| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 4^3, 2. a, b, c are relatively coprime, and 3. -1a^4 + 1b^4 + 0c^4 is divisible by 4^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "197", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 4^2 and a^4 + b^4 + c^4 + d^4 is divisible by 4^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "256", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 3^4 and a^3 + b^3 + c^3 + d^3 is divisible by 3^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. 1a^2 + -1b^2 + 1c^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^2, 2. a, b, c are relatively coprime, and 3. -1a^3 + 0b^3 + -1c^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "105", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^2, 2. a, b, c are relatively coprime, and 3. 1a^3 + 0b^3 + 0c^3 is divisible by 3^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "201", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^4, 2. a, b, c are relatively coprime, and 3. 0a^2 + -1b^2 + -1c^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "125", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 4^3 and a^4 + b^4 + c^4 + d^4 is divisible by 4^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "576", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. 0a^2 + -1b^2 + 1c^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "155", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 2^2 and a^2 + b^2 + c^2 + d^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^4, 2. a, b, c are relatively coprime, and 3. 1a^2 + 1b^2 + 0c^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "247", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^3, 2. a, b, c are relatively coprime, and 3. 0a^3 + 0b^3 + 1c^3 is divisible by 3^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "188", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 3^4 and a^3 + b^3 + c^3 + d^3 is divisible by 3^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "379", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^2, 2. a, b, c are relatively coprime, and 3. -1a^3 + -1b^3 + 0c^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "105", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 3^3 and a^3 + b^3 + c^3 + d^3 is divisible by 3^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "659", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^4, 2. a, b, c are relatively coprime, and 3. -1a^2 + 0b^2 + -1c^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "247", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^4, 2. a, b, c are relatively coprime, and 3. -1a^3 + 0b^3 + -1c^3 is divisible by 3^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 4^2 and a^4 + b^4 + c^4 + d^4 is divisible by 4^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "96", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. 1a^2 + -1b^2 + 1c^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. -1a^2 + -1b^2 + 1c^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^3, 2. a, b, c are relatively coprime, and 3. 0a^3 + 1b^3 + 0c^3 is divisible by 3^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "188", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 2^4 and a^2 + b^2 + c^2 + d^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "192", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^2, 2. a, b, c are relatively coprime, and 3. -1a^3 + -1b^3 + 0c^3 is divisible by 3^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "197", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. 0a^2 + -1b^2 + -1c^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 2^2 and a^2 + b^2 + c^2 + d^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^3, 2. a, b, c are relatively coprime, and 3. 0a^3 + -1b^3 + 1c^3 is divisible by 3^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. 1a^2 + -1b^2 + -1c^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "126", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 3^2 and a^3 + b^3 + c^3 + d^3 is divisible by 3^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "539", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 3^4 and a^3 + b^3 + c^3 + d^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "87", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^3, 2. a, b, c are relatively coprime, and 3. 1a^3 + 0b^3 + 0c^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "188", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. 1a^2 + 1b^2 + 0c^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^3, 2. a, b, c are relatively coprime, and 3. 1a^3 + 1b^3 + 0c^3 is divisible by 3^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "112", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. 0a^2 + 0b^2 + 0c^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. 1a^2 + 0b^2 + -1c^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^2, 2. a, b, c are relatively coprime, and 3. 0a^3 + -1b^3 + 1c^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^2, 2. a, b, c are relatively coprime, and 3. 1a^3 + 1b^3 + 0c^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "105", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^3, 2. a, b, c are relatively coprime, and 3. 0a^3 + -1b^3 + 0c^3 is divisible by 3^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "188", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. -1a^2 + 0b^2 + -1c^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 4^3 and a^4 + b^4 + c^4 + d^4 is divisible by 4^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. 1a^2 + -1b^2 + -1c^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. -1a^2 + 0b^2 + -1c^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 2^4 and a^2 + b^2 + c^2 + d^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. -1a^2 + 0b^2 + -1c^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. 0a^2 + -1b^2 + -1c^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 3^3 and a^3 + b^3 + c^3 + d^3 is divisible by 3^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "309", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^2, 2. a, b, c are relatively coprime, and 3. -1a^3 + 1b^3 + 0c^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. 0a^2 + -1b^2 + 0c^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "189", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 2^3 and a^2 + b^2 + c^2 + d^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "128", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^2, 2. a, b, c are relatively coprime, and 3. 0a^3 + 1b^3 + 1c^3 is divisible by 3^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. -1a^2 + -1b^2 + 0c^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^4, 2. a, b, c are relatively coprime, and 3. 1a^2 + -1b^2 + 1c^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "246", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. 0a^2 + 0b^2 + -1c^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 2^4 and a^2 + b^2 + c^2 + d^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "512", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^2, 2. a, b, c are relatively coprime, and 3. 1a^3 + -1b^3 + 0c^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 4^2, 2. a, b, c are relatively coprime, and 3. 1a^4 + 1b^4 + 0c^4 is divisible by 4^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "125", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 3^3 and a^3 + b^3 + c^3 + d^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "927", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. 1a^2 + 1b^2 + -1c^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "126", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 2^3 and a^2 + b^2 + c^2 + d^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 4^3 and a^4 + b^4 + c^4 + d^4 is divisible by 4^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "536", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. -1a^2 + 1b^2 + -1c^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "126", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 3^2 and a^3 + b^3 + c^3 + d^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "567", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 3^2 and a^3 + b^3 + c^3 + d^3 is divisible by 3^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "159", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^2, 2. a, b, c are relatively coprime, and 3. 1a^3 + 0b^3 + 0c^3 is divisible by 3^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. -1a^2 + 1b^2 + 1c^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. 1a^2 + 0b^2 + 1c^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 2^2 and a^2 + b^2 + c^2 + d^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^2, 2. a, b, c are relatively coprime, and 3. 0a^3 + 0b^3 + 1c^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "201", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. 1a^2 + 1b^2 + -1c^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^2, 2. a, b, c are relatively coprime, and 3. -1a^2 + 0b^2 + 0c^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. -1a^2 + -1b^2 + 1c^2 is divisible by 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 4^2 and a^4 + b^4 + c^4 + d^4 is divisible by 4^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "512", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^4, 2. a, b, c are relatively coprime, and 3. 1a^3 + 0b^3 + 1c^3 is divisible by 3^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^3, 2. a, b, c are relatively coprime, and 3. 0a^3 + 0b^3 + -1c^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "188", "dataset": "MATH"} |
| {"messages": [{"content": "Let N be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 2^3 and a^2 + b^2 + c^2 + d^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "512", "dataset": "MATH"} |
| {"messages": [{"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 2^3, 2. a, b, c are relatively coprime, and 3. -1a^2 + -1b^2 + 0c^2 is divisible by 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"} |
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