messages listlengths 1 1 | ground_truth stringlengths 1 6 | dataset stringclasses 1
value |
|---|---|---|
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(-2) is divisible by p^6. Find the least positive integer m such that m^4+(-2) is divisible by p^6.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21488 | MATH |
[
{
"content": "Let p be the smallest prime where n^6+1=0 (mod p^5) has a solution, then find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1068 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(0) is divisible by p^2, and that n^4+(0) is also divisible by p^6. Find the smallest positive integer m such that m^4+(0) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^6+(1) is divisible by p^4, and that n^6+(1) is also divisible by p^5. Find the smallest positive integer m such that m^6+(1) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1068 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^6+(-2) is divisible by p^2, and that n^6+(-2) is also divisible by p^3. Find the smallest positive integer m such that m^6+(-2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 2290 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(0) is divisible by p^5. Find the least positive integer m such that m^4+(0) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 4). For the least p where n^4=-1 (mod p^4) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20051 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(2) is divisible by p^5. Find the least positive integer m such that m^5+(2) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 220 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^3+(-1) is divisible by p^2, and that n^3+(-1) is also divisible by p^5. Find the smallest positive integer m such that m^3+(-1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 1 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^3+(-2) is divisible by p^2, and that n^3+(-2) is also divisible by p^5. Find the smallest positive integer m such that m^3+(-2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 2178 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(0) is divisible by p^5, and that n^5+(0) is also divisible by p^6. Find the smallest positive integer m such that m^5+(0) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "let p=5. How many solutions m in [1, 2, ..., 5^2] satisfy m^1+1=0 (mod 5^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^3+(2) is divisible by p^4. Find the least positive integer m such that m^3+(2) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 322 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^6+(2) is divisible by p^4, and that n^6+(2) is also divisible by p^6. Find the smallest positive integer m such that m^6+(2) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 584201 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^4+(0) is a multiple of p^5, and n^4+(0) is also a multiple of p^6. Find the largest positive integer m such that m^4+(0) is a multiple of p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 60 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 2). For the least p where n^1=-1 (mod p^2) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^2+(1) is divisible by p^3, and that n^2+(1) is also divisible by p^5. Find the smallest positive integer m such that m^2+(1) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1068 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^3+(-1) is a multiple of p^4, and n^3+(-1) is also a multiple of p^6. Find the largest positive integer m such that m^3+(-1) is a multiple of p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(-2) is divisible by p^2, and that n^5+(-2) is also divisible by p^5. Find the smallest positive integer m such that m^5+(-2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 23 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(-1) is divisible by p^6. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "let p=31. How many solutions m in [1, 2, ..., 31^1] satisfy m^1+1=0 (mod 31^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^5+(-1) is a multiple of p^2, and n^5+(-1) is also a multiple of p^5. Find the largest positive integer m such that m^5+(-1) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(1) is divisible by p^2, and that n^4+(1) is also divisible by p^3. Find the smallest positive integer m such that m^4+(1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 399 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(-2) is divisible by p^5, and that n^4+(-2) is also divisible by p^6. Find the smallest positive integer m such that m^4+(-2) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 21488 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^3+(1) is divisible by p^6. Find the least positive integer m such that m^3+(1) is divisible by p^6.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 63 | MATH |
[
{
"content": "Let p be the smallest prime where n^4+-1=0 (mod p^3) has a solution, then find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 5). For the least p where n^5=-1 (mod p^3) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 161 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(-2) is divisible by p^3, and that n^4+(-2) is also divisible by p^5. Find the smallest positive integer m such that m^4+(-2) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 4681 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(0) is divisible by p^2, and that n^5+(0) is also divisible by p^5. Find the smallest positive integer m such that m^5+(0) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 2 | MATH |
[
{
"content": "let p=41. How many solutions m in [1, 2, ..., 41^1] satisfy m^2+1=0 (mod 41^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^3+(-2) is divisible by p^2, and that n^3+(-2) is also divisible by p^3. Find the smallest positive integer m such that m^3+(-2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 53 | MATH |
[
{
"content": "let p=11. How many solutions m in [1, 2, ..., 11^2] satisfy m^3+1=0 (mod 11^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 8). For the least p where n^7=-1 (mod p^4) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 83520 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 5). For the least p where n^1=-1 (mod p^4) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14640 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^2+(0) is a multiple of p^2, and n^2+(0) is also a multiple of p^5. Find the largest positive integer m such that m^2+(0) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 24 | MATH |
[
{
"content": "let p=59. How many solutions m in [1, 2, ..., 59^1] satisfy m^3+1=0 (mod 59^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^2+(0) is divisible by p^4. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(2) is divisible by p^3, and that n^5+(2) is also divisible by p^6. Find the smallest positive integer m such that m^5+(2) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 706 | MATH |
[
{
"content": "let p=29. How many solutions m in [1, 2, ..., 29^4] satisfy m^2+1=0 (mod 29^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(0) is divisible by p^4, and that n^4+(0) is also divisible by p^6. Find the smallest positive integer m such that m^4+(0) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 2). For the least p where n^6=-1 (mod p^4) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 182 | MATH |
[
{
"content": "Let p be the smallest prime where n^7+-1=0 (mod p^5) has a solution, then find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 7). For the least p where n^2=-1 (mod p^4) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34522 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^5+(2) is a multiple of p^4, and n^5+(2) is also a multiple of p^6. Find the largest positive integer m such that m^5+(2) is a multiple of p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 706 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^6+(1) is divisible by p^4. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(-1) is divisible by p^2, and that n^5+(-1) is also divisible by p^6. Find the smallest positive integer m such that m^5+(-1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 1 | MATH |
[
{
"content": "Let p be the smallest prime where n^6+2=0 (mod p^2) has a solution, then find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(-1) is divisible by p^4, and that n^5+(-1) is also divisible by p^6. Find the smallest positive integer m such that m^5+(-1) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 1 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^5+(-1) is a multiple of p^3, and n^5+(-1) is also a multiple of p^4. Find the largest positive integer m such that m^5+(-1) is a multiple of p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^6+(-1) is divisible by p^6. Find the least positive integer m such that m^6+(-1) is divisible by p^6.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^6+(1) is divisible by p^6. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 3). For the least p where n^3=-1 (mod p^4) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1048 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 5). For the least p where n^2=-1 (mod p^3) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11389 | MATH |
[
{
"content": "let p=3. How many solutions m in [1, 2, ..., 3^4] satisfy m^3+1=0 (mod 3^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^6+(-1) is a multiple of p^4, and n^6+(-1) is also a multiple of p^5. Find the largest positive integer m such that m^6+(-1) is a multiple of p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 31 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^6+(-1) is a multiple of p^2, and n^6+(-1) is also a multiple of p^4. Find the largest positive integer m such that m^6+(-1) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 15 | MATH |
[
{
"content": "let p=43. How many solutions m in [1, 2, ..., 43^1] satisfy m^3+1=0 (mod 43^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 3). For the least p where n^7=-1 (mod p^4) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 342 | MATH |
[
{
"content": "let p=41. How many solutions m in [1, 2, ..., 41^4] satisfy m^2+1=0 (mod 41^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(-2) is divisible by p^3, and that n^4+(-2) is also divisible by p^6. Find the smallest positive integer m such that m^4+(-2) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 21488 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 2). For the least p where n^3=-1 (mod p^4) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^5+(-2) is a multiple of p^2, and n^5+(-2) is also a multiple of p^3. Find the largest positive integer m such that m^5+(-2) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 23 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 4). For the least p where n^6=-1 (mod p^2) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^3+(-2) is a multiple of p^2, and n^3+(-2) is also a multiple of p^6. Find the largest positive integer m such that m^3+(-2) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 5303 | MATH |
[
{
"content": "let p=53. How many solutions m in [1, 2, ..., 53^4] satisfy m^2+1=0 (mod 53^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 3). For the least p where n^6=-1 (mod p^2) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 6). For the least p where n^7=-1 (mod p^2) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(2) is divisible by p^5. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^3+(0) is divisible by p^5. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(-2) is divisible by p^2, and that n^5+(-2) is also divisible by p^6. Find the smallest positive integer m such that m^5+(-2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 23 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(0) is divisible by p^6. Find the least positive integer m such that m^4+(0) is divisible by p^6.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(2) is divisible by p^2, and that n^4+(2) is also divisible by p^4. Find the smallest positive integer m such that m^4+(2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 34 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^2+(-2) is a multiple of p^4, and n^2+(-2) is also a multiple of p^5. Find the largest positive integer m such that m^2+(-2) is a multiple of p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 12240 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^2+(2) is a multiple of p^2, and n^2+(2) is also a multiple of p^4. Find the largest positive integer m such that m^2+(2) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 59 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^4+(-2) is a multiple of p^3, and n^4+(-2) is also a multiple of p^5. Find the largest positive integer m such that m^4+(-2) is a multiple of p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 12126 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^3+(2) is divisible by p^3, and that n^3+(2) is also divisible by p^5. Find the smallest positive integer m such that m^3+(2) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 947 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^6+(0) is divisible by p^2, and that n^6+(0) is also divisible by p^3. Find the smallest positive integer m such that m^6+(0) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 2 | MATH |
[
{
"content": "Let p be a prime with p=1 (mod 4). For the least p where n^1=-1 (mod p^2) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(-1) is divisible by p^5, and that n^4+(-1) is also divisible by p^6. Find the smallest positive integer m such that m^4+(-1) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 1 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^6+(1) is divisible by p^3. Find the least positive integer m such that m^6+(1) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 57 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^3+(0) is a multiple of p^4, and n^3+(0) is also a multiple of p^5. Find the largest positive integer m such that m^3+(0) is a multiple of p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 28 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^6+(1) is divisible by p^3, and that n^6+(1) is also divisible by p^6. Find the smallest positive integer m such that m^6+(1) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1068 | MATH |
[
{
"content": "Let p be the smallest prime where n^7+1=0 (mod p^4) has a solution, then find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^3+(-2) is divisible by p^3. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^2+(-2) is divisible by p^3, and that n^2+(-2) is also divisible by p^5. Find the smallest positive integer m such that m^2+(-2) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 4567 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(-2) is divisible by p^2. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^6+(-1) is a multiple of p^3, and n^6+(-1) is also a multiple of p^5. Find the largest positive integer m such that m^6+(-1) is a multiple of p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 31 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^3+(-2) is divisible by p^5, and that n^3+(-2) is also divisible by p^6. Find the smallest positive integer m such that m^3+(-2) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 5303 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(-1) is divisible by p^5, and that n^5+(-1) is also divisible by p^6. Find the smallest positive integer m such that m^5+(-1) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 1 | MATH |
[
{
"content": "Let p be the smallest prime where n^6+2=0 (mod p^5) has a solution, then find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60003 | MATH |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^6+(0) is a multiple of p^4, and n^6+(0) is also a multiple of p^5. Find the largest positive integer m such that m^6+(0) is a multiple of p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 30 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^6+(0) is divisible by p^5. Find the least positive integer m such that m^6+(0) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^3+(0) is divisible by p^5. Find the least positive integer m such that m^3+(0) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "let p=47. How many solutions m in [1, 2, ..., 47^1] satisfy m^3+1=0 (mod 47^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "let p=43. How many solutions m in [1, 2, ..., 43^3] satisfy m^1+1=0 (mod 43^3)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(-1) is divisible by p^2, and that n^4+(-1) is also divisible by p^5. Find the smallest positive integer m such that m^4+(-1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 1 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^6+(1) is divisible by p^4, and that n^6+(1) is also divisible by p^6. Find the smallest positive integer m such that m^6+(1) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1068 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(-2) is divisible by p^2, and that n^4+(-2) is also divisible by p^4. Find the smallest positive integer m such that m^4+(-2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 121 | MATH |
[
{
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(2) is divisible by p^4, and that n^5+(2) is also divisible by p^5. Find the smallest positive integer m such that m^5+(2) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 220 | MATH |
[
{
"content": "Let p be the smallest prime where n^7+1=0 (mod p^1) has a solution, then find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
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