messages listlengths 1 1 | ground_truth stringlengths 1 3 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and 2a^3 + -1b^3 + 1c^3 = 0 (mod 3^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 2a^3 + -2b^3 + -1c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 215 | MATH |
[
{
"content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 2^3 and a^2 + b^2 is a multiple of 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 1a^2 + 1b^2 + 1c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and -1a^4 + -2b^4 + 2c^4 = 0 (mod 4^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 160 | MATH |
[
{
"content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 2^4 and a^2 + b^2 is a multiple of 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and 2a^2 + -1b^2 + -2c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 384 | MATH |
[
{
"content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 3^3 and a^3 + b^3 is a multiple of 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 135 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and 2a^4 + -2b^4 + 1c^4 = 0 (mod 4^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 160 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 2a^2 + -1b^2 + -1c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^2 and -1a^2 + 2b^2 + 0c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 2^4 and a^2 + b^2 is a multiple of 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and 1a^3 + 0b^3 + 1c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 243 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and -1a^4 + 2b^4 + 0c^4 = 0 (mod 4^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 256 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^2 and 1a^2 + 0b^2 + 1c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and 0a^4 + 2b^4 + 2c^4 = 0 (mod 4^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and -2a^3 + 0b^3 + 2c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 561 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and 2a^2 + -2b^2 + 0c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 536 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 2a^3 + -1b^3 + -2c^3 = 0 (mod 3^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 405 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and 1a^4 + 0b^4 + -1c^4 = 0 (mod 4^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 1a^2 + 0b^2 + -2c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and -2a^2 + 2b^2 + 2c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and -1a^2 + 0b^2 + 1c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 192 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and 0a^2 + 2b^2 + -2c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 2^3 and a^2 + b^2 is a multiple of 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and -2a^3 + -2b^3 + 2c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 81 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and -1a^4 + 0b^4 + -1c^4 = 0 (mod 4^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 256 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and 1a^2 + -2b^2 + 2c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 384 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 0a^2 + 2b^2 + 0c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 256 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and -2a^3 + 2b^3 + 2c^3 = 0 (mod 3^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 729 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^2 and -2a^2 + -1b^2 + 2c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and -1a^3 + -1b^3 + 0c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 561 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 1a^2 + -1b^2 + 0c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 256 | MATH |
[
{
"content": "Assume N is the number of integers a which are >= 0 where a <= 2^2 and a^2 is divisible by 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^2 and 0a^2 + 2b^2 + -2c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 1a^2 + 2b^2 + 2c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and -2a^2 + -1b^2 + 1c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 128 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 0a^3 + -2b^3 + -2c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 561 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and 2a^2 + 1b^2 + 2c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 512 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 1a^3 + 2b^3 + 2c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 215 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and 1a^4 + 1b^4 + 2c^4 = 0 (mod 4^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 2^2 and a^2 + b^2 is a multiple of 2^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 0a^3 + 2b^3 + 2c^3 = 0 (mod 3^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 215 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and -2a^3 + -2b^3 + 2c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 189 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 2a^3 + -1b^3 + 1c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 645 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and 0a^4 + -2b^4 + -1c^4 = 0 (mod 4^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 256 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^2 and 0a^2 + -2b^2 + -2c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and -2a^3 + -1b^3 + 2c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 187 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^2 and 2a^2 + 0b^2 + 1c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and 2a^2 + 0b^2 + 2c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 512 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and -2a^2 + -2b^2 + -1c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 0a^3 + 2b^3 + 2c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 645 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and -2a^4 + 2b^4 + 1c^4 = 0 (mod 4^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 160 | MATH |
[
{
"content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 2^3 and a^2 + b^2 is a multiple of 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 3^2 and a^3 + b^3 is a multiple of 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and 1a^3 + -2b^3 + 2c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 45 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and 0a^2 + 2b^2 + -2c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and -1a^2 + 2b^2 + 0c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 128 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and 0a^3 + -2b^3 + 2c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 243 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 1a^3 + 0b^3 + -2c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 187 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and 2a^4 + 1b^4 + -1c^4 = 0 (mod 4^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^2 and 1a^2 + -1b^2 + -2c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^2 and 1a^2 + 1b^2 + 1c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and -1a^4 + 2b^4 + 2c^4 = 0 (mod 4^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 512 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 0a^3 + 2b^3 + -2c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 645 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and -1a^4 + 0b^4 + -2c^4 = 0 (mod 4^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and 0a^4 + 0b^4 + -1c^4 = 0 (mod 4^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and -1a^2 + 1b^2 + 2c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 384 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and -2a^3 + -2b^3 + 0c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 645 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and 2a^3 + 1b^3 + -2c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 81 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and 1a^2 + -1b^2 + 2c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 768 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 2a^2 + 0b^2 + -2c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 256 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 2a^3 + 0b^3 + -2c^3 = 0 (mod 3^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 215 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^2 and 1a^2 + -1b^2 + 2c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 1a^2 + -1b^2 + -2c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 2a^3 + 2b^3 + 1c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 187 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 1a^3 + 2b^3 + 2c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 187 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and 1a^2 + 1b^2 + -1c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 448 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and -2a^3 + -1b^3 + 0c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 187 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and 1a^3 + -1b^3 + 0c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 135 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and 1a^4 + 1b^4 + 1c^4 = 0 (mod 4^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 2a^3 + -1b^3 + -1c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 645 | MATH |
[
{
"content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 4^2 and a^4 + b^4 is a multiple of 4^4. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 0a^2 + 1b^2 + 0c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 128 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 1a^3 + 0b^3 + -1c^3 = 0 (mod 3^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 215 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 2a^3 + 1b^3 + 0c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 187 | MATH |
[
{
"content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 2^3 and a^2 + b^2 is a multiple of 2^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^2 and -2a^2 + 2b^2 + 0c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and -2a^3 + 2b^3 + 0c^3 = 0 (mod 3^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 215 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and -1a^3 + -1b^3 + -2c^3 = 0 (mod 3^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and -2a^3 + 0b^3 + 0c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 243 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and -2a^2 + -2b^2 + 2c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 192 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and -1a^4 + -2b^4 + -2c^4 = 0 (mod 4^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 512 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and -1a^2 + -2b^2 + 0c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 256 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 2a^2 + -2b^2 + -2c^2 = 0 (mod 2^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 192 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 2a^2 + 1b^2 + 1c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and -1a^3 + -2b^3 + -1c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 63 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and -1a^3 + 2b^3 + -1c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 645 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and 1a^3 + 1b^3 + 2c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 135 | MATH |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^2 and -2a^2 + 1b^2 + 0c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
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