messages listlengths 1 1 | ground_truth stringlengths 1 3 | dataset stringclasses 1 value |
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{
"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"
... | 197 | MATH |
[
{
"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"
}
] | 256 | MATH |
[
{
"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"
}
] | 29 | MATH |
[
{
"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"
... | 16 | MATH |
[
{
"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"
... | 105 | MATH |
[
{
"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"
... | 201 | MATH |
[
{
"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"
... | 125 | MATH |
[
{
"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"
}
] | 576 | MATH |
[
{
"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"
... | 155 | MATH |
[
{
"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"
}
] | 32 | MATH |
[
{
"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"
... | 247 | MATH |
[
{
"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"
... | 188 | MATH |
[
{
"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"
}
] | 379 | MATH |
[
{
"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"
... | 105 | MATH |
[
{
"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"
}
] | 659 | MATH |
[
{
"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"
... | 247 | MATH |
[
{
"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"
... | 99 | MATH |
[
{
"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"
}
] | 96 | MATH |
[
{
"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"
... | 8 | MATH |
[
{
"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"
... | 32 | MATH |
[
{
"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"
... | 188 | MATH |
[
{
"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"
}
] | 192 | MATH |
[
{
"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"
... | 197 | MATH |
[
{
"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"
... | 16 | MATH |
[
{
"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"
}
] | 8 | MATH |
[
{
"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"
... | 99 | MATH |
[
{
"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"
... | 126 | MATH |
[
{
"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"
}
] | 539 | MATH |
[
{
"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"
}
] | 87 | MATH |
[
{
"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"
... | 188 | MATH |
[
{
"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"
... | 2 | MATH |
[
{
"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"
... | 112 | MATH |
[
{
"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"
... | 55 | MATH |
[
{
"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"
... | 19 | MATH |
[
{
"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"
... | 91 | MATH |
[
{
"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"
... | 105 | MATH |
[
{
"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"
... | 188 | MATH |
[
{
"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"
... | 16 | MATH |
[
{
"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"
}
] | 72 | MATH |
[
{
"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"
... | 64 | MATH |
[
{
"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"
... | 63 | MATH |
[
{
"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"
}
] | 48 | MATH |
[
{
"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"
... | 8 | MATH |
[
{
"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"
... | 2 | MATH |
[
{
"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"
}
] | 309 | MATH |
[
{
"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"
... | 91 | MATH |
[
{
"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"
... | 189 | MATH |
[
{
"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"
}
] | 128 | MATH |
[
{
"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"
... | 18 | MATH |
[
{
"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"
... | 8 | MATH |
[
{
"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"
... | 246 | MATH |
[
{
"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"
... | 12 | MATH |
[
{
"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"
}
] | 512 | MATH |
[
{
"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"
... | 91 | MATH |
[
{
"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"
... | 125 | MATH |
[
{
"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"
}
] | 927 | MATH |
[
{
"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"
... | 126 | MATH |
[
{
"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"
}
] | 32 | MATH |
[
{
"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"
}
] | 536 | MATH |
[
{
"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"
... | 126 | MATH |
[
{
"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"
}
] | 567 | MATH |
[
{
"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"
}
] | 159 | MATH |
[
{
"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"
... | 72 | MATH |
[
{
"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"
... | 16 | MATH |
[
{
"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"
... | 4 | MATH |
[
{
"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"
}
] | 2 | MATH |
[
{
"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"
... | 201 | MATH |
[
{
"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"
... | 8 | MATH |
[
{
"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"
... | 24 | MATH |
[
{
"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"
... | 64 | MATH |
[
{
"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"
}
] | 512 | MATH |
[
{
"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"
... | 99 | MATH |
[
{
"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"
... | 188 | MATH |
[
{
"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"
}
] | 512 | MATH |
[
{
"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"
... | 16 | MATH |
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