K.", "query": "J#M", "answer": 5, "cot": "L#J = 5 -> L#J = 5\nI#M = L#J = 5\nE#K = 2 + I#M\n = 2 + 5\n 2 + 5 = (2 + 5) mod 7 = 0\n => E#K = 0\nJ#M = L#J - E#K\n = 5 - 0\n 5 - 0 = (5 - 0) mod 7 = 5\n => J#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#I": 1, "L#J": 1, "I#J": 2, "I#M": 2, "E#O": 2, "E#K": 3, "E#L": 3, "J#M": 4}, "id": "mod7_arith_d4_0128"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := L#J / H#I. G#N := L#L * H#I. L#L := 4. E#L := G#N * L#L. K#O := K#I + L#L. K#I := 6. L#J := L#L * 4. J#I := K#O - 5. H#I := L#L * 5 + K#I. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := L#J / H#I. G#N := L#L * H#I. L#L := 4. E#L := G#N * L#L. K#O := K#I + L#L. K#I := 6. L#J := L#L * 4. J#I := K#O - 5. H#I := L#L * 5 + K#I.", "query": "E#L", "answer": 3, "cot": "L#L = 4 -> L#L = 4\nK#I = 6 -> K#I = 6\nH#I = L#L * 5 + K#I\n = 4 * 5 + 6\n 4 * 5 = (4 × 5) mod 7 = 6\n 6 + 6 = (6 + 6) mod 7 = 5\n => H#I = 5\nG#N = L#L * H#I\n = 4 * 5\n 4 * 5 = (4 × 5) mod 7 = 6\n => G#N = 6\nE#L = G#N * L#L\n = 6 * 4\n 6 * 4 = (6 × 4) mod 7 = 3\n => E#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#L": 1, "K#I": 1, "L#J": 2, "H#I": 2, "K#O": 2, "G#N": 3, "H#J": 3, "J#I": 3, "E#L": 4}, "id": "mod7_arith_d4_0129"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#N := K#O - 3 * L#L. G#O := 5. K#O := L#O * L#L. L#L := 5. L#O := 3. J#N := L#O / L#L. L#M := L#O * K#O. J#I := G#O / L#M * K#O. J#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#N := K#O - 3 * L#L. G#O := 5. K#O := L#O * L#L. L#L := 5. L#O := 3. J#N := L#O / L#L. L#M := L#O * K#O. J#I := G#O / L#M * K#O.", "query": "J#I", "answer": 4, "cot": "G#O = 5 -> G#O = 5\nL#L = 5 -> L#L = 5\nL#O = 3 -> L#O = 3\nK#O = L#O * L#L\n = 3 * 5\n 3 * 5 = (3 × 5) mod 7 = 1\n => K#O = 1\nL#M = L#O * K#O\n = 3 * 1\n 3 * 1 = (3 × 1) mod 7 = 3\n => L#M = 3\nJ#I = G#O / L#M * K#O\n = 5 / 3 * 1\n 5 / 3 = (5 × 3⁻¹) mod 7 = 4\n 4 * 1 = (4 × 1) mod 7 = 4\n => J#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#O": 1, "L#L": 1, "L#O": 1, "J#N": 2, "K#O": 2, "L#M": 3, "K#N": 3, "J#I": 4}, "id": "mod7_arith_d4_0130"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#M := 3. H#L := 1. H#K := F#O + 4 - H#L. H#J := L#L * H#L - H#O. E#O := L#L / H#L * E#M. F#O := 4 / H#O. L#L := E#M. H#O := E#M. H#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#M := 3. H#L := 1. H#K := F#O + 4 - H#L. H#J := L#L * H#L - H#O. E#O := L#L / H#L * E#M. F#O := 4 / H#O. L#L := E#M. H#O := E#M.", "query": "H#K", "answer": 2, "cot": "H#L = 1 -> H#L = 1\nE#M = 3 -> E#M = 3\nH#O = E#M = 3\nF#O = 4 / H#O\n = 4 / 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => F#O = 6\nH#K = F#O + 4 - H#L\n = 6 + 4 - 1\n 6 + 4 = (6 + 4) mod 7 = 3\n 3 - 1 = (3 - 1) mod 7 = 2\n => H#K = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#L": 1, "E#M": 1, "L#L": 2, "H#O": 2, "E#O": 3, "F#O": 3, "H#J": 3, "H#K": 4}, "id": "mod7_arith_d4_0131"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := 4. H#P := 6. G#I := 5 - K#P. E#L := I#N / G#I. K#I := 6. L#K := G#I / K#O. I#N := K#L / K#O - H#P. H#I := K#P + G#I. K#O := K#P + 5 - K#I. K#P := 3. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := 4. H#P := 6. G#I := 5 - K#P. E#L := I#N / G#I. K#I := 6. L#K := G#I / K#O. I#N := K#L / K#O - H#P. H#I := K#P + G#I. K#O := K#P + 5 - K#I. K#P := 3.", "query": "E#L", "answer": 5, "cot": "H#P = 6 -> H#P = 6\nK#I = 6 -> K#I = 6\nK#L = 4 -> K#L = 4\nK#P = 3 -> K#P = 3\nK#O = K#P + 5 - K#I\n = 3 + 5 - 6\n 3 + 5 = (3 + 5) mod 7 = 1\n 1 - 6 = (1 - 6) mod 7 = 2\n => K#O = 2\nG#I = 5 - K#P\n = 5 - 3\n 5 - 3 = (5 - 3) mod 7 = 2\n => G#I = 2\nI#N = K#L / K#O - H#P\n = 4 / 2 - 6\n 4 / 2 = (4 × 2⁻¹) mod 7 = 2\n 2 - 6 = (2 - 6) mod 7 = 3\n => I#N = 3\nE#L = I#N / G#I\n = 3 / 2\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => E#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 8, "num_distractors": 0, "var_layer": {"H#P": 1, "K#I": 1, "K#L": 1, "K#P": 1, "K#O": 2, "G#I": 2, "I#N": 3, "L#K": 3, "H#I": 3, "E#L": 4}, "id": "mod7_arith_d4_0132"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#N := 5. G#I := K#L + F#N. H#O := I#K. K#L := E#M. E#P := 6. F#N := 1. I#K := F#N / L#N. E#M := F#N + 4. G#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#N := 5. G#I := K#L + F#N. H#O := I#K. K#L := E#M. E#P := 6. F#N := 1. I#K := F#N / L#N. E#M := F#N + 4.", "query": "G#I", "answer": 6, "cot": "F#N = 1 -> F#N = 1\nE#M = F#N + 4\n = 1 + 4\n 1 + 4 = (1 + 4) mod 7 = 5\n => E#M = 5\nK#L = E#M = 5\nG#I = K#L + F#N\n = 5 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n => G#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#N": 1, "L#N": 1, "E#P": 1, "E#M": 2, "I#K": 2, "H#O": 3, "K#L": 3, "G#I": 4}, "id": "mod7_arith_d4_0133"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 1. K#P := G#J - G#K * K#J. J#I := 6. F#I := K#J. G#K := J#N * E#I - J#I. E#M := 3 / K#J / J#N. E#I := K#J / J#N. G#J := J#I + F#I - 6. K#J := 1. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 1. K#P := G#J - G#K * K#J. J#I := 6. F#I := K#J. G#K := J#N * E#I - J#I. E#M := 3 / K#J / J#N. E#I := K#J / J#N. G#J := J#I + F#I - 6. K#J := 1.", "query": "K#P", "answer": 6, "cot": "J#N = 1 -> J#N = 1\nK#J = 1 -> K#J = 1\nJ#I = 6 -> J#I = 6\nF#I = K#J = 1\nE#I = K#J / J#N\n = 1 / 1\n 1 / 1 = (1 × 1⁻¹) mod 7 = 1\n => E#I = 1\nG#K = J#N * E#I - J#I\n = 1 * 1 - 6\n 1 * 1 = (1 × 1) mod 7 = 1\n 1 - 6 = (1 - 6) mod 7 = 2\n => G#K = 2\nG#J = J#I + F#I - 6\n = 6 + 1 - 6\n 6 + 1 = (6 + 1) mod 7 = 0\n 0 - 6 = (0 - 6) mod 7 = 1\n => G#J = 1\nK#P = G#J - G#K * K#J\n = 1 - 2 * 1\n 1 - 2 = (1 - 2) mod 7 = 6\n 6 * 1 = (6 × 1) mod 7 = 6\n => K#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 8, "num_distractors": 0, "var_layer": {"J#N": 1, "K#J": 1, "J#I": 1, "F#I": 2, "E#I": 2, "E#M": 2, "G#K": 3, "G#J": 3, "K#P": 4}, "id": "mod7_arith_d4_0134"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := J#N - J#L / 3. L#M := H#K + I#O. I#O := 2. I#I := I#O + H#K. I#P := I#J * L#M. H#K := 1. I#J := H#K + I#O. J#L := L#M - H#K. J#N := L#M / I#J - 2. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := J#N - J#L / 3. L#M := H#K + I#O. I#O := 2. I#I := I#O + H#K. I#P := I#J * L#M. H#K := 1. I#J := H#K + I#O. J#L := L#M - H#K. J#N := L#M / I#J - 2.", "query": "E#K", "answer": 6, "cot": "I#O = 2 -> I#O = 2\nH#K = 1 -> H#K = 1\nI#J = H#K + I#O\n = 1 + 2\n 1 + 2 = (1 + 2) mod 7 = 3\n => I#J = 3\nL#M = H#K + I#O\n = 1 + 2\n 1 + 2 = (1 + 2) mod 7 = 3\n => L#M = 3\nJ#N = L#M / I#J - 2\n = 3 / 3 - 2\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n 1 - 2 = (1 - 2) mod 7 = 6\n => J#N = 6\nJ#L = L#M - H#K\n = 3 - 1\n 3 - 1 = (3 - 1) mod 7 = 2\n => J#L = 2\nE#K = J#N - J#L / 3\n = 6 - 2 / 3\n 6 - 2 = (6 - 2) mod 7 = 4\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => E#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"I#O": 1, "H#K": 1, "I#J": 2, "I#I": 2, "L#M": 2, "J#N": 3, "J#L": 3, "I#P": 3, "E#K": 4}, "id": "mod7_arith_d4_0135"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#L := 5. E#O := F#O. K#O := 5 / E#O. H#K := 2. F#O := 6. G#O := G#K - I#N / 4. H#P := E#O + H#L. G#K := 4. L#K := K#O - I#N * G#O. I#N := 2 * F#O. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#L := 5. E#O := F#O. K#O := 5 / E#O. H#K := 2. F#O := 6. G#O := G#K - I#N / 4. H#P := E#O + H#L. G#K := 4. L#K := K#O - I#N * G#O. I#N := 2 * F#O.", "query": "L#K", "answer": 6, "cot": "G#K = 4 -> G#K = 4\nF#O = 6 -> F#O = 6\nE#O = F#O = 6\nI#N = 2 * F#O\n = 2 * 6\n 2 * 6 = (2 × 6) mod 7 = 5\n => I#N = 5\nG#O = G#K - I#N / 4\n = 4 - 5 / 4\n 4 - 5 = (4 - 5) mod 7 = 6\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n => G#O = 5\nK#O = 5 / E#O\n = 5 / 6\n 5 / 6 = (5 × 6⁻¹) mod 7 = 2\n => K#O = 2\nL#K = K#O - I#N * G#O\n = 2 - 5 * 5\n 2 - 5 = (2 - 5) mod 7 = 4\n 4 * 5 = (4 × 5) mod 7 = 6\n => L#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"G#K": 1, "F#O": 1, "H#K": 1, "H#L": 1, "E#O": 2, "I#N": 2, "G#O": 3, "K#O": 3, "H#P": 3, "L#K": 4}, "id": "mod7_arith_d4_0136"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := 1. G#K := K#L. I#P := E#P / 2. F#P := G#N. E#P := F#P * G#N * 6. G#N := 6. K#O := K#L - F#K. K#L := G#N + K#M. F#K := K#M - G#N - 2. I#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := 1. G#K := K#L. I#P := E#P / 2. F#P := G#N. E#P := F#P * G#N * 6. G#N := 6. K#O := K#L - F#K. K#L := G#N + K#M. F#K := K#M - G#N - 2.", "query": "I#P", "answer": 3, "cot": "G#N = 6 -> G#N = 6\nF#P = G#N = 6\nE#P = F#P * G#N * 6\n = 6 * 6 * 6\n 6 * 6 = (6 × 6) mod 7 = 1\n 1 * 6 = (1 × 6) mod 7 = 6\n => E#P = 6\nI#P = E#P / 2\n = 6 / 2\n 6 / 2 = (6 × 2⁻¹) mod 7 = 3\n => I#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#M": 1, "G#N": 1, "F#P": 2, "F#K": 2, "K#L": 2, "G#K": 3, "K#O": 3, "E#P": 3, "I#P": 4}, "id": "mod7_arith_d4_0137"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#M := 2. J#O := G#P - H#M. I#M := J#P. F#M := L#J. J#P := 4 + J#L. L#J := J#O / 2 * 4. G#P := 1. H#J := 3. J#L := 6. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#M := 2. J#O := G#P - H#M. I#M := J#P. F#M := L#J. J#P := 4 + J#L. L#J := J#O / 2 * 4. G#P := 1. H#J := 3. J#L := 6.", "query": "F#M", "answer": 5, "cot": "H#M = 2 -> H#M = 2\nG#P = 1 -> G#P = 1\nJ#O = G#P - H#M\n = 1 - 2\n 1 - 2 = (1 - 2) mod 7 = 6\n => J#O = 6\nL#J = J#O / 2 * 4\n = 6 / 2 * 4\n 6 / 2 = (6 × 2⁻¹) mod 7 = 3\n 3 * 4 = (3 × 4) mod 7 = 5\n => L#J = 5\nF#M = L#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#M": 1, "G#P": 1, "J#L": 1, "H#J": 1, "J#O": 2, "J#P": 2, "I#M": 3, "L#J": 3, "F#M": 4}, "id": "mod7_arith_d4_0138"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := F#K. J#K := 6. F#M := 4. I#P := 3 * I#N. E#O := 6. H#L := I#P. I#N := J#K + E#O / 5. F#K := E#O + J#K. H#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := F#K. J#K := 6. F#M := 4. I#P := 3 * I#N. E#O := 6. H#L := I#P. I#N := J#K + E#O / 5. F#K := E#O + J#K.", "query": "H#L", "answer": 3, "cot": "E#O = 6 -> E#O = 6\nJ#K = 6 -> J#K = 6\nI#N = J#K + E#O / 5\n = 6 + 6 / 5\n 6 + 6 = (6 + 6) mod 7 = 5\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n => I#N = 1\nI#P = 3 * I#N\n = 3 * 1\n 3 * 1 = (3 × 1) mod 7 = 3\n => I#P = 3\nH#L = I#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"E#O": 1, "F#M": 1, "J#K": 1, "F#K": 2, "I#N": 2, "I#P": 3, "L#J": 3, "H#L": 4}, "id": "mod7_arith_d4_0139"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := K#L - E#I * L#M. F#P := F#K / L#M. E#P := I#L - L#M. L#M := 4. E#I := 4. K#M := F#K / L#M. J#L := E#P - 3. I#L := L#M. K#L := 2. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := K#L - E#I * L#M. F#P := F#K / L#M. E#P := I#L - L#M. L#M := 4. E#I := 4. K#M := F#K / L#M. J#L := E#P - 3. I#L := L#M. K#L := 2.", "query": "J#L", "answer": 4, "cot": "L#M = 4 -> L#M = 4\nI#L = L#M = 4\nE#P = I#L - L#M\n = 4 - 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => E#P = 0\nJ#L = E#P - 3\n = 0 - 3\n 0 - 3 = (0 - 3) mod 7 = 4\n => J#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 4, "num_distractors": 0, "var_layer": {"E#I": 1, "K#L": 1, "L#M": 1, "F#K": 2, "I#L": 2, "K#M": 3, "E#P": 3, "F#P": 3, "J#L": 4}, "id": "mod7_arith_d4_0140"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := K#N * F#O. I#N := F#O / F#K - E#O. F#K := H#N + F#O. F#O := 5. H#N := 1. G#J := E#O / 6 - J#P. K#N := F#O + H#N. E#O := F#O. J#P := F#K / F#O. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := K#N * F#O. I#N := F#O / F#K - E#O. F#K := H#N + F#O. F#O := 5. H#N := 1. G#J := E#O / 6 - J#P. K#N := F#O + H#N. E#O := F#O. J#P := F#K / F#O.", "query": "G#J", "answer": 5, "cot": "F#O = 5 -> F#O = 5\nH#N = 1 -> H#N = 1\nF#K = H#N + F#O\n = 1 + 5\n 1 + 5 = (1 + 5) mod 7 = 6\n => F#K = 6\nE#O = F#O = 5\nJ#P = F#K / F#O\n = 6 / 5\n 6 / 5 = (6 × 5⁻¹) mod 7 = 4\n => J#P = 4\nG#J = E#O / 6 - J#P\n = 5 / 6 - 4\n 5 / 6 = (5 × 6⁻¹) mod 7 = 2\n 2 - 4 = (2 - 4) mod 7 = 5\n => G#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#O": 1, "H#N": 1, "F#K": 2, "E#O": 2, "K#N": 2, "L#I": 3, "J#P": 3, "I#N": 3, "G#J": 4}, "id": "mod7_arith_d4_0141"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#M := K#I. F#I := F#J * L#M. F#J := 3. I#O := L#M * F#J. L#M := 2. F#O := I#O. K#I := F#I * 3. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#M := K#I. F#I := F#J * L#M. F#J := 3. I#O := L#M * F#J. L#M := 2. F#O := I#O. K#I := F#I * 3.", "query": "G#M", "answer": 4, "cot": "F#J = 3 -> F#J = 3\nL#M = 2 -> L#M = 2\nF#I = F#J * L#M\n = 3 * 2\n 3 * 2 = (3 × 2) mod 7 = 6\n => F#I = 6\nK#I = F#I * 3\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => K#I = 4\nG#M = K#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#J": 1, "L#M": 1, "I#O": 2, "F#I": 2, "K#I": 3, "F#O": 3, "G#M": 4}, "id": "mod7_arith_d4_0142"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := J#L * I#I + H#I. H#J := J#P - L#I. K#O := 6. I#I := J#O. J#O := 4. J#P := 3. J#L := J#P. L#I := J#P. H#I := I#I * J#P. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := J#L * I#I + H#I. H#J := J#P - L#I. K#O := 6. I#I := J#O. J#O := 4. J#P := 3. J#L := J#P. L#I := J#P. H#I := I#I * J#P.", "query": "L#M", "answer": 3, "cot": "J#P = 3 -> J#P = 3\nJ#O = 4 -> J#O = 4\nJ#L = J#P = 3\nI#I = J#O = 4\nH#I = I#I * J#P\n = 4 * 3\n 4 * 3 = (4 × 3) mod 7 = 5\n => H#I = 5\nL#M = J#L * I#I + H#I\n = 3 * 4 + 5\n 3 * 4 = (3 × 4) mod 7 = 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => L#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#P": 1, "K#O": 1, "J#O": 1, "J#L": 2, "L#I": 2, "I#I": 2, "H#I": 3, "H#J": 3, "L#M": 4}, "id": "mod7_arith_d4_0143"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#L := E#P - H#M - I#P. K#O := H#M. L#M := 4 / I#P. I#P := J#O / H#M. J#O := 1. H#M := 2. E#P := K#O + 4 - 6. K#L := J#O / 5. I#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#L := E#P - H#M - I#P. K#O := H#M. L#M := 4 / I#P. I#P := J#O / H#M. J#O := 1. H#M := 2. E#P := K#O + 4 - 6. K#L := J#O / 5.", "query": "I#L", "answer": 1, "cot": "J#O = 1 -> J#O = 1\nH#M = 2 -> H#M = 2\nI#P = J#O / H#M\n = 1 / 2\n 1 / 2 = (1 × 2⁻¹) mod 7 = 4\n => I#P = 4\nK#O = H#M = 2\nE#P = K#O + 4 - 6\n = 2 + 4 - 6\n 2 + 4 = (2 + 4) mod 7 = 6\n 6 - 6 = (6 - 6) mod 7 = 0\n => E#P = 0\nI#L = E#P - H#M - I#P\n = 0 - 2 - 4\n 0 - 2 = (0 - 2) mod 7 = 5\n 5 - 4 = (5 - 4) mod 7 = 1\n => I#L = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#O": 1, "H#M": 1, "I#P": 2, "K#O": 2, "K#L": 2, "L#M": 3, "E#P": 3, "I#L": 4}, "id": "mod7_arith_d4_0144"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := H#N / F#L. L#N := F#L / J#J. J#J := E#L / K#O. E#L := I#K + F#L. F#L := 6. I#K := 3. H#N := F#L / K#O. K#O := 6. L#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := H#N / F#L. L#N := F#L / J#J. J#J := E#L / K#O. E#L := I#K + F#L. F#L := 6. I#K := 3. H#N := F#L / K#O. K#O := 6.", "query": "L#N", "answer": 4, "cot": "I#K = 3 -> I#K = 3\nF#L = 6 -> F#L = 6\nK#O = 6 -> K#O = 6\nE#L = I#K + F#L\n = 3 + 6\n 3 + 6 = (3 + 6) mod 7 = 2\n => E#L = 2\nJ#J = E#L / K#O\n = 2 / 6\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => J#J = 5\nL#N = F#L / J#J\n = 6 / 5\n 6 / 5 = (6 × 5⁻¹) mod 7 = 4\n => L#N = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"I#K": 1, "F#L": 1, "K#O": 1, "H#N": 2, "E#L": 2, "J#J": 3, "J#O": 3, "L#N": 4}, "id": "mod7_arith_d4_0145"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := L#O * L#M. K#P := H#O - 5. L#M := G#K * L#O / 6. G#K := 3. L#O := 6. H#O := G#K. E#J := K#P. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := L#O * L#M. K#P := H#O - 5. L#M := G#K * L#O / 6. G#K := 3. L#O := 6. H#O := G#K. E#J := K#P.", "query": "E#J", "answer": 5, "cot": "G#K = 3 -> G#K = 3\nH#O = G#K = 3\nK#P = H#O - 5\n = 3 - 5\n 3 - 5 = (3 - 5) mod 7 = 5\n => K#P = 5\nE#J = K#P = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#O": 1, "G#K": 1, "L#M": 2, "H#O": 2, "K#P": 3, "G#J": 3, "E#J": 4}, "id": "mod7_arith_d4_0146"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#K := 4. E#O := K#K. E#K := E#O - H#P. F#M := F#N - E#I * H#P. H#P := K#K * 4. F#N := E#O + 2. E#I := 2. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#K := 4. E#O := K#K. E#K := E#O - H#P. F#M := F#N - E#I * H#P. H#P := K#K * 4. F#N := E#O + 2. E#I := 2.", "query": "F#M", "answer": 1, "cot": "K#K = 4 -> K#K = 4\nE#I = 2 -> E#I = 2\nH#P = K#K * 4\n = 4 * 4\n 4 * 4 = (4 × 4) mod 7 = 2\n => H#P = 2\nE#O = K#K = 4\nF#N = E#O + 2\n = 4 + 2\n 4 + 2 = (4 + 2) mod 7 = 6\n => F#N = 6\nF#M = F#N - E#I * H#P\n = 6 - 2 * 2\n 6 - 2 = (6 - 2) mod 7 = 4\n 4 * 2 = (4 × 2) mod 7 = 1\n => F#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#K": 1, "E#I": 1, "H#P": 2, "E#O": 2, "E#K": 3, "F#N": 3, "F#M": 4}, "id": "mod7_arith_d4_0147"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := G#J + E#N. G#L := F#M + G#J. I#N := F#M * G#J. F#O := H#O * E#N. G#J := 4. E#N := 6. F#M := 2. F#N := F#M - F#O. E#I := G#L. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := G#J + E#N. G#L := F#M + G#J. I#N := F#M * G#J. F#O := H#O * E#N. G#J := 4. E#N := 6. F#M := 2. F#N := F#M - F#O. E#I := G#L.", "query": "F#N", "answer": 5, "cot": "E#N = 6 -> E#N = 6\nF#M = 2 -> F#M = 2\nG#J = 4 -> G#J = 4\nH#O = G#J + E#N\n = 4 + 6\n 4 + 6 = (4 + 6) mod 7 = 3\n => H#O = 3\nF#O = H#O * E#N\n = 3 * 6\n 3 * 6 = (3 × 6) mod 7 = 4\n => F#O = 4\nF#N = F#M - F#O\n = 2 - 4\n 2 - 4 = (2 - 4) mod 7 = 5\n => F#N = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#N": 1, "F#M": 1, "G#J": 1, "H#O": 2, "I#N": 2, "G#L": 2, "E#I": 3, "F#O": 3, "F#N": 4}, "id": "mod7_arith_d4_0148"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#I := F#K + H#M. E#N := J#I + H#M - F#N. F#K := 3. G#L := 4. K#L := E#L - H#M. H#M := 3. L#I := F#N - H#M - F#K. G#O := F#N + F#K. F#N := 1. E#L := G#O - H#M. J#M := G#O. K#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#I := F#K + H#M. E#N := J#I + H#M - F#N. F#K := 3. G#L := 4. K#L := E#L - H#M. H#M := 3. L#I := F#N - H#M - F#K. G#O := F#N + F#K. F#N := 1. E#L := G#O - H#M. J#M := G#O.", "query": "K#L", "answer": 5, "cot": "F#K = 3 -> F#K = 3\nF#N = 1 -> F#N = 1\nH#M = 3 -> H#M = 3\nG#O = F#N + F#K\n = 1 + 3\n 1 + 3 = (1 + 3) mod 7 = 4\n => G#O = 4\nE#L = G#O - H#M\n = 4 - 3\n 4 - 3 = (4 - 3) mod 7 = 1\n => E#L = 1\nK#L = E#L - H#M\n = 1 - 3\n 1 - 3 = (1 - 3) mod 7 = 5\n => K#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#K": 1, "F#N": 1, "G#L": 1, "H#M": 1, "G#O": 2, "J#I": 2, "L#I": 2, "J#M": 3, "E#L": 3, "E#N": 3, "K#L": 4}, "id": "mod7_arith_d4_0149"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#M := 1. J#N := K#M. H#O := F#M + G#P. J#I := J#N * G#P. G#P := 2. K#M := G#P - F#M. K#O := F#I + K#M. H#L := K#M - F#M. F#I := F#M * G#P. J#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#M := 1. J#N := K#M. H#O := F#M + G#P. J#I := J#N * G#P. G#P := 2. K#M := G#P - F#M. K#O := F#I + K#M. H#L := K#M - F#M. F#I := F#M * G#P.", "query": "J#I", "answer": 2, "cot": "G#P = 2 -> G#P = 2\nF#M = 1 -> F#M = 1\nK#M = G#P - F#M\n = 2 - 1\n 2 - 1 = (2 - 1) mod 7 = 1\n => K#M = 1\nJ#N = K#M = 1\nJ#I = J#N * G#P\n = 1 * 2\n 1 * 2 = (1 × 2) mod 7 = 2\n => J#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#P": 1, "F#M": 1, "F#I": 2, "K#M": 2, "H#O": 2, "J#N": 3, "K#O": 3, "H#L": 3, "J#I": 4}, "id": "mod7_arith_d4_0150"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#P := 5. F#K := 6. E#J := K#P - F#I. G#M := 1. G#K := I#L + E#J + 3. F#L := 3 - I#P * F#K. F#I := 5. L#N := G#M. I#P := F#K + E#J. K#I := I#L - 3. I#L := G#M + F#I. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#P := 5. F#K := 6. E#J := K#P - F#I. G#M := 1. G#K := I#L + E#J + 3. F#L := 3 - I#P * F#K. F#I := 5. L#N := G#M. I#P := F#K + E#J. K#I := I#L - 3. I#L := G#M + F#I.", "query": "F#L", "answer": 3, "cot": "K#P = 5 -> K#P = 5\nF#I = 5 -> F#I = 5\nF#K = 6 -> F#K = 6\nE#J = K#P - F#I\n = 5 - 5\n 5 - 5 = (5 - 5) mod 7 = 0\n => E#J = 0\nI#P = F#K + E#J\n = 6 + 0\n 6 + 0 = (6 + 0) mod 7 = 6\n => I#P = 6\nF#L = 3 - I#P * F#K\n = 3 - 6 * 6\n 3 - 6 = (3 - 6) mod 7 = 4\n 4 * 6 = (4 × 6) mod 7 = 3\n => F#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#P": 1, "G#M": 1, "F#I": 1, "F#K": 1, "E#J": 2, "I#L": 2, "L#N": 2, "G#K": 3, "K#I": 3, "I#P": 3, "F#L": 4}, "id": "mod7_arith_d4_0151"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := H#P. F#L := E#N + G#N. H#P := F#J / 5 - G#N. F#J := 5. L#O := F#L. K#M := L#O. J#I := 4 * E#N + F#J. G#N := 6. E#N := 2. K#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := H#P. F#L := E#N + G#N. H#P := F#J / 5 - G#N. F#J := 5. L#O := F#L. K#M := L#O. J#I := 4 * E#N + F#J. G#N := 6. E#N := 2.", "query": "K#M", "answer": 1, "cot": "E#N = 2 -> E#N = 2\nG#N = 6 -> G#N = 6\nF#L = E#N + G#N\n = 2 + 6\n 2 + 6 = (2 + 6) mod 7 = 1\n => F#L = 1\nL#O = F#L = 1\nK#M = L#O = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#J": 1, "E#N": 1, "G#N": 1, "J#I": 2, "H#P": 2, "F#L": 2, "L#P": 3, "L#O": 3, "K#M": 4}, "id": "mod7_arith_d4_0152"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#L := G#M + 5. K#I := 4. E#L := 5. I#I := 1. G#N := G#M. H#K := 5. G#M := E#L + K#I. G#J := E#L. L#J := G#N. L#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#L := G#M + 5. K#I := 4. E#L := 5. I#I := 1. G#N := G#M. H#K := 5. G#M := E#L + K#I. G#J := E#L. L#J := G#N.", "query": "L#J", "answer": 2, "cot": "K#I = 4 -> K#I = 4\nE#L = 5 -> E#L = 5\nG#M = E#L + K#I\n = 5 + 4\n 5 + 4 = (5 + 4) mod 7 = 2\n => G#M = 2\nG#N = G#M = 2\nL#J = G#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#I": 1, "E#L": 1, "I#I": 1, "H#K": 1, "G#J": 2, "G#M": 2, "G#L": 3, "G#N": 3, "L#J": 4}, "id": "mod7_arith_d4_0153"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#N := G#M - 6 * I#L. L#K := G#M / F#K - 6. L#O := 3. G#M := H#O * I#L. J#M := E#J. H#O := 3. F#K := L#O / I#L. I#L := 4. L#L := L#O / K#N. E#J := 2. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#N := G#M - 6 * I#L. L#K := G#M / F#K - 6. L#O := 3. G#M := H#O * I#L. J#M := E#J. H#O := 3. F#K := L#O / I#L. I#L := 4. L#L := L#O / K#N. E#J := 2.", "query": "L#L", "answer": 1, "cot": "L#O = 3 -> L#O = 3\nH#O = 3 -> H#O = 3\nI#L = 4 -> I#L = 4\nG#M = H#O * I#L\n = 3 * 4\n 3 * 4 = (3 × 4) mod 7 = 5\n => G#M = 5\nK#N = G#M - 6 * I#L\n = 5 - 6 * 4\n 5 - 6 = (5 - 6) mod 7 = 6\n 6 * 4 = (6 × 4) mod 7 = 3\n => K#N = 3\nL#L = L#O / K#N\n = 3 / 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => L#L = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"L#O": 1, "E#J": 1, "H#O": 1, "I#L": 1, "G#M": 2, "J#M": 2, "F#K": 2, "L#K": 3, "K#N": 3, "L#L": 4}, "id": "mod7_arith_d4_0154"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#N := 2 / J#M. J#J := E#I + 2 - 5. E#I := E#L / G#M. G#M := 1. J#M := G#M / F#I. F#I := 2. K#I := K#N. K#L := F#I. E#L := 2. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#N := 2 / J#M. J#J := E#I + 2 - 5. E#I := E#L / G#M. G#M := 1. J#M := G#M / F#I. F#I := 2. K#I := K#N. K#L := F#I. E#L := 2.", "query": "K#I", "answer": 4, "cot": "F#I = 2 -> F#I = 2\nG#M = 1 -> G#M = 1\nJ#M = G#M / F#I\n = 1 / 2\n 1 / 2 = (1 × 2⁻¹) mod 7 = 4\n => J#M = 4\nK#N = 2 / J#M\n = 2 / 4\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n => K#N = 4\nK#I = K#N = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#I": 1, "E#L": 1, "G#M": 1, "K#L": 2, "E#I": 2, "J#M": 2, "J#J": 3, "K#N": 3, "K#I": 4}, "id": "mod7_arith_d4_0155"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := 2 * I#L. I#L := 3. K#P := 6 - I#L * H#P. H#P := 4. H#K := I#L - H#P. H#L := H#P / K#J. K#L := H#K - K#P. G#N := K#P - K#J. G#P := K#L * G#N. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := 2 * I#L. I#L := 3. K#P := 6 - I#L * H#P. H#P := 4. H#K := I#L - H#P. H#L := H#P / K#J. K#L := H#K - K#P. G#N := K#P - K#J. G#P := K#L * G#N.", "query": "G#P", "answer": 6, "cot": "I#L = 3 -> I#L = 3\nH#P = 4 -> H#P = 4\nK#P = 6 - I#L * H#P\n = 6 - 3 * 4\n 6 - 3 = (6 - 3) mod 7 = 3\n 3 * 4 = (3 × 4) mod 7 = 5\n => K#P = 5\nH#K = I#L - H#P\n = 3 - 4\n 3 - 4 = (3 - 4) mod 7 = 6\n => H#K = 6\nK#J = 2 * I#L\n = 2 * 3\n 2 * 3 = (2 × 3) mod 7 = 6\n => K#J = 6\nK#L = H#K - K#P\n = 6 - 5\n 6 - 5 = (6 - 5) mod 7 = 1\n => K#L = 1\nG#N = K#P - K#J\n = 5 - 6\n 5 - 6 = (5 - 6) mod 7 = 6\n => G#N = 6\nG#P = K#L * G#N\n = 1 * 6\n 1 * 6 = (1 × 6) mod 7 = 6\n => G#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 8, "num_distractors": 0, "var_layer": {"I#L": 1, "H#P": 1, "K#P": 2, "H#K": 2, "K#J": 2, "K#L": 3, "H#L": 3, "G#N": 3, "G#P": 4}, "id": "mod7_arith_d4_0156"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#I := H#L * 5. H#L := 5 / F#K. G#O := H#K - L#N / F#K. L#L := G#O * H#L * F#K. J#M := H#L - L#N. I#J := F#K + G#I. H#K := 2. L#N := 2. F#K := 2. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#I := H#L * 5. H#L := 5 / F#K. G#O := H#K - L#N / F#K. L#L := G#O * H#L * F#K. J#M := H#L - L#N. I#J := F#K + G#I. H#K := 2. L#N := 2. F#K := 2.", "query": "I#J", "answer": 4, "cot": "F#K = 2 -> F#K = 2\nH#L = 5 / F#K\n = 5 / 2\n 5 / 2 = (5 × 2⁻¹) mod 7 = 6\n => H#L = 6\nG#I = H#L * 5\n = 6 * 5\n 6 * 5 = (6 × 5) mod 7 = 2\n => G#I = 2\nI#J = F#K + G#I\n = 2 + 2\n 2 + 2 = (2 + 2) mod 7 = 4\n => I#J = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#K": 1, "H#K": 1, "L#N": 1, "G#O": 2, "H#L": 2, "L#L": 3, "G#I": 3, "J#M": 3, "I#J": 4}, "id": "mod7_arith_d4_0157"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := 6 + F#L. F#L := 4 - K#N. L#O := J#I * 6. K#N := 3. J#M := K#N. J#I := J#M * J#O - K#N. J#O := 6. E#J := J#O * K#N. H#N := F#L / J#O * J#M. L#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := 6 + F#L. F#L := 4 - K#N. L#O := J#I * 6. K#N := 3. J#M := K#N. J#I := J#M * J#O - K#N. J#O := 6. E#J := J#O * K#N. H#N := F#L / J#O * J#M.", "query": "L#O", "answer": 6, "cot": "K#N = 3 -> K#N = 3\nJ#O = 6 -> J#O = 6\nJ#M = K#N = 3\nJ#I = J#M * J#O - K#N\n = 3 * 6 - 3\n 3 * 6 = (3 × 6) mod 7 = 4\n 4 - 3 = (4 - 3) mod 7 = 1\n => J#I = 1\nL#O = J#I * 6\n = 1 * 6\n 1 * 6 = (1 × 6) mod 7 = 6\n => L#O = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#N": 1, "J#O": 1, "F#L": 2, "E#J": 2, "J#M": 2, "K#J": 3, "H#N": 3, "J#I": 3, "L#O": 4}, "id": "mod7_arith_d4_0158"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := 5. G#M := G#O. J#K := E#K. I#K := 1. G#O := 2. F#O := J#K. L#J := G#M - L#K. F#I := 3. L#K := 6 + F#J. F#J := G#O + E#K. L#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := 5. G#M := G#O. J#K := E#K. I#K := 1. G#O := 2. F#O := J#K. L#J := G#M - L#K. F#I := 3. L#K := 6 + F#J. F#J := G#O + E#K.", "query": "L#J", "answer": 3, "cot": "E#K = 5 -> E#K = 5\nG#O = 2 -> G#O = 2\nF#J = G#O + E#K\n = 2 + 5\n 2 + 5 = (2 + 5) mod 7 = 0\n => F#J = 0\nG#M = G#O = 2\nL#K = 6 + F#J\n = 6 + 0\n 6 + 0 = (6 + 0) mod 7 = 6\n => L#K = 6\nL#J = G#M - L#K\n = 2 - 6\n 2 - 6 = (2 - 6) mod 7 = 3\n => L#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#K": 1, "F#I": 1, "I#K": 1, "G#O": 1, "F#J": 2, "J#K": 2, "G#M": 2, "F#O": 3, "L#K": 3, "L#J": 4}, "id": "mod7_arith_d4_0159"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#L := 1. G#J := L#N * J#L * G#N. E#P := 1. J#L := L#N. L#M := L#L. L#K := E#P * L#L. L#N := L#L + E#P. G#N := L#M - E#P + L#K. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#L := 1. G#J := L#N * J#L * G#N. E#P := 1. J#L := L#N. L#M := L#L. L#K := E#P * L#L. L#N := L#L + E#P. G#N := L#M - E#P + L#K.", "query": "G#J", "answer": 4, "cot": "L#L = 1 -> L#L = 1\nE#P = 1 -> E#P = 1\nL#N = L#L + E#P\n = 1 + 1\n 1 + 1 = (1 + 1) mod 7 = 2\n => L#N = 2\nL#M = L#L = 1\nL#K = E#P * L#L\n = 1 * 1\n 1 * 1 = (1 × 1) mod 7 = 1\n => L#K = 1\nG#N = L#M - E#P + L#K\n = 1 - 1 + 1\n 1 - 1 = (1 - 1) mod 7 = 0\n 0 + 1 = (0 + 1) mod 7 = 1\n => G#N = 1\nJ#L = L#N = 2\nG#J = L#N * J#L * G#N\n = 2 * 2 * 1\n 2 * 2 = (2 × 2) mod 7 = 4\n 4 * 1 = (4 × 1) mod 7 = 4\n => G#J = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 8, "num_distractors": 0, "var_layer": {"L#L": 1, "E#P": 1, "L#N": 2, "L#M": 2, "L#K": 2, "G#N": 3, "J#L": 3, "G#J": 4}, "id": "mod7_arith_d4_0160"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#N := 5 / L#P - 6. L#J := K#K + 6 - J#N. L#P := 6. G#K := J#N. G#P := G#K. J#K := 1. I#K := K#K. K#K := 2. H#K := J#N - I#K. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#N := 5 / L#P - 6. L#J := K#K + 6 - J#N. L#P := 6. G#K := J#N. G#P := G#K. J#K := 1. I#K := K#K. K#K := 2. H#K := J#N - I#K.", "query": "G#P", "answer": 3, "cot": "L#P = 6 -> L#P = 6\nJ#N = 5 / L#P - 6\n = 5 / 6 - 6\n 5 / 6 = (5 × 6⁻¹) mod 7 = 2\n 2 - 6 = (2 - 6) mod 7 = 3\n => J#N = 3\nG#K = J#N = 3\nG#P = G#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 4, "num_distractors": 0, "var_layer": {"K#K": 1, "J#K": 1, "L#P": 1, "I#K": 2, "J#N": 2, "H#K": 3, "G#K": 3, "L#J": 3, "G#P": 4}, "id": "mod7_arith_d4_0161"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#N := I#J - G#N. H#P := 6. I#J := L#I * H#P. G#K := L#I / G#N. L#I := 5. J#M := G#K * G#N. G#N := H#P / 5. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#N := I#J - G#N. H#P := 6. I#J := L#I * H#P. G#K := L#I / G#N. L#I := 5. J#M := G#K * G#N. G#N := H#P / 5.", "query": "J#M", "answer": 5, "cot": "L#I = 5 -> L#I = 5\nH#P = 6 -> H#P = 6\nG#N = H#P / 5\n = 6 / 5\n 6 / 5 = (6 × 5⁻¹) mod 7 = 4\n => G#N = 4\nG#K = L#I / G#N\n = 5 / 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => G#K = 3\nJ#M = G#K * G#N\n = 3 * 4\n 3 * 4 = (3 × 4) mod 7 = 5\n => J#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#I": 1, "H#P": 1, "G#N": 2, "I#J": 2, "I#N": 3, "G#K": 3, "J#M": 4}, "id": "mod7_arith_d4_0162"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#P := 3. H#K := 6. I#O := H#K * G#M. F#M := I#L. E#K := 3 + I#O. G#M := K#P / I#L - H#K. E#N := 6. I#L := 5. H#M := K#P - G#M. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#P := 3. H#K := 6. I#O := H#K * G#M. F#M := I#L. E#K := 3 + I#O. G#M := K#P / I#L - H#K. E#N := 6. I#L := 5. H#M := K#P - G#M.", "query": "E#K", "answer": 0, "cot": "H#K = 6 -> H#K = 6\nK#P = 3 -> K#P = 3\nI#L = 5 -> I#L = 5\nG#M = K#P / I#L - H#K\n = 3 / 5 - 6\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n 2 - 6 = (2 - 6) mod 7 = 3\n => G#M = 3\nI#O = H#K * G#M\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => I#O = 4\nE#K = 3 + I#O\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => E#K = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"H#K": 1, "K#P": 1, "I#L": 1, "E#N": 1, "F#M": 2, "G#M": 2, "I#O": 3, "H#M": 3, "E#K": 4}, "id": "mod7_arith_d4_0163"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := L#L / F#N - 3. I#J := F#N + L#L. F#J := G#O. G#O := K#P + G#N. F#N := 5. I#P := E#N. G#N := 4. K#P := 5 + F#N. L#N := 2. H#M := K#P + E#N * F#N. L#L := 1. F#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := L#L / F#N - 3. I#J := F#N + L#L. F#J := G#O. G#O := K#P + G#N. F#N := 5. I#P := E#N. G#N := 4. K#P := 5 + F#N. L#N := 2. H#M := K#P + E#N * F#N. L#L := 1.", "query": "F#J", "answer": 0, "cot": "G#N = 4 -> G#N = 4\nF#N = 5 -> F#N = 5\nK#P = 5 + F#N\n = 5 + 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => K#P = 3\nG#O = K#P + G#N\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => G#O = 0\nF#J = G#O = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#N": 1, "L#L": 1, "F#N": 1, "L#N": 1, "I#J": 2, "K#P": 2, "E#N": 2, "I#P": 3, "G#O": 3, "H#M": 3, "F#J": 4}, "id": "mod7_arith_d4_0164"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#O := 2. J#P := H#O. K#K := J#I * E#I + I#K. H#O := E#I. I#M := L#L / J#P - H#O. E#I := 3. L#L := J#I * H#O. J#I := I#O * E#I. I#K := I#O - E#I. I#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#O := 2. J#P := H#O. K#K := J#I * E#I + I#K. H#O := E#I. I#M := L#L / J#P - H#O. E#I := 3. L#L := J#I * H#O. J#I := I#O * E#I. I#K := I#O - E#I.", "query": "I#M", "answer": 3, "cot": "E#I = 3 -> E#I = 3\nI#O = 2 -> I#O = 2\nH#O = E#I = 3\nJ#I = I#O * E#I\n = 2 * 3\n 2 * 3 = (2 × 3) mod 7 = 6\n => J#I = 6\nL#L = J#I * H#O\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => L#L = 4\nJ#P = H#O = 3\nI#M = L#L / J#P - H#O\n = 4 / 3 - 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n 6 - 3 = (6 - 3) mod 7 = 3\n => I#M = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"E#I": 1, "I#O": 1, "I#K": 2, "H#O": 2, "J#I": 2, "K#K": 3, "L#L": 3, "J#P": 3, "I#M": 4}, "id": "mod7_arith_d4_0165"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#M := I#P. I#P := 5. G#K := 3. H#O := L#I + I#P. J#I := I#K / 5 / 6. L#I := 5. E#M := G#M - L#L. I#K := H#O. L#L := L#I + I#P. G#J := G#M. J#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#M := I#P. I#P := 5. G#K := 3. H#O := L#I + I#P. J#I := I#K / 5 / 6. L#I := 5. E#M := G#M - L#L. I#K := H#O. L#L := L#I + I#P. G#J := G#M.", "query": "J#I", "answer": 5, "cot": "L#I = 5 -> L#I = 5\nI#P = 5 -> I#P = 5\nH#O = L#I + I#P\n = 5 + 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => H#O = 3\nI#K = H#O = 3\nJ#I = I#K / 5 / 6\n = 3 / 5 / 6\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => J#I = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#I": 1, "G#K": 1, "I#P": 1, "H#O": 2, "L#L": 2, "G#M": 2, "G#J": 3, "I#K": 3, "E#M": 3, "J#I": 4}, "id": "mod7_arith_d4_0166"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#M := K#N / H#J. K#L := 6 + F#J. K#N := 4. H#J := 4. I#O := J#O - 6. I#P := K#N. J#O := I#P + F#J. F#J := H#L + H#J + K#N. G#J := I#P * K#N * H#L. H#L := 1. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#M := K#N / H#J. K#L := 6 + F#J. K#N := 4. H#J := 4. I#O := J#O - 6. I#P := K#N. J#O := I#P + F#J. F#J := H#L + H#J + K#N. G#J := I#P * K#N * H#L. H#L := 1.", "query": "I#O", "answer": 0, "cot": "K#N = 4 -> K#N = 4\nH#L = 1 -> H#L = 1\nH#J = 4 -> H#J = 4\nF#J = H#L + H#J + K#N\n = 1 + 4 + 4\n 1 + 4 = (1 + 4) mod 7 = 5\n 5 + 4 = (5 + 4) mod 7 = 2\n => F#J = 2\nI#P = K#N = 4\nJ#O = I#P + F#J\n = 4 + 2\n 4 + 2 = (4 + 2) mod 7 = 6\n => J#O = 6\nI#O = J#O - 6\n = 6 - 6\n 6 - 6 = (6 - 6) mod 7 = 0\n => I#O = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"K#N": 1, "H#L": 1, "H#J": 1, "I#M": 2, "F#J": 2, "I#P": 2, "K#L": 3, "J#O": 3, "G#J": 3, "I#O": 4}, "id": "mod7_arith_d4_0167"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := G#P / J#P. L#K := K#P / E#K. K#P := E#K. J#P := 1. E#M := 2. F#L := E#K. I#N := 3 - G#P. I#M := 6. G#P := 4. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := G#P / J#P. L#K := K#P / E#K. K#P := E#K. J#P := 1. E#M := 2. F#L := E#K. I#N := 3 - G#P. I#M := 6. G#P := 4.", "query": "L#K", "answer": 1, "cot": "J#P = 1 -> J#P = 1\nG#P = 4 -> G#P = 4\nE#K = G#P / J#P\n = 4 / 1\n 4 / 1 = (4 × 1⁻¹) mod 7 = 4\n => E#K = 4\nK#P = E#K = 4\nL#K = K#P / E#K\n = 4 / 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => L#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#P": 1, "E#M": 1, "G#P": 1, "I#M": 1, "I#N": 2, "E#K": 2, "F#L": 3, "K#P": 3, "L#K": 4}, "id": "mod7_arith_d4_0168"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := 4 * F#K. L#J := 4 * K#L. F#K := K#L. E#K := K#L - F#K. K#L := 3. G#J := 3. H#I := 2. E#I := G#P * E#K - K#L. E#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := 4 * F#K. L#J := 4 * K#L. F#K := K#L. E#K := K#L - F#K. K#L := 3. G#J := 3. H#I := 2. E#I := G#P * E#K - K#L.", "query": "E#I", "answer": 4, "cot": "K#L = 3 -> K#L = 3\nF#K = K#L = 3\nG#P = 4 * F#K\n = 4 * 3\n 4 * 3 = (4 × 3) mod 7 = 5\n => G#P = 5\nE#K = K#L - F#K\n = 3 - 3\n 3 - 3 = (3 - 3) mod 7 = 0\n => E#K = 0\nE#I = G#P * E#K - K#L\n = 5 * 0 - 3\n 5 * 0 = (5 × 0) mod 7 = 0\n 0 - 3 = (0 - 3) mod 7 = 4\n => E#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#J": 1, "H#I": 1, "K#L": 1, "L#J": 2, "F#K": 2, "G#P": 3, "E#K": 3, "E#I": 4}, "id": "mod7_arith_d4_0169"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#K := 4. G#I := K#P. F#L := F#M - I#N. K#I := E#K * 6. L#P := 4 * K#I. E#J := 2. H#N := 6. F#M := 1. I#N := K#P * 2. K#P := H#N + 4. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#K := 4. G#I := K#P. F#L := F#M - I#N. K#I := E#K * 6. L#P := 4 * K#I. E#J := 2. H#N := 6. F#M := 1. I#N := K#P * 2. K#P := H#N + 4.", "query": "F#L", "answer": 2, "cot": "F#M = 1 -> F#M = 1\nH#N = 6 -> H#N = 6\nK#P = H#N + 4\n = 6 + 4\n 6 + 4 = (6 + 4) mod 7 = 3\n => K#P = 3\nI#N = K#P * 2\n = 3 * 2\n 3 * 2 = (3 × 2) mod 7 = 6\n => I#N = 6\nF#L = F#M - I#N\n = 1 - 6\n 1 - 6 = (1 - 6) mod 7 = 2\n => F#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"E#K": 1, "E#J": 1, "F#M": 1, "H#N": 1, "K#I": 2, "K#P": 2, "I#N": 3, "G#I": 3, "L#P": 3, "F#L": 4}, "id": "mod7_arith_d4_0170"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#N := 2 - I#K. F#N := 1. E#K := 3. H#L := 1. I#M := H#P / H#L + G#L. I#K := F#N. G#M := I#M. F#O := I#K. H#P := F#N / 6. G#L := 3. G#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#N := 2 - I#K. F#N := 1. E#K := 3. H#L := 1. I#M := H#P / H#L + G#L. I#K := F#N. G#M := I#M. F#O := I#K. H#P := F#N / 6. G#L := 3.", "query": "G#M", "answer": 2, "cot": "G#L = 3 -> G#L = 3\nF#N = 1 -> F#N = 1\nH#L = 1 -> H#L = 1\nH#P = F#N / 6\n = 1 / 6\n 1 / 6 = (1 × 6⁻¹) mod 7 = 6\n => H#P = 6\nI#M = H#P / H#L + G#L\n = 6 / 1 + 3\n 6 / 1 = (6 × 1⁻¹) mod 7 = 6\n 6 + 3 = (6 + 3) mod 7 = 2\n => I#M = 2\nG#M = I#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#K": 1, "G#L": 1, "F#N": 1, "H#L": 1, "I#K": 2, "H#P": 2, "F#O": 3, "I#N": 3, "I#M": 3, "G#M": 4}, "id": "mod7_arith_d4_0171"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#L := 5. J#N := L#K. L#K := J#L - L#L. H#M := 4. E#N := 5. E#K := J#N / L#L / I#N. I#N := L#L - H#K. L#L := 4. H#K := E#N - H#M. E#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#L := 5. J#N := L#K. L#K := J#L - L#L. H#M := 4. E#N := 5. E#K := J#N / L#L / I#N. I#N := L#L - H#K. L#L := 4. H#K := E#N - H#M.", "query": "E#K", "answer": 3, "cot": "L#L = 4 -> L#L = 4\nH#M = 4 -> H#M = 4\nE#N = 5 -> E#N = 5\nJ#L = 5 -> J#L = 5\nH#K = E#N - H#M\n = 5 - 4\n 5 - 4 = (5 - 4) mod 7 = 1\n => H#K = 1\nL#K = J#L - L#L\n = 5 - 4\n 5 - 4 = (5 - 4) mod 7 = 1\n => L#K = 1\nJ#N = L#K = 1\nI#N = L#L - H#K\n = 4 - 1\n 4 - 1 = (4 - 1) mod 7 = 3\n => I#N = 3\nE#K = J#N / L#L / I#N\n = 1 / 4 / 3\n 1 / 4 = (1 × 4⁻¹) mod 7 = 2\n 2 / 3 = (2 × 3⁻¹) mod 7 = 3\n => E#K = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 9, "num_distractors": 0, "var_layer": {"L#L": 1, "H#M": 1, "E#N": 1, "J#L": 1, "H#K": 2, "L#K": 2, "J#N": 3, "I#N": 3, "E#K": 4}, "id": "mod7_arith_d4_0172"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#P := K#O + K#J. J#P := K#J + K#P. K#O := 2. G#N := K#O. L#K := H#P / K#M / K#J. K#M := 3 + K#J - L#J. L#J := 3. K#J := K#O + L#J. K#P := L#J. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#P := K#O + K#J. J#P := K#J + K#P. K#O := 2. G#N := K#O. L#K := H#P / K#M / K#J. K#M := 3 + K#J - L#J. L#J := 3. K#J := K#O + L#J. K#P := L#J.", "query": "L#K", "answer": 0, "cot": "K#O = 2 -> K#O = 2\nL#J = 3 -> L#J = 3\nK#J = K#O + L#J\n = 2 + 3\n 2 + 3 = (2 + 3) mod 7 = 5\n => K#J = 5\nK#M = 3 + K#J - L#J\n = 3 + 5 - 3\n 3 + 5 = (3 + 5) mod 7 = 1\n 1 - 3 = (1 - 3) mod 7 = 5\n => K#M = 5\nH#P = K#O + K#J\n = 2 + 5\n 2 + 5 = (2 + 5) mod 7 = 0\n => H#P = 0\nL#K = H#P / K#M / K#J\n = 0 / 5 / 5\n 0 / 5 = (0 × 5⁻¹) mod 7 = 0\n 0 / 5 = (0 × 5⁻¹) mod 7 = 0\n => L#K = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#O": 1, "L#J": 1, "K#J": 2, "G#N": 2, "K#P": 2, "J#P": 3, "K#M": 3, "H#P": 3, "L#K": 4}, "id": "mod7_arith_d4_0173"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#L := J#O + 6 / F#M. G#L := J#O * L#P + F#M. L#I := 5. J#O := 5. I#N := F#K * 3. K#J := L#P. I#O := 1. F#K := J#O. F#M := 4. L#P := J#O / F#M. F#N := I#N - J#L. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#L := J#O + 6 / F#M. G#L := J#O * L#P + F#M. L#I := 5. J#O := 5. I#N := F#K * 3. K#J := L#P. I#O := 1. F#K := J#O. F#M := 4. L#P := J#O / F#M. F#N := I#N - J#L.", "query": "F#N", "answer": 0, "cot": "F#M = 4 -> F#M = 4\nJ#O = 5 -> J#O = 5\nJ#L = J#O + 6 / F#M\n = 5 + 6 / 4\n 5 + 6 = (5 + 6) mod 7 = 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => J#L = 1\nF#K = J#O = 5\nI#N = F#K * 3\n = 5 * 3\n 5 * 3 = (5 × 3) mod 7 = 1\n => I#N = 1\nF#N = I#N - J#L\n = 1 - 1\n 1 - 1 = (1 - 1) mod 7 = 0\n => F#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 6, "num_distractors": 0, "var_layer": {"L#I": 1, "F#M": 1, "J#O": 1, "I#O": 1, "J#L": 2, "L#P": 2, "F#K": 2, "I#N": 3, "K#J": 3, "G#L": 3, "F#N": 4}, "id": "mod7_arith_d4_0174"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#P := 4 - K#L - J#L. I#M := 6. J#L := 2 * K#K. K#K := 4. K#L := J#L. E#L := I#M + K#K / J#L. G#L := 5 * K#K / 2. L#P := G#L. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#P := 4 - K#L - J#L. I#M := 6. J#L := 2 * K#K. K#K := 4. K#L := J#L. E#L := I#M + K#K / J#L. G#L := 5 * K#K / 2. L#P := G#L.", "query": "K#P", "answer": 2, "cot": "K#K = 4 -> K#K = 4\nJ#L = 2 * K#K\n = 2 * 4\n 2 * 4 = (2 × 4) mod 7 = 1\n => J#L = 1\nK#L = J#L = 1\nK#P = 4 - K#L - J#L\n = 4 - 1 - 1\n 4 - 1 = (4 - 1) mod 7 = 3\n 3 - 1 = (3 - 1) mod 7 = 2\n => K#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#M": 1, "K#K": 1, "J#L": 2, "G#L": 2, "E#L": 3, "K#L": 3, "L#P": 3, "K#P": 4}, "id": "mod7_arith_d4_0175"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := E#L / J#K. J#K := 2. F#I := K#P. K#K := L#J. E#L := 4. L#J := 6. K#P := H#O. F#J := E#L + K#K. F#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := E#L / J#K. J#K := 2. F#I := K#P. K#K := L#J. E#L := 4. L#J := 6. K#P := H#O. F#J := E#L + K#K.", "query": "F#I", "answer": 2, "cot": "J#K = 2 -> J#K = 2\nE#L = 4 -> E#L = 4\nH#O = E#L / J#K\n = 4 / 2\n 4 / 2 = (4 × 2⁻¹) mod 7 = 2\n => H#O = 2\nK#P = H#O = 2\nF#I = K#P = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#K": 1, "L#J": 1, "E#L": 1, "K#K": 2, "H#O": 2, "K#P": 3, "F#J": 3, "F#I": 4}, "id": "mod7_arith_d4_0176"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#P := H#I - H#K. F#J := I#N. E#I := 6. I#N := 3. F#K := 4 / I#M. I#M := K#P. E#J := H#I. H#I := 5. H#K := 2. I#K := E#J / 6. F#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#P := H#I - H#K. F#J := I#N. E#I := 6. I#N := 3. F#K := 4 / I#M. I#M := K#P. E#J := H#I. H#I := 5. H#K := 2. I#K := E#J / 6.", "query": "F#K", "answer": 6, "cot": "H#I = 5 -> H#I = 5\nH#K = 2 -> H#K = 2\nK#P = H#I - H#K\n = 5 - 2\n 5 - 2 = (5 - 2) mod 7 = 3\n => K#P = 3\nI#M = K#P = 3\nF#K = 4 / I#M\n = 4 / 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => F#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#I": 1, "H#K": 1, "I#N": 1, "E#I": 1, "E#J": 2, "K#P": 2, "F#J": 2, "I#M": 3, "I#K": 3, "F#K": 4}, "id": "mod7_arith_d4_0177"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := J#K - E#P. G#N := L#J * 6. J#K := G#N * J#J / L#O. J#J := 1. F#I := L#J * L#O. K#P := 4. L#O := J#J * K#P. L#J := 3. E#P := E#L + L#O. E#L := 2. F#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := J#K - E#P. G#N := L#J * 6. J#K := G#N * J#J / L#O. J#J := 1. F#I := L#J * L#O. K#P := 4. L#O := J#J * K#P. L#J := 3. E#P := E#L + L#O. E#L := 2.", "query": "F#N", "answer": 2, "cot": "L#J = 3 -> L#J = 3\nK#P = 4 -> K#P = 4\nE#L = 2 -> E#L = 2\nJ#J = 1 -> J#J = 1\nG#N = L#J * 6\n = 3 * 6\n 3 * 6 = (3 × 6) mod 7 = 4\n => G#N = 4\nL#O = J#J * K#P\n = 1 * 4\n 1 * 4 = (1 × 4) mod 7 = 4\n => L#O = 4\nE#P = E#L + L#O\n = 2 + 4\n 2 + 4 = (2 + 4) mod 7 = 6\n => E#P = 6\nJ#K = G#N * J#J / L#O\n = 4 * 1 / 4\n 4 * 1 = (4 × 1) mod 7 = 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => J#K = 1\nF#N = J#K - E#P\n = 1 - 6\n 1 - 6 = (1 - 6) mod 7 = 2\n => F#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 9, "num_distractors": 0, "var_layer": {"L#J": 1, "K#P": 1, "E#L": 1, "J#J": 1, "G#N": 2, "L#O": 2, "F#I": 3, "E#P": 3, "J#K": 3, "F#N": 4}, "id": "mod7_arith_d4_0178"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := 3. H#K := 4. F#O := 3. E#I := 6 - J#N. I#I := I#P. G#O := H#K / F#L. J#N := 3 - F#O. H#P := J#N / 5 - E#I. I#P := F#L. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := 3. H#K := 4. F#O := 3. E#I := 6 - J#N. I#I := I#P. G#O := H#K / F#L. J#N := 3 - F#O. H#P := J#N / 5 - E#I. I#P := F#L.", "query": "H#P", "answer": 1, "cot": "F#O = 3 -> F#O = 3\nJ#N = 3 - F#O\n = 3 - 3\n 3 - 3 = (3 - 3) mod 7 = 0\n => J#N = 0\nE#I = 6 - J#N\n = 6 - 0\n 6 - 0 = (6 - 0) mod 7 = 6\n => E#I = 6\nH#P = J#N / 5 - E#I\n = 0 / 5 - 6\n 0 / 5 = (0 × 5⁻¹) mod 7 = 0\n 0 - 6 = (0 - 6) mod 7 = 1\n => H#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#L": 1, "F#O": 1, "H#K": 1, "I#P": 2, "J#N": 2, "G#O": 2, "E#I": 3, "I#I": 3, "H#P": 4}, "id": "mod7_arith_d4_0179"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := 6 - K#N + G#O. G#O := 4 * G#M. G#M := 1. K#J := G#M - F#M. F#K := 4. H#L := 5. E#K := 3 - G#O / G#M. E#N := H#L + G#M. K#N := F#K * H#L. F#M := K#N - E#N. K#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := 6 - K#N + G#O. G#O := 4 * G#M. G#M := 1. K#J := G#M - F#M. F#K := 4. H#L := 5. E#K := 3 - G#O / G#M. E#N := H#L + G#M. K#N := F#K * H#L. F#M := K#N - E#N.", "query": "K#J", "answer": 1, "cot": "F#K = 4 -> F#K = 4\nH#L = 5 -> H#L = 5\nG#M = 1 -> G#M = 1\nE#N = H#L + G#M\n = 5 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n => E#N = 6\nK#N = F#K * H#L\n = 4 * 5\n 4 * 5 = (4 × 5) mod 7 = 6\n => K#N = 6\nF#M = K#N - E#N\n = 6 - 6\n 6 - 6 = (6 - 6) mod 7 = 0\n => F#M = 0\nK#J = G#M - F#M\n = 1 - 0\n 1 - 0 = (1 - 0) mod 7 = 1\n => K#J = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"F#K": 1, "H#L": 1, "G#M": 1, "G#O": 2, "E#N": 2, "K#N": 2, "J#O": 3, "E#K": 3, "F#M": 3, "K#J": 4}, "id": "mod7_arith_d4_0180"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#L := L#K * J#M. K#J := I#L - H#N. J#L := J#I / I#L. J#M := 4. H#N := J#M * L#K. F#I := H#N + I#J. I#J := L#K - J#M. J#I := I#L * J#M. L#K := 4. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#L := L#K * J#M. K#J := I#L - H#N. J#L := J#I / I#L. J#M := 4. H#N := J#M * L#K. F#I := H#N + I#J. I#J := L#K - J#M. J#I := I#L * J#M. L#K := 4.", "query": "J#L", "answer": 4, "cot": "L#K = 4 -> L#K = 4\nJ#M = 4 -> J#M = 4\nI#L = L#K * J#M\n = 4 * 4\n 4 * 4 = (4 × 4) mod 7 = 2\n => I#L = 2\nJ#I = I#L * J#M\n = 2 * 4\n 2 * 4 = (2 × 4) mod 7 = 1\n => J#I = 1\nJ#L = J#I / I#L\n = 1 / 2\n 1 / 2 = (1 × 2⁻¹) mod 7 = 4\n => J#L = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#K": 1, "J#M": 1, "H#N": 2, "I#J": 2, "I#L": 2, "J#I": 3, "K#J": 3, "F#I": 3, "J#L": 4}, "id": "mod7_arith_d4_0181"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := I#L. F#N := 4. L#O := F#O - 3 + L#J. I#L := 1. F#O := H#O. L#L := 1. G#J := K#K. J#M := H#O. K#K := 4 * L#L. L#J := 3. L#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := I#L. F#N := 4. L#O := F#O - 3 + L#J. I#L := 1. F#O := H#O. L#L := 1. G#J := K#K. J#M := H#O. K#K := 4 * L#L. L#J := 3.", "query": "L#O", "answer": 1, "cot": "I#L = 1 -> I#L = 1\nL#J = 3 -> L#J = 3\nH#O = I#L = 1\nF#O = H#O = 1\nL#O = F#O - 3 + L#J\n = 1 - 3 + 3\n 1 - 3 = (1 - 3) mod 7 = 5\n 5 + 3 = (5 + 3) mod 7 = 1\n => L#O = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#L": 1, "I#L": 1, "F#N": 1, "L#J": 1, "K#K": 2, "H#O": 2, "J#M": 3, "G#J": 3, "F#O": 3, "L#O": 4}, "id": "mod7_arith_d4_0182"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := 6. F#N := H#J * 3. H#J := G#J / E#O / 5. E#O := 3. E#L := G#N * H#J * F#N. G#N := I#I + E#O / H#J. G#K := G#O. I#I := G#J - 5. G#J := 6. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := 6. F#N := H#J * 3. H#J := G#J / E#O / 5. E#O := 3. E#L := G#N * H#J * F#N. G#N := I#I + E#O / H#J. G#K := G#O. I#I := G#J - 5. G#J := 6.", "query": "E#L", "answer": 2, "cot": "E#O = 3 -> E#O = 3\nG#J = 6 -> G#J = 6\nH#J = G#J / E#O / 5\n = 6 / 3 / 5\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n 2 / 5 = (2 × 5⁻¹) mod 7 = 6\n => H#J = 6\nI#I = G#J - 5\n = 6 - 5\n 6 - 5 = (6 - 5) mod 7 = 1\n => I#I = 1\nF#N = H#J * 3\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => F#N = 4\nG#N = I#I + E#O / H#J\n = 1 + 3 / 6\n 1 + 3 = (1 + 3) mod 7 = 4\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n => G#N = 3\nE#L = G#N * H#J * F#N\n = 3 * 6 * 4\n 3 * 6 = (3 × 6) mod 7 = 4\n 4 * 4 = (4 × 4) mod 7 = 2\n => E#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"E#O": 1, "G#J": 1, "G#O": 1, "G#K": 2, "H#J": 2, "I#I": 2, "F#N": 3, "G#N": 3, "E#L": 4}, "id": "mod7_arith_d4_0183"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := 4. F#P := L#O - H#I / E#O. J#P := 2. F#M := 3 * 4 * H#I. K#N := I#I + E#O * L#O. H#M := 1. H#I := L#O * H#M. L#O := 3. L#P := F#P. E#O := I#I. L#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := 4. F#P := L#O - H#I / E#O. J#P := 2. F#M := 3 * 4 * H#I. K#N := I#I + E#O * L#O. H#M := 1. H#I := L#O * H#M. L#O := 3. L#P := F#P. E#O := I#I.", "query": "L#P", "answer": 0, "cot": "L#O = 3 -> L#O = 3\nI#I = 4 -> I#I = 4\nH#M = 1 -> H#M = 1\nH#I = L#O * H#M\n = 3 * 1\n 3 * 1 = (3 × 1) mod 7 = 3\n => H#I = 3\nE#O = I#I = 4\nF#P = L#O - H#I / E#O\n = 3 - 3 / 4\n 3 - 3 = (3 - 3) mod 7 = 0\n 0 / 4 = (0 × 4⁻¹) mod 7 = 0\n => F#P = 0\nL#P = F#P = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"L#O": 1, "I#I": 1, "J#P": 1, "H#M": 1, "H#I": 2, "E#O": 2, "F#M": 3, "K#N": 3, "F#P": 3, "L#P": 4}, "id": "mod7_arith_d4_0184"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#K := J#N * K#P. K#P := K#K * J#P + H#J. J#L := L#P - 4. K#K := 5. J#P := K#K * J#N - H#J. J#N := 2. H#J := 3. L#P := H#J. G#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#K := J#N * K#P. K#P := K#K * J#P + H#J. J#L := L#P - 4. K#K := 5. J#P := K#K * J#N - H#J. J#N := 2. H#J := 3. L#P := H#J.", "query": "G#K", "answer": 6, "cot": "K#K = 5 -> K#K = 5\nJ#N = 2 -> J#N = 2\nH#J = 3 -> H#J = 3\nJ#P = K#K * J#N - H#J\n = 5 * 2 - 3\n 5 * 2 = (5 × 2) mod 7 = 3\n 3 - 3 = (3 - 3) mod 7 = 0\n => J#P = 0\nK#P = K#K * J#P + H#J\n = 5 * 0 + 3\n 5 * 0 = (5 × 0) mod 7 = 0\n 0 + 3 = (0 + 3) mod 7 = 3\n => K#P = 3\nG#K = J#N * K#P\n = 2 * 3\n 2 * 3 = (2 × 3) mod 7 = 6\n => G#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#K": 1, "J#N": 1, "H#J": 1, "L#P": 2, "J#P": 2, "K#P": 3, "J#L": 3, "G#K": 4}, "id": "mod7_arith_d4_0185"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := E#I * I#N. H#N := I#N. L#O := 5 + G#K / H#N. G#K := H#N. I#N := 1. E#I := I#N / J#I. I#I := E#I / 5. J#L := 2. J#I := 5. L#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := E#I * I#N. H#N := I#N. L#O := 5 + G#K / H#N. G#K := H#N. I#N := 1. E#I := I#N / J#I. I#I := E#I / 5. J#L := 2. J#I := 5.", "query": "L#O", "answer": 6, "cot": "I#N = 1 -> I#N = 1\nH#N = I#N = 1\nG#K = H#N = 1\nL#O = 5 + G#K / H#N\n = 5 + 1 / 1\n 5 + 1 = (5 + 1) mod 7 = 6\n 6 / 1 = (6 × 1⁻¹) mod 7 = 6\n => L#O = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 4, "num_distractors": 0, "var_layer": {"I#N": 1, "J#L": 1, "J#I": 1, "H#N": 2, "E#I": 2, "I#I": 3, "G#K": 3, "K#J": 3, "L#O": 4}, "id": "mod7_arith_d4_0186"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#P := I#I - J#O * E#L. I#I := 4. I#M := 2 - G#I. E#L := 4. L#J := J#N / 6. G#I := E#L * J#O * I#I. J#O := 2. J#N := E#L * G#I - L#P. E#K := I#I * J#O. L#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#P := I#I - J#O * E#L. I#I := 4. I#M := 2 - G#I. E#L := 4. L#J := J#N / 6. G#I := E#L * J#O * I#I. J#O := 2. J#N := E#L * G#I - L#P. E#K := I#I * J#O.", "query": "L#J", "answer": 6, "cot": "J#O = 2 -> J#O = 2\nE#L = 4 -> E#L = 4\nI#I = 4 -> I#I = 4\nG#I = E#L * J#O * I#I\n = 4 * 2 * 4\n 4 * 2 = (4 × 2) mod 7 = 1\n 1 * 4 = (1 × 4) mod 7 = 4\n => G#I = 4\nL#P = I#I - J#O * E#L\n = 4 - 2 * 4\n 4 - 2 = (4 - 2) mod 7 = 2\n 2 * 4 = (2 × 4) mod 7 = 1\n => L#P = 1\nJ#N = E#L * G#I - L#P\n = 4 * 4 - 1\n 4 * 4 = (4 × 4) mod 7 = 2\n 2 - 1 = (2 - 1) mod 7 = 1\n => J#N = 1\nL#J = J#N / 6\n = 1 / 6\n 1 / 6 = (1 × 6⁻¹) mod 7 = 6\n => L#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"J#O": 1, "E#L": 1, "I#I": 1, "G#I": 2, "E#K": 2, "L#P": 2, "J#N": 3, "I#M": 3, "L#J": 4}, "id": "mod7_arith_d4_0187"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := L#J + 3. G#P := 6. K#J := 4. F#P := G#P * L#J + 6. G#N := 2. K#O := F#J * G#L. F#J := 4 / 3 / K#I. G#L := F#P - G#K + K#I. G#K := G#N / 5. L#J := 2. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := L#J + 3. G#P := 6. K#J := 4. F#P := G#P * L#J + 6. G#N := 2. K#O := F#J * G#L. F#J := 4 / 3 / K#I. G#L := F#P - G#K + K#I. G#K := G#N / 5. L#J := 2.", "query": "K#O", "answer": 5, "cot": "L#J = 2 -> L#J = 2\nG#N = 2 -> G#N = 2\nG#P = 6 -> G#P = 6\nF#P = G#P * L#J + 6\n = 6 * 2 + 6\n 6 * 2 = (6 × 2) mod 7 = 5\n 5 + 6 = (5 + 6) mod 7 = 4\n => F#P = 4\nK#I = L#J + 3\n = 2 + 3\n 2 + 3 = (2 + 3) mod 7 = 5\n => K#I = 5\nG#K = G#N / 5\n = 2 / 5\n 2 / 5 = (2 × 5⁻¹) mod 7 = 6\n => G#K = 6\nG#L = F#P - G#K + K#I\n = 4 - 6 + 5\n 4 - 6 = (4 - 6) mod 7 = 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => G#L = 3\nF#J = 4 / 3 / K#I\n = 4 / 3 / 5\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n 6 / 5 = (6 × 5⁻¹) mod 7 = 4\n => F#J = 4\nK#O = F#J * G#L\n = 4 * 3\n 4 * 3 = (4 × 3) mod 7 = 5\n => K#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 9, "num_distractors": 0, "var_layer": {"L#J": 1, "K#J": 1, "G#N": 1, "G#P": 1, "F#P": 2, "K#I": 2, "G#K": 2, "G#L": 3, "F#J": 3, "K#O": 4}, "id": "mod7_arith_d4_0188"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#P := I#I / 4. F#I := 4. L#L := 3. J#I := 6. I#I := F#P - L#L + J#I. I#J := F#I * E#M. K#M := I#J. F#P := 5. J#P := F#P. E#M := J#I * L#L. K#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#P := I#I / 4. F#I := 4. L#L := 3. J#I := 6. I#I := F#P - L#L + J#I. I#J := F#I * E#M. K#M := I#J. F#P := 5. J#P := F#P. E#M := J#I * L#L.", "query": "K#M", "answer": 2, "cot": "F#I = 4 -> F#I = 4\nL#L = 3 -> L#L = 3\nJ#I = 6 -> J#I = 6\nE#M = J#I * L#L\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => E#M = 4\nI#J = F#I * E#M\n = 4 * 4\n 4 * 4 = (4 × 4) mod 7 = 2\n => I#J = 2\nK#M = I#J = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#P": 1, "F#I": 1, "L#L": 1, "J#I": 1, "E#M": 2, "I#I": 2, "J#P": 2, "I#J": 3, "I#P": 3, "K#M": 4}, "id": "mod7_arith_d4_0189"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#O := 5. F#I := G#I * 3. G#L := 1. J#P := H#O - F#P + 6. K#I := 2. G#P := E#M + G#I. L#I := 3. G#I := K#I. E#M := L#I. F#P := E#M / L#I. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#O := 5. F#I := G#I * 3. G#L := 1. J#P := H#O - F#P + 6. K#I := 2. G#P := E#M + G#I. L#I := 3. G#I := K#I. E#M := L#I. F#P := E#M / L#I.", "query": "J#P", "answer": 3, "cot": "L#I = 3 -> L#I = 3\nH#O = 5 -> H#O = 5\nE#M = L#I = 3\nF#P = E#M / L#I\n = 3 / 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => F#P = 1\nJ#P = H#O - F#P + 6\n = 5 - 1 + 6\n 5 - 1 = (5 - 1) mod 7 = 4\n 4 + 6 = (4 + 6) mod 7 = 3\n => J#P = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#I": 1, "G#L": 1, "K#I": 1, "H#O": 1, "E#M": 2, "G#I": 2, "F#I": 3, "F#P": 3, "G#P": 3, "J#P": 4}, "id": "mod7_arith_d4_0190"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := L#I + K#J. G#I := 3. K#O := F#K / G#I - 4. F#K := 6. L#I := G#I / F#K. J#P := I#K + L#I. K#J := G#I - 3 - F#K. I#K := K#J / K#O. H#N := L#I * K#O. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := L#I + K#J. G#I := 3. K#O := F#K / G#I - 4. F#K := 6. L#I := G#I / F#K. J#P := I#K + L#I. K#J := G#I - 3 - F#K. I#K := K#J / K#O. H#N := L#I * K#O.", "query": "J#P", "answer": 0, "cot": "F#K = 6 -> F#K = 6\nG#I = 3 -> G#I = 3\nL#I = G#I / F#K\n = 3 / 6\n 3 / 6 = (3 × 6⁻¹) mod 7 = 4\n => L#I = 4\nK#O = F#K / G#I - 4\n = 6 / 3 - 4\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n 2 - 4 = (2 - 4) mod 7 = 5\n => K#O = 5\nK#J = G#I - 3 - F#K\n = 3 - 3 - 6\n 3 - 3 = (3 - 3) mod 7 = 0\n 0 - 6 = (0 - 6) mod 7 = 1\n => K#J = 1\nI#K = K#J / K#O\n = 1 / 5\n 1 / 5 = (1 × 5⁻¹) mod 7 = 3\n => I#K = 3\nJ#P = I#K + L#I\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => J#P = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"F#K": 1, "G#I": 1, "L#I": 2, "K#O": 2, "K#J": 2, "H#N": 3, "I#K": 3, "E#I": 3, "J#P": 4}, "id": "mod7_arith_d4_0191"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#M := F#M - E#J. H#M := 5. J#N := I#I * F#K. G#J := 4 + F#M. F#M := 6 * E#K / I#I. F#N := I#I + L#K. E#K := 6. L#K := 5. E#J := L#K - I#I - H#M. F#K := F#N - F#M / 2. I#I := 2. J#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#M := F#M - E#J. H#M := 5. J#N := I#I * F#K. G#J := 4 + F#M. F#M := 6 * E#K / I#I. F#N := I#I + L#K. E#K := 6. L#K := 5. E#J := L#K - I#I - H#M. F#K := F#N - F#M / 2. I#I := 2.", "query": "J#N", "answer": 3, "cot": "E#K = 6 -> E#K = 6\nI#I = 2 -> I#I = 2\nL#K = 5 -> L#K = 5\nF#N = I#I + L#K\n = 2 + 5\n 2 + 5 = (2 + 5) mod 7 = 0\n => F#N = 0\nF#M = 6 * E#K / I#I\n = 6 * 6 / 2\n 6 * 6 = (6 × 6) mod 7 = 1\n 1 / 2 = (1 × 2⁻¹) mod 7 = 4\n => F#M = 4\nF#K = F#N - F#M / 2\n = 0 - 4 / 2\n 0 - 4 = (0 - 4) mod 7 = 3\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => F#K = 5\nJ#N = I#I * F#K\n = 2 * 5\n 2 * 5 = (2 × 5) mod 7 = 3\n => J#N = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 7, "num_distractors": 0, "var_layer": {"H#M": 1, "E#K": 1, "I#I": 1, "L#K": 1, "F#N": 2, "E#J": 2, "F#M": 2, "G#J": 3, "F#K": 3, "L#M": 3, "J#N": 4}, "id": "mod7_arith_d4_0192"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#I := G#P - I#N. L#K := I#N * E#M. G#P := E#M. K#I := E#M * L#P. I#N := 1. E#M := 1. L#M := I#N - E#M. L#P := L#K. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#I := G#P - I#N. L#K := I#N * E#M. G#P := E#M. K#I := E#M * L#P. I#N := 1. E#M := 1. L#M := I#N - E#M. L#P := L#K.", "query": "K#I", "answer": 1, "cot": "E#M = 1 -> E#M = 1\nI#N = 1 -> I#N = 1\nL#K = I#N * E#M\n = 1 * 1\n 1 * 1 = (1 × 1) mod 7 = 1\n => L#K = 1\nL#P = L#K = 1\nK#I = E#M * L#P\n = 1 * 1\n 1 * 1 = (1 × 1) mod 7 = 1\n => K#I = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"E#M": 1, "I#N": 1, "L#K": 2, "G#P": 2, "L#M": 2, "E#I": 3, "L#P": 3, "K#I": 4}, "id": "mod7_arith_d4_0193"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := J#K + I#J. E#J := 1. G#N := G#J. K#N := G#N / 3. J#K := 3. I#J := 3. L#P := E#J * 5. G#M := L#P + I#J. K#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := J#K + I#J. E#J := 1. G#N := G#J. K#N := G#N / 3. J#K := 3. I#J := 3. L#P := E#J * 5. G#M := L#P + I#J.", "query": "K#N", "answer": 2, "cot": "J#K = 3 -> J#K = 3\nI#J = 3 -> I#J = 3\nG#J = J#K + I#J\n = 3 + 3\n 3 + 3 = (3 + 3) mod 7 = 6\n => G#J = 6\nG#N = G#J = 6\nK#N = G#N / 3\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => K#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#K": 1, "I#J": 1, "E#J": 1, "G#J": 2, "L#P": 2, "G#M": 3, "G#N": 3, "K#N": 4}, "id": "mod7_arith_d4_0194"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#L := F#N + I#J + K#N. I#M := 3 / G#P. K#N := 3. I#J := K#P - 6. G#P := F#N + 2 - 4. K#P := K#N. F#N := 5. E#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#L := F#N + I#J + K#N. I#M := 3 / G#P. K#N := 3. I#J := K#P - 6. G#P := F#N + 2 - 4. K#P := K#N. F#N := 5.", "query": "E#L", "answer": 5, "cot": "K#N = 3 -> K#N = 3\nF#N = 5 -> F#N = 5\nK#P = K#N = 3\nI#J = K#P - 6\n = 3 - 6\n 3 - 6 = (3 - 6) mod 7 = 4\n => I#J = 4\nE#L = F#N + I#J + K#N\n = 5 + 4 + 3\n 5 + 4 = (5 + 4) mod 7 = 2\n 2 + 3 = (2 + 3) mod 7 = 5\n => E#L = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"K#N": 1, "F#N": 1, "K#P": 2, "G#P": 2, "I#J": 3, "I#M": 3, "E#L": 4}, "id": "mod7_arith_d4_0195"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#K := 2. E#P := 2. L#O := E#P / L#K * K#M. F#K := E#M / 5 / 5. E#M := H#K - L#K. E#K := E#M + L#O + K#M. K#M := 4. K#I := F#K - E#P / L#K. H#K := 3. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#K := 2. E#P := 2. L#O := E#P / L#K * K#M. F#K := E#M / 5 / 5. E#M := H#K - L#K. E#K := E#M + L#O + K#M. K#M := 4. K#I := F#K - E#P / L#K. H#K := 3.", "query": "K#I", "answer": 0, "cot": "L#K = 2 -> L#K = 2\nH#K = 3 -> H#K = 3\nE#P = 2 -> E#P = 2\nE#M = H#K - L#K\n = 3 - 2\n 3 - 2 = (3 - 2) mod 7 = 1\n => E#M = 1\nF#K = E#M / 5 / 5\n = 1 / 5 / 5\n 1 / 5 = (1 × 5⁻¹) mod 7 = 3\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n => F#K = 2\nK#I = F#K - E#P / L#K\n = 2 - 2 / 2\n 2 - 2 = (2 - 2) mod 7 = 0\n 0 / 2 = (0 × 2⁻¹) mod 7 = 0\n => K#I = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#M": 1, "L#K": 1, "H#K": 1, "E#P": 1, "E#M": 2, "L#O": 2, "E#K": 3, "F#K": 3, "K#I": 4}, "id": "mod7_arith_d4_0196"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#N := 4. G#I := 6. K#K := H#P + H#N / G#I. F#M := G#I + H#P. H#P := 3. G#J := K#J. K#J := G#I - H#N. I#L := 5 / K#J. L#M := G#I / F#M * K#K. E#N := 6 + G#J. E#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#N := 4. G#I := 6. K#K := H#P + H#N / G#I. F#M := G#I + H#P. H#P := 3. G#J := K#J. K#J := G#I - H#N. I#L := 5 / K#J. L#M := G#I / F#M * K#K. E#N := 6 + G#J.", "query": "E#N", "answer": 1, "cot": "G#I = 6 -> G#I = 6\nH#N = 4 -> H#N = 4\nK#J = G#I - H#N\n = 6 - 4\n 6 - 4 = (6 - 4) mod 7 = 2\n => K#J = 2\nG#J = K#J = 2\nE#N = 6 + G#J\n = 6 + 2\n 6 + 2 = (6 + 2) mod 7 = 1\n => E#N = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#I": 1, "H#N": 1, "H#P": 1, "K#K": 2, "F#M": 2, "K#J": 2, "L#M": 3, "I#L": 3, "G#J": 3, "E#N": 4}, "id": "mod7_arith_d4_0197"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := 3. H#O := K#O. H#P := J#L + K#L. G#L := 4. K#K := K#O. H#L := 4. J#L := 3 - E#I + 4. E#I := G#L / K#L. K#O := G#L / H#L. F#L := 4. I#P := G#L / 2 + K#L. H#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := 3. H#O := K#O. H#P := J#L + K#L. G#L := 4. K#K := K#O. H#L := 4. J#L := 3 - E#I + 4. E#I := G#L / K#L. K#O := G#L / H#L. F#L := 4. I#P := G#L / 2 + K#L.", "query": "H#P", "answer": 4, "cot": "G#L = 4 -> G#L = 4\nK#L = 3 -> K#L = 3\nE#I = G#L / K#L\n = 4 / 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => E#I = 6\nJ#L = 3 - E#I + 4\n = 3 - 6 + 4\n 3 - 6 = (3 - 6) mod 7 = 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => J#L = 1\nH#P = J#L + K#L\n = 1 + 3\n 1 + 3 = (1 + 3) mod 7 = 4\n => H#P = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#L": 1, "G#L": 1, "F#L": 1, "K#L": 1, "K#O": 2, "E#I": 2, "I#P": 2, "K#K": 3, "H#O": 3, "J#L": 3, "H#P": 4}, "id": "mod7_arith_d4_0198"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#K := E#O * 2 - 6. E#K := H#L / 4 * L#P. H#L := 6. E#O := E#P - L#P. E#P := H#L + L#P. L#P := 1. G#I := E#P - E#K. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#K := E#O * 2 - 6. E#K := H#L / 4 * L#P. H#L := 6. E#O := E#P - L#P. E#P := H#L + L#P. L#P := 1. G#I := E#P - E#K.", "query": "J#K", "answer": 6, "cot": "H#L = 6 -> H#L = 6\nL#P = 1 -> L#P = 1\nE#P = H#L + L#P\n = 6 + 1\n 6 + 1 = (6 + 1) mod 7 = 0\n => E#P = 0\nE#O = E#P - L#P\n = 0 - 1\n 0 - 1 = (0 - 1) mod 7 = 6\n => E#O = 6\nJ#K = E#O * 2 - 6\n = 6 * 2 - 6\n 6 * 2 = (6 × 2) mod 7 = 5\n 5 - 6 = (5 - 6) mod 7 = 6\n => J#K = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#L": 1, "L#P": 1, "E#K": 2, "E#P": 2, "G#I": 3, "E#O": 3, "J#K": 4}, "id": "mod7_arith_d4_0199"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := F#M. H#M := 6. H#O := 4. E#J := E#K - H#O * 3. L#M := 6 / E#K / F#M. G#O := H#M + E#J. H#I := F#J. F#M := 2. L#I := G#O. E#K := 6. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := F#M. H#M := 6. H#O := 4. E#J := E#K - H#O * 3. L#M := 6 / E#K / F#M. G#O := H#M + E#J. H#I := F#J. F#M := 2. L#I := G#O. E#K := 6.", "query": "L#I", "answer": 5, "cot": "H#M = 6 -> H#M = 6\nH#O = 4 -> H#O = 4\nE#K = 6 -> E#K = 6\nE#J = E#K - H#O * 3\n = 6 - 4 * 3\n 6 - 4 = (6 - 4) mod 7 = 2\n 2 * 3 = (2 × 3) mod 7 = 6\n => E#J = 6\nG#O = H#M + E#J\n = 6 + 6\n 6 + 6 = (6 + 6) mod 7 = 5\n => G#O = 5\nL#I = G#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"H#M": 1, "H#O": 1, "F#M": 1, "E#K": 1, "F#J": 2, "L#M": 2, "E#J": 2, "H#I": 3, "G#O": 3, "L#I": 4}, "id": "mod7_arith_d4_0200"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#J := E#M. G#P := H#N + G#I. H#N := E#M. E#J := 1. G#I := E#M. J#P := 6. E#M := E#J / J#P / 5. J#L := E#J. G#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#J := E#M. G#P := H#N + G#I. H#N := E#M. E#J := 1. G#I := E#M. J#P := 6. E#M := E#J / J#P / 5. J#L := E#J.", "query": "G#P", "answer": 1, "cot": "E#J = 1 -> E#J = 1\nJ#P = 6 -> J#P = 6\nE#M = E#J / J#P / 5\n = 1 / 6 / 5\n 1 / 6 = (1 × 6⁻¹) mod 7 = 6\n 6 / 5 = (6 × 5⁻¹) mod 7 = 4\n => E#M = 4\nG#I = E#M = 4\nH#N = E#M = 4\nG#P = H#N + G#I\n = 4 + 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => G#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#J": 1, "J#P": 1, "E#M": 2, "J#L": 2, "G#I": 3, "K#J": 3, "H#N": 3, "G#P": 4}, "id": "mod7_arith_d4_0201"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#L := J#I / I#N. F#L := 5. K#O := K#L * J#I. I#L := J#K + F#L. J#I := J#K - F#L. J#K := 6. F#J := J#I + F#L / I#L. I#N := J#K * F#L. G#P := J#I + I#L. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#L := J#I / I#N. F#L := 5. K#O := K#L * J#I. I#L := J#K + F#L. J#I := J#K - F#L. J#K := 6. F#J := J#I + F#L / I#L. I#N := J#K * F#L. G#P := J#I + I#L.", "query": "K#O", "answer": 4, "cot": "J#K = 6 -> J#K = 6\nF#L = 5 -> F#L = 5\nJ#I = J#K - F#L\n = 6 - 5\n 6 - 5 = (6 - 5) mod 7 = 1\n => J#I = 1\nI#N = J#K * F#L\n = 6 * 5\n 6 * 5 = (6 × 5) mod 7 = 2\n => I#N = 2\nK#L = J#I / I#N\n = 1 / 2\n 1 / 2 = (1 × 2⁻¹) mod 7 = 4\n => K#L = 4\nK#O = K#L * J#I\n = 4 * 1\n 4 * 1 = (4 × 1) mod 7 = 4\n => K#O = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"J#K": 1, "F#L": 1, "J#I": 2, "I#L": 2, "I#N": 2, "K#L": 3, "G#P": 3, "F#J": 3, "K#O": 4}, "id": "mod7_arith_d4_0202"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#I := I#N * F#J. L#L := 1. F#I := E#O - L#L. J#O := F#I. E#O := 6. E#P := F#J / E#O. I#N := E#P. K#P := F#I + F#J. F#J := 6. K#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#I := I#N * F#J. L#L := 1. F#I := E#O - L#L. J#O := F#I. E#O := 6. E#P := F#J / E#O. I#N := E#P. K#P := F#I + F#J. F#J := 6.", "query": "K#I", "answer": 6, "cot": "F#J = 6 -> F#J = 6\nE#O = 6 -> E#O = 6\nE#P = F#J / E#O\n = 6 / 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => E#P = 1\nI#N = E#P = 1\nK#I = I#N * F#J\n = 1 * 6\n 1 * 6 = (1 × 6) mod 7 = 6\n => K#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#J": 1, "E#O": 1, "L#L": 1, "F#I": 2, "E#P": 2, "J#O": 3, "I#N": 3, "K#P": 3, "K#I": 4}, "id": "mod7_arith_d4_0203"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#L := 2 - L#I. L#J := 3. L#I := L#J / G#I. G#I := 3. E#O := 3 - J#K. J#K := L#J / G#I. K#L := F#L. L#L := G#I. H#P := L#J - L#L. K#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#L := 2 - L#I. L#J := 3. L#I := L#J / G#I. G#I := 3. E#O := 3 - J#K. J#K := L#J / G#I. K#L := F#L. L#L := G#I. H#P := L#J - L#L.", "query": "K#L", "answer": 1, "cot": "L#J = 3 -> L#J = 3\nG#I = 3 -> G#I = 3\nL#I = L#J / G#I\n = 3 / 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => L#I = 1\nF#L = 2 - L#I\n = 2 - 1\n 2 - 1 = (2 - 1) mod 7 = 1\n => F#L = 1\nK#L = F#L = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 5, "num_distractors": 0, "var_layer": {"L#J": 1, "G#I": 1, "L#I": 2, "J#K": 2, "L#L": 2, "F#L": 3, "E#O": 3, "H#P": 3, "K#L": 4}, "id": "mod7_arith_d4_0204"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#O := H#N + I#P. K#M := 4. F#J := 4. H#M := J#K - E#P + K#M. E#P := F#J - K#M. I#P := H#N * K#M. H#N := 2. G#K := G#P / H#N. J#K := G#P * H#N. G#P := H#N. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#O := H#N + I#P. K#M := 4. F#J := 4. H#M := J#K - E#P + K#M. E#P := F#J - K#M. I#P := H#N * K#M. H#N := 2. G#K := G#P / H#N. J#K := G#P * H#N. G#P := H#N.", "query": "H#M", "answer": 1, "cot": "K#M = 4 -> K#M = 4\nF#J = 4 -> F#J = 4\nH#N = 2 -> H#N = 2\nG#P = H#N = 2\nE#P = F#J - K#M\n = 4 - 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => E#P = 0\nJ#K = G#P * H#N\n = 2 * 2\n 2 * 2 = (2 × 2) mod 7 = 4\n => J#K = 4\nH#M = J#K - E#P + K#M\n = 4 - 0 + 4\n 4 - 0 = (4 - 0) mod 7 = 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => H#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"K#M": 1, "F#J": 1, "H#N": 1, "G#P": 2, "E#P": 2, "I#P": 2, "G#K": 3, "J#K": 3, "F#O": 3, "H#M": 4}, "id": "mod7_arith_d4_0205"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := G#P + E#N. H#N := H#I / G#P. J#L := L#J / 2. I#M := F#J. F#J := H#N / J#M - G#P. J#M := 6. K#I := H#I + H#N + L#J. H#I := 5. E#N := 3. G#P := 4. I#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := G#P + E#N. H#N := H#I / G#P. J#L := L#J / 2. I#M := F#J. F#J := H#N / J#M - G#P. J#M := 6. K#I := H#I + H#N + L#J. H#I := 5. E#N := 3. G#P := 4.", "query": "I#M", "answer": 0, "cot": "G#P = 4 -> G#P = 4\nH#I = 5 -> H#I = 5\nJ#M = 6 -> J#M = 6\nH#N = H#I / G#P\n = 5 / 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => H#N = 3\nF#J = H#N / J#M - G#P\n = 3 / 6 - 4\n 3 / 6 = (3 × 6⁻¹) mod 7 = 4\n 4 - 4 = (4 - 4) mod 7 = 0\n => F#J = 0\nI#M = F#J = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#P": 1, "E#N": 1, "H#I": 1, "J#M": 1, "H#N": 2, "L#J": 2, "K#I": 3, "J#L": 3, "F#J": 3, "I#M": 4}, "id": "mod7_arith_d4_0206"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#K := G#I. G#L := 6. E#M := G#L * H#J. E#N := G#L * E#M. E#P := L#P * H#J / 4. L#P := G#I. G#I := H#J / G#L. H#J := 5. E#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#K := G#I. G#L := 6. E#M := G#L * H#J. E#N := G#L * E#M. E#P := L#P * H#J / 4. L#P := G#I. G#I := H#J / G#L. H#J := 5.", "query": "E#P", "answer": 6, "cot": "G#L = 6 -> G#L = 6\nH#J = 5 -> H#J = 5\nG#I = H#J / G#L\n = 5 / 6\n 5 / 6 = (5 × 6⁻¹) mod 7 = 2\n => G#I = 2\nL#P = G#I = 2\nE#P = L#P * H#J / 4\n = 2 * 5 / 4\n 2 * 5 = (2 × 5) mod 7 = 3\n 3 / 4 = (3 × 4⁻¹) mod 7 = 6\n => E#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#L": 1, "H#J": 1, "E#M": 2, "G#I": 2, "L#P": 3, "F#K": 3, "E#N": 3, "E#P": 4}, "id": "mod7_arith_d4_0207"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := 6. H#L := 3. L#K := J#M. K#K := H#L - G#J. H#M := H#N / G#J / K#K. H#N := 5 + G#J. J#M := K#K / H#N. L#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := 6. H#L := 3. L#K := J#M. K#K := H#L - G#J. H#M := H#N / G#J / K#K. H#N := 5 + G#J. J#M := K#K / H#N.", "query": "L#K", "answer": 1, "cot": "G#J = 6 -> G#J = 6\nH#L = 3 -> H#L = 3\nK#K = H#L - G#J\n = 3 - 6\n 3 - 6 = (3 - 6) mod 7 = 4\n => K#K = 4\nH#N = 5 + G#J\n = 5 + 6\n 5 + 6 = (5 + 6) mod 7 = 4\n => H#N = 4\nJ#M = K#K / H#N\n = 4 / 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => J#M = 1\nL#K = J#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"G#J": 1, "H#L": 1, "K#K": 2, "H#N": 2, "J#M": 3, "H#M": 3, "L#K": 4}, "id": "mod7_arith_d4_0208"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := 3 / I#J - 5. E#P := 5. E#O := I#I / H#K. I#J := I#I * J#N. J#N := E#P. I#I := E#P - 4 + H#K. H#K := 3. L#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := 3 / I#J - 5. E#P := 5. E#O := I#I / H#K. I#J := I#I * J#N. J#N := E#P. I#I := E#P - 4 + H#K. H#K := 3.", "query": "L#I", "answer": 6, "cot": "H#K = 3 -> H#K = 3\nE#P = 5 -> E#P = 5\nJ#N = E#P = 5\nI#I = E#P - 4 + H#K\n = 5 - 4 + 3\n 5 - 4 = (5 - 4) mod 7 = 1\n 1 + 3 = (1 + 3) mod 7 = 4\n => I#I = 4\nI#J = I#I * J#N\n = 4 * 5\n 4 * 5 = (4 × 5) mod 7 = 6\n => I#J = 6\nL#I = 3 / I#J - 5\n = 3 / 6 - 5\n 3 / 6 = (3 × 6⁻¹) mod 7 = 4\n 4 - 5 = (4 - 5) mod 7 = 6\n => L#I = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"H#K": 1, "E#P": 1, "J#N": 2, "I#I": 2, "I#J": 3, "E#O": 3, "L#I": 4}, "id": "mod7_arith_d4_0209"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := 4 + I#N. E#P := L#N + K#O. G#N := E#O. L#N := 6. I#N := 4. K#O := E#O. E#O := 3 * I#N. E#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := 4 + I#N. E#P := L#N + K#O. G#N := E#O. L#N := 6. I#N := 4. K#O := E#O. E#O := 3 * I#N.", "query": "E#P", "answer": 4, "cot": "I#N = 4 -> I#N = 4\nL#N = 6 -> L#N = 6\nE#O = 3 * I#N\n = 3 * 4\n 3 * 4 = (3 × 4) mod 7 = 5\n => E#O = 5\nK#O = E#O = 5\nE#P = L#N + K#O\n = 6 + 5\n 6 + 5 = (6 + 5) mod 7 = 4\n => E#P = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#N": 1, "L#N": 1, "E#O": 2, "I#I": 2, "G#N": 3, "K#O": 3, "E#P": 4}, "id": "mod7_arith_d4_0210"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := K#P / E#O + I#O. K#L := E#O. F#J := K#P. I#O := 5. H#M := I#O * K#L. K#P := I#O * E#O. F#M := F#J / H#J / H#M. E#O := 6. F#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := K#P / E#O + I#O. K#L := E#O. F#J := K#P. I#O := 5. H#M := I#O * K#L. K#P := I#O * E#O. F#M := F#J / H#J / H#M. E#O := 6.", "query": "F#M", "answer": 5, "cot": "E#O = 6 -> E#O = 6\nI#O = 5 -> I#O = 5\nK#P = I#O * E#O\n = 5 * 6\n 5 * 6 = (5 × 6) mod 7 = 2\n => K#P = 2\nK#L = E#O = 6\nH#J = K#P / E#O + I#O\n = 2 / 6 + 5\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n 5 + 5 = (5 + 5) mod 7 = 3\n => H#J = 3\nF#J = K#P = 2\nH#M = I#O * K#L\n = 5 * 6\n 5 * 6 = (5 × 6) mod 7 = 2\n => H#M = 2\nF#M = F#J / H#J / H#M\n = 2 / 3 / 2\n 2 / 3 = (2 × 3⁻¹) mod 7 = 3\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => F#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 8, "num_distractors": 0, "var_layer": {"E#O": 1, "I#O": 1, "K#P": 2, "K#L": 2, "H#J": 3, "F#J": 3, "H#M": 3, "F#M": 4}, "id": "mod7_arith_d4_0211"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#O := F#M / L#O + 4. I#J := L#O. F#M := 2. J#K := 5. E#O := 6. I#N := E#O * I#O. L#O := F#M + K#I / E#O. L#J := L#O + I#I. I#I := 4 + K#I. K#I := 2. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#O := F#M / L#O + 4. I#J := L#O. F#M := 2. J#K := 5. E#O := 6. I#N := E#O * I#O. L#O := F#M + K#I / E#O. L#J := L#O + I#I. I#I := 4 + K#I. K#I := 2.", "query": "I#N", "answer": 0, "cot": "F#M = 2 -> F#M = 2\nE#O = 6 -> E#O = 6\nK#I = 2 -> K#I = 2\nL#O = F#M + K#I / E#O\n = 2 + 2 / 6\n 2 + 2 = (2 + 2) mod 7 = 4\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n => L#O = 3\nI#O = F#M / L#O + 4\n = 2 / 3 + 4\n 2 / 3 = (2 × 3⁻¹) mod 7 = 3\n 3 + 4 = (3 + 4) mod 7 = 0\n => I#O = 0\nI#N = E#O * I#O\n = 6 * 0\n 6 * 0 = (6 × 0) mod 7 = 0\n => I#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#M": 1, "E#O": 1, "J#K": 1, "K#I": 1, "I#I": 2, "L#O": 2, "I#J": 3, "L#J": 3, "I#O": 3, "I#N": 4}, "id": "mod7_arith_d4_0212"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#O := G#O / 4. K#M := H#K + L#K. G#O := K#I + K#M. H#K := 6. L#K := 3. I#K := H#K / L#K - 3. K#I := H#K * L#K. K#O := K#I + I#K. E#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#O := G#O / 4. K#M := H#K + L#K. G#O := K#I + K#M. H#K := 6. L#K := 3. I#K := H#K / L#K - 3. K#I := H#K * L#K. K#O := K#I + I#K.", "query": "E#O", "answer": 5, "cot": "H#K = 6 -> H#K = 6\nL#K = 3 -> L#K = 3\nK#M = H#K + L#K\n = 6 + 3\n 6 + 3 = (6 + 3) mod 7 = 2\n => K#M = 2\nK#I = H#K * L#K\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => K#I = 4\nG#O = K#I + K#M\n = 4 + 2\n 4 + 2 = (4 + 2) mod 7 = 6\n => G#O = 6\nE#O = G#O / 4\n = 6 / 4\n 6 / 4 = (6 × 4⁻¹) mod 7 = 5\n => E#O = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"H#K": 1, "L#K": 1, "K#M": 2, "I#K": 2, "K#I": 2, "G#O": 3, "K#O": 3, "E#O": 4}, "id": "mod7_arith_d4_0213"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#O := K#L. E#K := 6. E#O := G#O. L#I := K#L * E#K. J#K := I#L * 6. K#L := E#K + I#L. I#L := 1. I#P := 2. E#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#O := K#L. E#K := 6. E#O := G#O. L#I := K#L * E#K. J#K := I#L * 6. K#L := E#K + I#L. I#L := 1. I#P := 2.", "query": "E#O", "answer": 0, "cot": "I#L = 1 -> I#L = 1\nE#K = 6 -> E#K = 6\nK#L = E#K + I#L\n = 6 + 1\n 6 + 1 = (6 + 1) mod 7 = 0\n => K#L = 0\nG#O = K#L = 0\nE#O = G#O = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#L": 1, "E#K": 1, "I#P": 1, "K#L": 2, "J#K": 2, "L#I": 3, "G#O": 3, "E#O": 4}, "id": "mod7_arith_d4_0214"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#N := 4. E#O := 4. H#I := G#N * G#O. L#K := G#N / J#J. E#J := J#J + L#N + G#O. E#P := L#K. G#K := 6. J#J := 2. G#O := L#N + E#O. G#N := 4 + L#N. E#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#N := 4. E#O := 4. H#I := G#N * G#O. L#K := G#N / J#J. E#J := J#J + L#N + G#O. E#P := L#K. G#K := 6. J#J := 2. G#O := L#N + E#O. G#N := 4 + L#N.", "query": "E#P", "answer": 4, "cot": "J#J = 2 -> J#J = 2\nL#N = 4 -> L#N = 4\nG#N = 4 + L#N\n = 4 + 4\n 4 + 4 = (4 + 4) mod 7 = 1\n => G#N = 1\nL#K = G#N / J#J\n = 1 / 2\n 1 / 2 = (1 × 2⁻¹) mod 7 = 4\n => L#K = 4\nE#P = L#K = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#J": 1, "E#O": 1, "L#N": 1, "G#K": 1, "G#O": 2, "G#N": 2, "H#I": 3, "E#J": 3, "L#K": 3, "E#P": 4}, "id": "mod7_arith_d4_0215"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#M := H#L - J#N / G#J. H#L := 4. H#K := J#N. J#N := 5 + G#J * 6. L#L := G#M. F#N := G#M. G#J := 5. I#I := I#M. G#M := 2. L#O := L#L + J#N. I#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#M := H#L - J#N / G#J. H#L := 4. H#K := J#N. J#N := 5 + G#J * 6. L#L := G#M. F#N := G#M. G#J := 5. I#I := I#M. G#M := 2. L#O := L#L + J#N.", "query": "I#I", "answer": 0, "cot": "H#L = 4 -> H#L = 4\nG#J = 5 -> G#J = 5\nJ#N = 5 + G#J * 6\n = 5 + 5 * 6\n 5 + 5 = (5 + 5) mod 7 = 3\n 3 * 6 = (3 × 6) mod 7 = 4\n => J#N = 4\nI#M = H#L - J#N / G#J\n = 4 - 4 / 5\n 4 - 4 = (4 - 4) mod 7 = 0\n 0 / 5 = (0 × 5⁻¹) mod 7 = 0\n => I#M = 0\nI#I = I#M = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"G#M": 1, "H#L": 1, "G#J": 1, "F#N": 2, "J#N": 2, "L#L": 2, "L#O": 3, "I#M": 3, "H#K": 3, "I#I": 4}, "id": "mod7_arith_d4_0216"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#O := E#K * G#L. H#J := G#J + E#K. E#K := 3. K#K := 5. G#L := K#K. G#J := E#K. F#M := K#K. J#K := G#J / 5. G#O := K#O - H#J. G#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#O := E#K * G#L. H#J := G#J + E#K. E#K := 3. K#K := 5. G#L := K#K. G#J := E#K. F#M := K#K. J#K := G#J / 5. G#O := K#O - H#J.", "query": "G#O", "answer": 2, "cot": "E#K = 3 -> E#K = 3\nK#K = 5 -> K#K = 5\nG#J = E#K = 3\nG#L = K#K = 5\nH#J = G#J + E#K\n = 3 + 3\n 3 + 3 = (3 + 3) mod 7 = 6\n => H#J = 6\nK#O = E#K * G#L\n = 3 * 5\n 3 * 5 = (3 × 5) mod 7 = 1\n => K#O = 1\nG#O = K#O - H#J\n = 1 - 6\n 1 - 6 = (1 - 6) mod 7 = 2\n => G#O = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"E#K": 1, "K#K": 1, "F#M": 2, "G#J": 2, "G#L": 2, "H#J": 3, "K#O": 3, "J#K": 3, "G#O": 4}, "id": "mod7_arith_d4_0217"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#L := F#J. F#M := 4. J#L := 3. L#N := F#P. J#J := 6 - I#L. L#O := F#J / K#J. F#P := F#M - 5. K#J := 4. K#L := 2. F#J := 4 + 6 + K#J. J#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#L := F#J. F#M := 4. J#L := 3. L#N := F#P. J#J := 6 - I#L. L#O := F#J / K#J. F#P := F#M - 5. K#J := 4. K#L := 2. F#J := 4 + 6 + K#J.", "query": "J#J", "answer": 6, "cot": "K#J = 4 -> K#J = 4\nF#J = 4 + 6 + K#J\n = 4 + 6 + 4\n 4 + 6 = (4 + 6) mod 7 = 3\n 3 + 4 = (3 + 4) mod 7 = 0\n => F#J = 0\nI#L = F#J = 0\nJ#J = 6 - I#L\n = 6 - 0\n 6 - 0 = (6 - 0) mod 7 = 6\n => J#J = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 4, "num_distractors": 0, "var_layer": {"J#L": 1, "K#J": 1, "K#L": 1, "F#M": 1, "F#J": 2, "F#P": 2, "L#N": 3, "I#L": 3, "L#O": 3, "J#J": 4}, "id": "mod7_arith_d4_0218"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := J#I. K#K := L#P + J#I. H#M := K#K. K#J := J#I. J#I := K#L. E#L := 6. L#P := K#L / E#I - E#L. E#I := 4. G#L := 6. K#L := 2. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := J#I. K#K := L#P + J#I. H#M := K#K. K#J := J#I. J#I := K#L. E#L := 6. L#P := K#L / E#I - E#L. E#I := 4. G#L := 6. K#L := 2.", "query": "H#M", "answer": 0, "cot": "E#L = 6 -> E#L = 6\nK#L = 2 -> K#L = 2\nE#I = 4 -> E#I = 4\nJ#I = K#L = 2\nL#P = K#L / E#I - E#L\n = 2 / 4 - 6\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n 4 - 6 = (4 - 6) mod 7 = 5\n => L#P = 5\nK#K = L#P + J#I\n = 5 + 2\n 5 + 2 = (5 + 2) mod 7 = 0\n => K#K = 0\nH#M = K#K = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"E#L": 1, "K#L": 1, "G#L": 1, "E#I": 1, "J#I": 2, "L#P": 2, "K#J": 3, "K#M": 3, "K#K": 3, "H#M": 4}, "id": "mod7_arith_d4_0219"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := H#N. G#O := 5. J#K := G#I / G#N. H#M := 4. J#O := 4. G#I := G#O * G#N - H#N. G#N := J#O. H#N := J#O + 2 - H#M. J#K?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := H#N. G#O := 5. J#K := G#I / G#N. H#M := 4. J#O := 4. G#I := G#O * G#N - H#N. G#N := J#O. H#N := J#O + 2 - H#M.", "query": "J#K", "answer": 1, "cot": "J#O = 4 -> J#O = 4\nH#M = 4 -> H#M = 4\nG#O = 5 -> G#O = 5\nG#N = J#O = 4\nH#N = J#O + 2 - H#M\n = 4 + 2 - 4\n 4 + 2 = (4 + 2) mod 7 = 6\n 6 - 4 = (6 - 4) mod 7 = 2\n => H#N = 2\nG#I = G#O * G#N - H#N\n = 5 * 4 - 2\n 5 * 4 = (5 × 4) mod 7 = 6\n 6 - 2 = (6 - 2) mod 7 = 4\n => G#I = 4\nJ#K = G#I / G#N\n = 4 / 4\n 4 / 4 = (4 × 4⁻¹) mod 7 = 1\n => J#K = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 7, "num_distractors": 0, "var_layer": {"J#O": 1, "H#M": 1, "G#O": 1, "G#N": 2, "H#N": 2, "G#J": 3, "G#I": 3, "J#K": 4}, "id": "mod7_arith_d4_0220"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#J := J#M. J#I := 4. E#M := I#P * 5. F#J := J#I + F#O. I#P := 5 * J#I + F#O. L#O := J#I - E#N. E#N := 2. J#M := E#N * G#L * L#O. F#O := 4. G#L := 2. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#J := J#M. J#I := 4. E#M := I#P * 5. F#J := J#I + F#O. I#P := 5 * J#I + F#O. L#O := J#I - E#N. E#N := 2. J#M := E#N * G#L * L#O. F#O := 4. G#L := 2.", "query": "G#J", "answer": 1, "cot": "E#N = 2 -> E#N = 2\nG#L = 2 -> G#L = 2\nJ#I = 4 -> J#I = 4\nL#O = J#I - E#N\n = 4 - 2\n 4 - 2 = (4 - 2) mod 7 = 2\n => L#O = 2\nJ#M = E#N * G#L * L#O\n = 2 * 2 * 2\n 2 * 2 = (2 × 2) mod 7 = 4\n 4 * 2 = (4 × 2) mod 7 = 1\n => J#M = 1\nG#J = J#M = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#N": 1, "F#O": 1, "G#L": 1, "J#I": 1, "L#O": 2, "F#J": 2, "I#P": 2, "E#M": 3, "J#M": 3, "G#J": 4}, "id": "mod7_arith_d4_0221"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := 2. E#K := 6. I#I := H#I + G#I. J#M := G#I - I#I * H#I. H#M := 4. H#I := H#M. G#I := E#K / J#O. K#N := E#K. E#I := H#I / H#M. J#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := 2. E#K := 6. I#I := H#I + G#I. J#M := G#I - I#I * H#I. H#M := 4. H#I := H#M. G#I := E#K / J#O. K#N := E#K. E#I := H#I / H#M.", "query": "J#M", "answer": 5, "cot": "H#M = 4 -> H#M = 4\nE#K = 6 -> E#K = 6\nJ#O = 2 -> J#O = 2\nG#I = E#K / J#O\n = 6 / 2\n 6 / 2 = (6 × 2⁻¹) mod 7 = 3\n => G#I = 3\nH#I = H#M = 4\nI#I = H#I + G#I\n = 4 + 3\n 4 + 3 = (4 + 3) mod 7 = 0\n => I#I = 0\nJ#M = G#I - I#I * H#I\n = 3 - 0 * 4\n 3 - 0 = (3 - 0) mod 7 = 3\n 3 * 4 = (3 × 4) mod 7 = 5\n => J#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 7, "num_distractors": 0, "var_layer": {"H#M": 1, "E#K": 1, "J#O": 1, "G#I": 2, "H#I": 2, "K#N": 2, "E#I": 3, "I#I": 3, "J#M": 4}, "id": "mod7_arith_d4_0222"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#O := 3 / J#K * H#P. H#P := 2. E#I := H#P. J#L := J#K / E#I / 5. L#J := 4. E#J := E#I + H#P. F#P := J#O * K#I. K#I := L#J - H#P / 6. J#K := H#P / L#J. F#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#O := 3 / J#K * H#P. H#P := 2. E#I := H#P. J#L := J#K / E#I / 5. L#J := 4. E#J := E#I + H#P. F#P := J#O * K#I. K#I := L#J - H#P / 6. J#K := H#P / L#J.", "query": "F#P", "answer": 4, "cot": "L#J = 4 -> L#J = 4\nH#P = 2 -> H#P = 2\nK#I = L#J - H#P / 6\n = 4 - 2 / 6\n 4 - 2 = (4 - 2) mod 7 = 2\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n => K#I = 5\nJ#K = H#P / L#J\n = 2 / 4\n 2 / 4 = (2 × 4⁻¹) mod 7 = 4\n => J#K = 4\nJ#O = 3 / J#K * H#P\n = 3 / 4 * 2\n 3 / 4 = (3 × 4⁻¹) mod 7 = 6\n 6 * 2 = (6 × 2) mod 7 = 5\n => J#O = 5\nF#P = J#O * K#I\n = 5 * 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => F#P = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"L#J": 1, "H#P": 1, "K#I": 2, "E#I": 2, "J#K": 2, "E#J": 3, "J#O": 3, "J#L": 3, "F#P": 4}, "id": "mod7_arith_d4_0223"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#O := E#I - L#L. I#P := K#O - L#P. G#L := L#J / L#P. H#N := 6. E#I := L#J / L#P. L#J := L#P / K#O + H#N. L#P := 4. L#L := I#P. K#O := 3. I#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#O := E#I - L#L. I#P := K#O - L#P. G#L := L#J / L#P. H#N := 6. E#I := L#J / L#P. L#J := L#P / K#O + H#N. L#P := 4. L#L := I#P. K#O := 3.", "query": "I#O", "answer": 4, "cot": "K#O = 3 -> K#O = 3\nL#P = 4 -> L#P = 4\nH#N = 6 -> H#N = 6\nI#P = K#O - L#P\n = 3 - 4\n 3 - 4 = (3 - 4) mod 7 = 6\n => I#P = 6\nL#J = L#P / K#O + H#N\n = 4 / 3 + 6\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n 6 + 6 = (6 + 6) mod 7 = 5\n => L#J = 5\nL#L = I#P = 6\nE#I = L#J / L#P\n = 5 / 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => E#I = 3\nI#O = E#I - L#L\n = 3 - 6\n 3 - 6 = (3 - 6) mod 7 = 4\n => I#O = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 8, "num_distractors": 0, "var_layer": {"K#O": 1, "L#P": 1, "H#N": 1, "I#P": 2, "L#J": 2, "L#L": 3, "G#L": 3, "E#I": 3, "I#O": 4}, "id": "mod7_arith_d4_0224"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#L := J#I / K#N. F#P := E#M. I#M := 4 / F#P. K#J := E#M. K#N := 3 * K#I + K#J. J#I := K#J. K#I := 2. E#M := 5. E#K := E#M * K#I. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#L := J#I / K#N. F#P := E#M. I#M := 4 / F#P. K#J := E#M. K#N := 3 * K#I + K#J. J#I := K#J. K#I := 2. E#M := 5. E#K := E#M * K#I.", "query": "L#L", "answer": 3, "cot": "E#M = 5 -> E#M = 5\nK#I = 2 -> K#I = 2\nK#J = E#M = 5\nK#N = 3 * K#I + K#J\n = 3 * 2 + 5\n 3 * 2 = (3 × 2) mod 7 = 6\n 6 + 5 = (6 + 5) mod 7 = 4\n => K#N = 4\nJ#I = K#J = 5\nL#L = J#I / K#N\n = 5 / 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => L#L = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#M": 1, "K#I": 1, "K#J": 2, "E#K": 2, "F#P": 2, "K#N": 3, "J#I": 3, "I#M": 3, "L#L": 4}, "id": "mod7_arith_d4_0225"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#P := 2 * I#K. F#P := 1. G#L := 4. K#O := G#L. G#M := K#O + J#J. G#O := J#J - K#O. I#K := J#O + J#J. J#J := 2 / 5 * J#O. J#O := 2. L#I := 3. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#P := 2 * I#K. F#P := 1. G#L := 4. K#O := G#L. G#M := K#O + J#J. G#O := J#J - K#O. I#K := J#O + J#J. J#J := 2 / 5 * J#O. J#O := 2. L#I := 3.", "query": "J#P", "answer": 0, "cot": "J#O = 2 -> J#O = 2\nJ#J = 2 / 5 * J#O\n = 2 / 5 * 2\n 2 / 5 = (2 × 5⁻¹) mod 7 = 6\n 6 * 2 = (6 × 2) mod 7 = 5\n => J#J = 5\nI#K = J#O + J#J\n = 2 + 5\n 2 + 5 = (2 + 5) mod 7 = 0\n => I#K = 0\nJ#P = 2 * I#K\n = 2 * 0\n 2 * 0 = (2 × 0) mod 7 = 0\n => J#P = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 4, "num_distractors": 0, "var_layer": {"G#L": 1, "J#O": 1, "F#P": 1, "L#I": 1, "K#O": 2, "J#J": 2, "G#M": 3, "I#K": 3, "G#O": 3, "J#P": 4}, "id": "mod7_arith_d4_0226"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := F#I / F#L. F#L := F#I / 5. H#M := I#O. F#I := 2. I#O := F#J. J#K := 3. I#K := 5. F#J := J#K. E#M := 5 / J#K - F#I. H#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := F#I / F#L. F#L := F#I / 5. H#M := I#O. F#I := 2. I#O := F#J. J#K := 3. I#K := 5. F#J := J#K. E#M := 5 / J#K - F#I.", "query": "H#M", "answer": 3, "cot": "J#K = 3 -> J#K = 3\nF#J = J#K = 3\nI#O = F#J = 3\nH#M = I#O = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 4, "num_distractors": 0, "var_layer": {"F#I": 1, "J#K": 1, "I#K": 1, "F#J": 2, "F#L": 2, "E#M": 2, "F#N": 3, "I#O": 3, "H#M": 4}, "id": "mod7_arith_d4_0227"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nE#N := F#O / L#I. L#I := I#K. E#J := 4. F#O := E#J. G#J := I#N. I#N := I#K * F#O * E#J. I#K := 1. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "E#N := F#O / L#I. L#I := I#K. E#J := 4. F#O := E#J. G#J := I#N. I#N := I#K * F#O * E#J. I#K := 1.", "query": "G#J", "answer": 2, "cot": "I#K = 1 -> I#K = 1\nE#J = 4 -> E#J = 4\nF#O = E#J = 4\nI#N = I#K * F#O * E#J\n = 1 * 4 * 4\n 1 * 4 = (1 × 4) mod 7 = 4\n 4 * 4 = (4 × 4) mod 7 = 2\n => I#N = 2\nG#J = I#N = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#K": 1, "E#J": 1, "L#I": 2, "F#O": 2, "I#N": 3, "E#N": 3, "G#J": 4}, "id": "mod7_arith_d4_0228"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := 5 * G#P. E#L := 1. F#M := 5. J#K := K#L + G#P * 6. K#L := 4 - F#M. L#P := J#K. G#L := K#L. G#P := F#M * E#L / 5. G#O := F#M. L#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := 5 * G#P. E#L := 1. F#M := 5. J#K := K#L + G#P * 6. K#L := 4 - F#M. L#P := J#K. G#L := K#L. G#P := F#M * E#L / 5. G#O := F#M.", "query": "L#P", "answer": 0, "cot": "F#M = 5 -> F#M = 5\nE#L = 1 -> E#L = 1\nG#P = F#M * E#L / 5\n = 5 * 1 / 5\n 5 * 1 = (5 × 1) mod 7 = 5\n 5 / 5 = (5 × 5⁻¹) mod 7 = 1\n => G#P = 1\nK#L = 4 - F#M\n = 4 - 5\n 4 - 5 = (4 - 5) mod 7 = 6\n => K#L = 6\nJ#K = K#L + G#P * 6\n = 6 + 1 * 6\n 6 + 1 = (6 + 1) mod 7 = 0\n 0 * 6 = (0 × 6) mod 7 = 0\n => J#K = 0\nL#P = J#K = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"F#M": 1, "E#L": 1, "G#P": 2, "K#L": 2, "G#O": 2, "J#K": 3, "G#L": 3, "F#P": 3, "L#P": 4}, "id": "mod7_arith_d4_0229"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := F#M. G#P := G#J * 2. I#J := 6. F#M := 5 * I#J. H#N := 3. F#K := 3 - I#N. G#J := H#N. I#N := I#J - H#N. L#L := G#P / G#J. L#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := F#M. G#P := G#J * 2. I#J := 6. F#M := 5 * I#J. H#N := 3. F#K := 3 - I#N. G#J := H#N. I#N := I#J - H#N. L#L := G#P / G#J.", "query": "L#L", "answer": 2, "cot": "H#N = 3 -> H#N = 3\nG#J = H#N = 3\nG#P = G#J * 2\n = 3 * 2\n 3 * 2 = (3 × 2) mod 7 = 6\n => G#P = 6\nL#L = G#P / G#J\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => L#L = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 4, "num_distractors": 0, "var_layer": {"H#N": 1, "I#J": 1, "I#N": 2, "G#J": 2, "F#M": 2, "F#K": 3, "L#I": 3, "G#P": 3, "L#L": 4}, "id": "mod7_arith_d4_0230"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#P := 5. F#K := 4. G#P := 3 - J#P. F#P := E#P * 3. J#P := 2 - F#K. F#O := F#N * F#K. E#P := F#N * I#K + I#O. H#N := 4. I#K := 2. F#N := H#N + F#K + 4. I#O := 3 * I#K. F#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#P := 5. F#K := 4. G#P := 3 - J#P. F#P := E#P * 3. J#P := 2 - F#K. F#O := F#N * F#K. E#P := F#N * I#K + I#O. H#N := 4. I#K := 2. F#N := H#N + F#K + 4. I#O := 3 * I#K.", "query": "F#P", "answer": 6, "cot": "I#K = 2 -> I#K = 2\nF#K = 4 -> F#K = 4\nH#N = 4 -> H#N = 4\nI#O = 3 * I#K\n = 3 * 2\n 3 * 2 = (3 × 2) mod 7 = 6\n => I#O = 6\nF#N = H#N + F#K + 4\n = 4 + 4 + 4\n 4 + 4 = (4 + 4) mod 7 = 1\n 1 + 4 = (1 + 4) mod 7 = 5\n => F#N = 5\nE#P = F#N * I#K + I#O\n = 5 * 2 + 6\n 5 * 2 = (5 × 2) mod 7 = 3\n 3 + 6 = (3 + 6) mod 7 = 2\n => E#P = 2\nF#P = E#P * 3\n = 2 * 3\n 2 * 3 = (2 × 3) mod 7 = 6\n => F#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 7, "num_distractors": 0, "var_layer": {"H#P": 1, "I#K": 1, "F#K": 1, "H#N": 1, "I#O": 2, "F#N": 2, "J#P": 2, "F#O": 3, "E#P": 3, "G#P": 3, "F#P": 4}, "id": "mod7_arith_d4_0231"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#N := I#L - G#O. H#O := K#K. L#I := G#K / I#L. I#L := 3. E#K := I#L + G#O. G#K := G#O / I#L. G#O := 4. K#K := G#K / I#L. H#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#N := I#L - G#O. H#O := K#K. L#I := G#K / I#L. I#L := 3. E#K := I#L + G#O. G#K := G#O / I#L. G#O := 4. K#K := G#K / I#L.", "query": "H#O", "answer": 2, "cot": "I#L = 3 -> I#L = 3\nG#O = 4 -> G#O = 4\nG#K = G#O / I#L\n = 4 / 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => G#K = 6\nK#K = G#K / I#L\n = 6 / 3\n 6 / 3 = (6 × 3⁻¹) mod 7 = 2\n => K#K = 2\nH#O = K#K = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#L": 1, "G#O": 1, "G#K": 2, "E#K": 2, "G#N": 2, "K#K": 3, "L#I": 3, "H#O": 4}, "id": "mod7_arith_d4_0232"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nK#M := 3 * G#I. E#J := K#M + 6 / L#J. L#J := E#O + K#L. E#O := E#I - K#L. E#I := 3. L#K := E#O - 4. H#J := L#O * K#L. L#O := 2. G#I := E#I. K#L := 2. E#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "K#M := 3 * G#I. E#J := K#M + 6 / L#J. L#J := E#O + K#L. E#O := E#I - K#L. E#I := 3. L#K := E#O - 4. H#J := L#O * K#L. L#O := 2. G#I := E#I. K#L := 2.", "query": "E#J", "answer": 5, "cot": "E#I = 3 -> E#I = 3\nK#L = 2 -> K#L = 2\nG#I = E#I = 3\nE#O = E#I - K#L\n = 3 - 2\n 3 - 2 = (3 - 2) mod 7 = 1\n => E#O = 1\nK#M = 3 * G#I\n = 3 * 3\n 3 * 3 = (3 × 3) mod 7 = 2\n => K#M = 2\nL#J = E#O + K#L\n = 1 + 2\n 1 + 2 = (1 + 2) mod 7 = 3\n => L#J = 3\nE#J = K#M + 6 / L#J\n = 2 + 6 / 3\n 2 + 6 = (2 + 6) mod 7 = 1\n 1 / 3 = (1 × 3⁻¹) mod 7 = 5\n => E#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"E#I": 1, "L#O": 1, "K#L": 1, "G#I": 2, "E#O": 2, "H#J": 2, "K#M": 3, "L#J": 3, "L#K": 3, "E#J": 4}, "id": "mod7_arith_d4_0233"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nG#P := 6. F#P := 4. G#I := F#P. F#I := G#I + L#O. J#L := F#P / G#I. H#O := 5. E#M := G#P + F#P / 6. L#O := G#I - G#P. E#P := F#P * E#M. L#M := 5. F#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "G#P := 6. F#P := 4. G#I := F#P. F#I := G#I + L#O. J#L := F#P / G#I. H#O := 5. E#M := G#P + F#P / 6. L#O := G#I - G#P. E#P := F#P * E#M. L#M := 5.", "query": "F#I", "answer": 2, "cot": "F#P = 4 -> F#P = 4\nG#P = 6 -> G#P = 6\nG#I = F#P = 4\nL#O = G#I - G#P\n = 4 - 6\n 4 - 6 = (4 - 6) mod 7 = 5\n => L#O = 5\nF#I = G#I + L#O\n = 4 + 5\n 4 + 5 = (4 + 5) mod 7 = 2\n => F#I = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"F#P": 1, "H#O": 1, "G#P": 1, "L#M": 1, "G#I": 2, "E#M": 2, "J#L": 3, "L#O": 3, "E#P": 3, "F#I": 4}, "id": "mod7_arith_d4_0234"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#N := 6 / J#M. J#J := H#I * G#L + H#J. H#I := 2. J#I := 1. I#J := 3 * E#K. J#M := H#I / K#N + J#J. G#L := 2. K#N := G#L. H#L := K#N. E#K := H#I * J#I. H#J := 1. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#N := 6 / J#M. J#J := H#I * G#L + H#J. H#I := 2. J#I := 1. I#J := 3 * E#K. J#M := H#I / K#N + J#J. G#L := 2. K#N := G#L. H#L := K#N. E#K := H#I * J#I. H#J := 1.", "query": "I#N", "answer": 1, "cot": "G#L = 2 -> G#L = 2\nH#I = 2 -> H#I = 2\nH#J = 1 -> H#J = 1\nJ#J = H#I * G#L + H#J\n = 2 * 2 + 1\n 2 * 2 = (2 × 2) mod 7 = 4\n 4 + 1 = (4 + 1) mod 7 = 5\n => J#J = 5\nK#N = G#L = 2\nJ#M = H#I / K#N + J#J\n = 2 / 2 + 5\n 2 / 2 = (2 × 2⁻¹) mod 7 = 1\n 1 + 5 = (1 + 5) mod 7 = 6\n => J#M = 6\nI#N = 6 / J#M\n = 6 / 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => I#N = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 7, "num_distractors": 0, "var_layer": {"G#L": 1, "H#I": 1, "H#J": 1, "J#I": 1, "J#J": 2, "E#K": 2, "K#N": 2, "H#L": 3, "J#M": 3, "I#J": 3, "I#N": 4}, "id": "mod7_arith_d4_0235"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#J := K#I * H#J. L#K := 2 / 4 - G#N. G#N := 4. K#I := H#N / J#M. H#K := F#O * J#M + G#N. H#N := 1. J#M := 3. H#J := H#N + H#K. F#O := 1. K#P := H#N - L#K. I#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#J := K#I * H#J. L#K := 2 / 4 - G#N. G#N := 4. K#I := H#N / J#M. H#K := F#O * J#M + G#N. H#N := 1. J#M := 3. H#J := H#N + H#K. F#O := 1. K#P := H#N - L#K.", "query": "I#J", "answer": 5, "cot": "G#N = 4 -> G#N = 4\nF#O = 1 -> F#O = 1\nJ#M = 3 -> J#M = 3\nH#N = 1 -> H#N = 1\nK#I = H#N / J#M\n = 1 / 3\n 1 / 3 = (1 × 3⁻¹) mod 7 = 5\n => K#I = 5\nH#K = F#O * J#M + G#N\n = 1 * 3 + 4\n 1 * 3 = (1 × 3) mod 7 = 3\n 3 + 4 = (3 + 4) mod 7 = 0\n => H#K = 0\nH#J = H#N + H#K\n = 1 + 0\n 1 + 0 = (1 + 0) mod 7 = 1\n => H#J = 1\nI#J = K#I * H#J\n = 5 * 1\n 5 * 1 = (5 × 1) mod 7 = 5\n => I#J = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 8, "num_distractors": 0, "var_layer": {"G#N": 1, "F#O": 1, "J#M": 1, "H#N": 1, "K#I": 2, "H#K": 2, "L#K": 2, "H#J": 3, "K#P": 3, "I#J": 4}, "id": "mod7_arith_d4_0236"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#J := 5 / K#P. H#P := 5. I#N := F#M. F#M := H#P / K#P. L#M := F#M * E#P - 5. K#P := 4. E#P := F#M - L#J * K#P. L#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#J := 5 / K#P. H#P := 5. I#N := F#M. F#M := H#P / K#P. L#M := F#M * E#P - 5. K#P := 4. E#P := F#M - L#J * K#P.", "query": "L#M", "answer": 2, "cot": "K#P = 4 -> K#P = 4\nH#P = 5 -> H#P = 5\nF#M = H#P / K#P\n = 5 / 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => F#M = 3\nL#J = 5 / K#P\n = 5 / 4\n 5 / 4 = (5 × 4⁻¹) mod 7 = 3\n => L#J = 3\nE#P = F#M - L#J * K#P\n = 3 - 3 * 4\n 3 - 3 = (3 - 3) mod 7 = 0\n 0 * 4 = (0 × 4) mod 7 = 0\n => E#P = 0\nL#M = F#M * E#P - 5\n = 3 * 0 - 5\n 3 * 0 = (3 × 0) mod 7 = 0\n 0 - 5 = (0 - 5) mod 7 = 2\n => L#M = 2", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 7, "num_necessary": 6, "num_distractors": 0, "var_layer": {"K#P": 1, "H#P": 1, "F#M": 2, "L#J": 2, "I#N": 3, "E#P": 3, "L#M": 4}, "id": "mod7_arith_d4_0237"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#P := F#O. I#P := 5 + J#K / H#L. I#M := H#M. F#O := H#I + H#L. J#L := I#M / G#N + H#L. H#M := H#L * H#I. H#L := 5. H#I := 5. J#K := 5. G#N := H#M - I#P. J#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#P := F#O. I#P := 5 + J#K / H#L. I#M := H#M. F#O := H#I + H#L. J#L := I#M / G#N + H#L. H#M := H#L * H#I. H#L := 5. H#I := 5. J#K := 5. G#N := H#M - I#P.", "query": "J#L", "answer": 0, "cot": "J#K = 5 -> J#K = 5\nH#L = 5 -> H#L = 5\nH#I = 5 -> H#I = 5\nI#P = 5 + J#K / H#L\n = 5 + 5 / 5\n 5 + 5 = (5 + 5) mod 7 = 3\n 3 / 5 = (3 × 5⁻¹) mod 7 = 2\n => I#P = 2\nH#M = H#L * H#I\n = 5 * 5\n 5 * 5 = (5 × 5) mod 7 = 4\n => H#M = 4\nI#M = H#M = 4\nG#N = H#M - I#P\n = 4 - 2\n 4 - 2 = (4 - 2) mod 7 = 2\n => G#N = 2\nJ#L = I#M / G#N + H#L\n = 4 / 2 + 5\n 4 / 2 = (4 × 2⁻¹) mod 7 = 2\n 2 + 5 = (2 + 5) mod 7 = 0\n => J#L = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 8, "num_distractors": 0, "var_layer": {"J#K": 1, "H#L": 1, "H#I": 1, "I#P": 2, "F#O": 2, "H#M": 2, "F#P": 3, "I#M": 3, "G#N": 3, "J#L": 4}, "id": "mod7_arith_d4_0238"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#J := 4. J#N := 6. F#M := 4 + J#N / E#I. F#N := J#J. H#L := G#M. F#P := 2 + E#I. E#I := J#J * 4. G#J := 4. G#M := J#N * G#J. K#J := F#M. K#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#J := 4. J#N := 6. F#M := 4 + J#N / E#I. F#N := J#J. H#L := G#M. F#P := 2 + E#I. E#I := J#J * 4. G#J := 4. G#M := J#N * G#J. K#J := F#M.", "query": "K#J", "answer": 5, "cot": "J#N = 6 -> J#N = 6\nJ#J = 4 -> J#J = 4\nE#I = J#J * 4\n = 4 * 4\n 4 * 4 = (4 × 4) mod 7 = 2\n => E#I = 2\nF#M = 4 + J#N / E#I\n = 4 + 6 / 2\n 4 + 6 = (4 + 6) mod 7 = 3\n 3 / 2 = (3 × 2⁻¹) mod 7 = 5\n => F#M = 5\nK#J = F#M = 5", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 5, "num_distractors": 0, "var_layer": {"J#N": 1, "J#J": 1, "G#J": 1, "E#I": 2, "G#M": 2, "F#N": 2, "H#L": 3, "F#P": 3, "F#M": 3, "K#J": 4}, "id": "mod7_arith_d4_0239"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#I := I#L + K#O - L#L. L#L := 6. K#L := 3. K#O := K#L - L#L. H#P := L#L * H#I. G#I := K#L / H#P. H#I := 3 / K#L. I#L := 4. G#I?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#I := I#L + K#O - L#L. L#L := 6. K#L := 3. K#O := K#L - L#L. H#P := L#L * H#I. G#I := K#L / H#P. H#I := 3 / K#L. I#L := 4.", "query": "G#I", "answer": 4, "cot": "L#L = 6 -> L#L = 6\nK#L = 3 -> K#L = 3\nH#I = 3 / K#L\n = 3 / 3\n 3 / 3 = (3 × 3⁻¹) mod 7 = 1\n => H#I = 1\nH#P = L#L * H#I\n = 6 * 1\n 6 * 1 = (6 × 1) mod 7 = 6\n => H#P = 6\nG#I = K#L / H#P\n = 3 / 6\n 3 / 6 = (3 × 6⁻¹) mod 7 = 4\n => G#I = 4", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#L": 1, "L#L": 1, "K#L": 1, "H#I": 2, "K#O": 2, "H#P": 3, "J#I": 3, "G#I": 4}, "id": "mod7_arith_d4_0240"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#J := 6. F#K := 2. E#M := 3. J#O := F#K / F#J + E#M. I#J := J#O / E#M. J#L := E#M. F#M := J#O / E#M. L#J := H#K + J#L. H#K := J#O + F#J. L#N := E#M * F#K. L#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#J := 6. F#K := 2. E#M := 3. J#O := F#K / F#J + E#M. I#J := J#O / E#M. J#L := E#M. F#M := J#O / E#M. L#J := H#K + J#L. H#K := J#O + F#J. L#N := E#M * F#K.", "query": "L#J", "answer": 3, "cot": "E#M = 3 -> E#M = 3\nF#J = 6 -> F#J = 6\nF#K = 2 -> F#K = 2\nJ#O = F#K / F#J + E#M\n = 2 / 6 + 3\n 2 / 6 = (2 × 6⁻¹) mod 7 = 5\n 5 + 3 = (5 + 3) mod 7 = 1\n => J#O = 1\nJ#L = E#M = 3\nH#K = J#O + F#J\n = 1 + 6\n 1 + 6 = (1 + 6) mod 7 = 0\n => H#K = 0\nL#J = H#K + J#L\n = 0 + 3\n 0 + 3 = (0 + 3) mod 7 = 3\n => L#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"E#M": 1, "F#J": 1, "F#K": 1, "J#O": 2, "L#N": 2, "J#L": 2, "I#J": 3, "F#M": 3, "H#K": 3, "L#J": 4}, "id": "mod7_arith_d4_0241"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nJ#I := 3 - E#K. E#L := J#I. H#M := 2. E#K := 4. H#O := I#M. K#O := 6. L#J := I#M - K#O. I#M := 2 + H#P. F#L := E#L / H#O. H#P := 4. F#L?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "J#I := 3 - E#K. E#L := J#I. H#M := 2. E#K := 4. H#O := I#M. K#O := 6. L#J := I#M - K#O. I#M := 2 + H#P. F#L := E#L / H#O. H#P := 4.", "query": "F#L", "answer": 1, "cot": "H#P = 4 -> H#P = 4\nE#K = 4 -> E#K = 4\nI#M = 2 + H#P\n = 2 + 4\n 2 + 4 = (2 + 4) mod 7 = 6\n => I#M = 6\nJ#I = 3 - E#K\n = 3 - 4\n 3 - 4 = (3 - 4) mod 7 = 6\n => J#I = 6\nH#O = I#M = 6\nE#L = J#I = 6\nF#L = E#L / H#O\n = 6 / 6\n 6 / 6 = (6 × 6⁻¹) mod 7 = 1\n => F#L = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 7, "num_distractors": 0, "var_layer": {"H#P": 1, "H#M": 1, "E#K": 1, "K#O": 1, "I#M": 2, "J#I": 2, "H#O": 3, "L#J": 3, "E#L": 3, "F#L": 4}, "id": "mod7_arith_d4_0242"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#J := 1. F#M := 5. E#M := 4. I#M := E#M. I#L := E#M + F#M / H#J. H#N := F#M * E#M. H#L := F#M / H#N * 3. K#M := H#L / J#L. J#L := H#N - F#M. K#M?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#J := 1. F#M := 5. E#M := 4. I#M := E#M. I#L := E#M + F#M / H#J. H#N := F#M * E#M. H#L := F#M / H#N * 3. K#M := H#L / J#L. J#L := H#N - F#M.", "query": "K#M", "answer": 6, "cot": "E#M = 4 -> E#M = 4\nF#M = 5 -> F#M = 5\nH#N = F#M * E#M\n = 5 * 4\n 5 * 4 = (5 × 4) mod 7 = 6\n => H#N = 6\nJ#L = H#N - F#M\n = 6 - 5\n 6 - 5 = (6 - 5) mod 7 = 1\n => J#L = 1\nH#L = F#M / H#N * 3\n = 5 / 6 * 3\n 5 / 6 = (5 × 6⁻¹) mod 7 = 2\n 2 * 3 = (2 × 3) mod 7 = 6\n => H#L = 6\nK#M = H#L / J#L\n = 6 / 1\n 6 / 1 = (6 × 1⁻¹) mod 7 = 6\n => K#M = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 9, "num_necessary": 6, "num_distractors": 0, "var_layer": {"E#M": 1, "F#M": 1, "H#J": 1, "H#N": 2, "I#M": 2, "I#L": 2, "J#L": 3, "H#L": 3, "K#M": 4}, "id": "mod7_arith_d4_0243"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nL#I := 1. G#O := J#N + L#I. K#L := 5 + L#I. J#P := E#L / L#I - 2. J#N := 5. J#I := K#J / J#N. E#L := 4 / K#L. K#J := L#I * J#N. J#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "L#I := 1. G#O := J#N + L#I. K#L := 5 + L#I. J#P := E#L / L#I - 2. J#N := 5. J#I := K#J / J#N. E#L := 4 / K#L. K#J := L#I * J#N.", "query": "J#P", "answer": 1, "cot": "L#I = 1 -> L#I = 1\nK#L = 5 + L#I\n = 5 + 1\n 5 + 1 = (5 + 1) mod 7 = 6\n => K#L = 6\nE#L = 4 / K#L\n = 4 / 6\n 4 / 6 = (4 × 6⁻¹) mod 7 = 3\n => E#L = 3\nJ#P = E#L / L#I - 2\n = 3 / 1 - 2\n 3 / 1 = (3 × 1⁻¹) mod 7 = 3\n 3 - 2 = (3 - 2) mod 7 = 1\n => J#P = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 4, "num_distractors": 0, "var_layer": {"L#I": 1, "J#N": 1, "K#L": 2, "G#O": 2, "K#J": 2, "E#L": 3, "J#I": 3, "J#P": 4}, "id": "mod7_arith_d4_0244"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#K := 1. G#J := I#K / G#P - J#M. G#P := 4. H#M := G#P * J#M. J#J := H#M. H#N := J#J. K#M := G#J * G#P. J#M := 6. H#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#K := 1. G#J := I#K / G#P - J#M. G#P := 4. H#M := G#P * J#M. J#J := H#M. H#N := J#J. K#M := G#J * G#P. J#M := 6.", "query": "H#N", "answer": 3, "cot": "G#P = 4 -> G#P = 4\nJ#M = 6 -> J#M = 6\nH#M = G#P * J#M\n = 4 * 6\n 4 * 6 = (4 × 6) mod 7 = 3\n => H#M = 3\nJ#J = H#M = 3\nH#N = J#J = 3", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"I#K": 1, "G#P": 1, "J#M": 1, "H#M": 2, "G#J": 2, "K#M": 3, "J#J": 3, "H#N": 4}, "id": "mod7_arith_d4_0245"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nI#I := H#J - H#M + H#K. H#M := 3. J#M := 3. H#K := 2 * H#M. K#K := H#K * H#M. H#J := J#M + L#N. K#P := K#K / J#M. L#N := 3. K#P?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "I#I := H#J - H#M + H#K. H#M := 3. J#M := 3. H#K := 2 * H#M. K#K := H#K * H#M. H#J := J#M + L#N. K#P := K#K / J#M. L#N := 3.", "query": "K#P", "answer": 6, "cot": "H#M = 3 -> H#M = 3\nJ#M = 3 -> J#M = 3\nH#K = 2 * H#M\n = 2 * 3\n 2 * 3 = (2 × 3) mod 7 = 6\n => H#K = 6\nK#K = H#K * H#M\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => K#K = 4\nK#P = K#K / J#M\n = 4 / 3\n 4 / 3 = (4 × 3⁻¹) mod 7 = 6\n => K#P = 6", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 5, "num_distractors": 0, "var_layer": {"H#M": 1, "L#N": 1, "J#M": 1, "H#J": 2, "H#K": 2, "K#K": 3, "I#I": 3, "K#P": 4}, "id": "mod7_arith_d4_0246"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nF#N := 1. F#I := J#K / E#I. L#K := E#I * F#N. H#J := J#P - 4. E#M := 3. I#N := L#K * F#I * L#O. J#K := I#P - E#M. J#P := F#N. E#I := 2. I#P := 2. L#O := L#K - E#I. I#N?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "F#N := 1. F#I := J#K / E#I. L#K := E#I * F#N. H#J := J#P - 4. E#M := 3. I#N := L#K * F#I * L#O. J#K := I#P - E#M. J#P := F#N. E#I := 2. I#P := 2. L#O := L#K - E#I.", "query": "I#N", "answer": 0, "cot": "I#P = 2 -> I#P = 2\nE#I = 2 -> E#I = 2\nF#N = 1 -> F#N = 1\nE#M = 3 -> E#M = 3\nL#K = E#I * F#N\n = 2 * 1\n 2 * 1 = (2 × 1) mod 7 = 2\n => L#K = 2\nJ#K = I#P - E#M\n = 2 - 3\n 2 - 3 = (2 - 3) mod 7 = 6\n => J#K = 6\nL#O = L#K - E#I\n = 2 - 2\n 2 - 2 = (2 - 2) mod 7 = 0\n => L#O = 0\nF#I = J#K / E#I\n = 6 / 2\n 6 / 2 = (6 × 2⁻¹) mod 7 = 3\n => F#I = 3\nI#N = L#K * F#I * L#O\n = 2 * 3 * 0\n 2 * 3 = (2 × 3) mod 7 = 6\n 6 * 0 = (6 × 0) mod 7 = 0\n => I#N = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 11, "num_necessary": 9, "num_distractors": 0, "var_layer": {"I#P": 1, "E#I": 1, "F#N": 1, "E#M": 1, "J#P": 2, "L#K": 2, "J#K": 2, "L#O": 3, "F#I": 3, "H#J": 3, "I#N": 4}, "id": "mod7_arith_d4_0247"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#N := 1. I#J := H#K - H#P. G#J := F#I + I#I. H#K := 3. H#P := 3. G#I := H#N - H#K. E#L := 6. F#I := H#P + L#M. L#M := E#L * H#K. I#I := I#J * E#L. G#J?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#N := 1. I#J := H#K - H#P. G#J := F#I + I#I. H#K := 3. H#P := 3. G#I := H#N - H#K. E#L := 6. F#I := H#P + L#M. L#M := E#L * H#K. I#I := I#J * E#L.", "query": "G#J", "answer": 0, "cot": "H#K = 3 -> H#K = 3\nH#P = 3 -> H#P = 3\nE#L = 6 -> E#L = 6\nL#M = E#L * H#K\n = 6 * 3\n 6 * 3 = (6 × 3) mod 7 = 4\n => L#M = 4\nI#J = H#K - H#P\n = 3 - 3\n 3 - 3 = (3 - 3) mod 7 = 0\n => I#J = 0\nF#I = H#P + L#M\n = 3 + 4\n 3 + 4 = (3 + 4) mod 7 = 0\n => F#I = 0\nI#I = I#J * E#L\n = 0 * 6\n 0 * 6 = (0 × 6) mod 7 = 0\n => I#I = 0\nG#J = F#I + I#I\n = 0 + 0\n 0 + 0 = (0 + 0) mod 7 = 0\n => G#J = 0", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 10, "num_necessary": 8, "num_distractors": 0, "var_layer": {"H#K": 1, "H#N": 1, "H#P": 1, "E#L": 1, "L#M": 2, "I#J": 2, "G#I": 2, "F#I": 3, "I#I": 3, "G#J": 4}, "id": "mod7_arith_d4_0248"} {"mod": 7, "ops_mode": "arith", "question": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].\n\nH#K := 3. H#J := G#M. L#O := G#J + H#J * 2. G#M := 3. G#J := H#K. L#L := H#K. K#I := L#L * 5. K#O := G#M + L#O. K#O?", "preamble": "Operator definitions (evaluate strictly left to right, all results mod 7): a + b = (a + b) mod 7 a - b = (a - b) mod 7 a * b = (a × b) mod 7 a / b = (a × b⁻¹) mod 7 [b≠0].", "equations": "H#K := 3. H#J := G#M. L#O := G#J + H#J * 2. G#M := 3. G#J := H#K. L#L := H#K. K#I := L#L * 5. K#O := G#M + L#O.", "query": "K#O", "answer": 1, "cot": "H#K = 3 -> H#K = 3\nG#M = 3 -> G#M = 3\nG#J = H#K = 3\nH#J = G#M = 3\nL#O = G#J + H#J * 2\n = 3 + 3 * 2\n 3 + 3 = (3 + 3) mod 7 = 6\n 6 * 2 = (6 × 2) mod 7 = 5\n => L#O = 5\nK#O = G#M + L#O\n = 3 + 5\n 3 + 5 = (3 + 5) mod 7 = 1\n => K#O = 1", "operator_table": [{"glyph": "+", "kind": "add", "params": {}, "definition": "a + b = (a + b) mod 7"}, {"glyph": "-", "kind": "sub", "params": {}, "definition": "a - b = (a - b) mod 7"}, {"glyph": "*", "kind": "mul", "params": {}, "definition": "a * b = (a × b) mod 7"}, {"glyph": "/", "kind": "div", "params": {}, "definition": "a / b = (a × b⁻¹) mod 7 [b≠0]"}], "target_depth": 4, "achieved_depth": 4, "num_vars": 8, "num_necessary": 6, "num_distractors": 0, "var_layer": {"H#K": 1, "G#M": 1, "L#L": 2, "G#J": 2, "H#J": 2, "K#I": 3, "L#O": 3, "K#O": 4}, "id": "mod7_arith_d4_0249"}