[User]
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.

Present the answer in LaTex format: \boxed{Your answer}

dataset: MATH


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}


[User]
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.

Present the answer in LaTex format: \boxed{Your answer}