id stringlengths 30 30 | setting_key stringclasses 1
value | setting_name stringclasses 0
values | messages listlengths 1 1 | ground_truth stringclasses 125
values | dataset stringclasses 1
value |
|---|---|---|---|---|---|
numbertheory_lte_qr_train_0000 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^5. Find the least positive integer m such that m^2+(0) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 8 | math |
numbertheory_lte_qr_train_0001 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=17. How many solutions m in [1, 2, ..., 17^4] satisfy m^4+1=0 (mod 17^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
numbertheory_lte_qr_train_0002 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^6+(-2) is a multiple of p^2, and n^6+(-2) is also a multiple of p^4. Find the largest positive integer m such that m^6+(-2) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxe... | 76318 | math |
numbertheory_lte_qr_train_0003 | numbertheory_lte_qr | null | [
{
"role": "user",
"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 p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0004 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^5, and n^6+(-1) is also a multiple of p^6. Find the largest positive integer m such that m^6+(-1) is a multiple of p^5.\n\nPresent the answer in LaTex format: \\boxe... | 63 | math |
numbertheory_lte_qr_train_0005 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=47. How many solutions m in [1, 2, ..., 47^4] satisfy m^1+1=0 (mod 47^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0006 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4, 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^4.\n\nPresent the answer in LaTex format: \\box... | 1068 | math |
numbertheory_lte_qr_train_0007 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^3. Find the smallest positive integer m such that m^4+(-2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\... | 121 | math |
numbertheory_lte_qr_train_0008 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4. Find the least positive integer m such that m^4+(1) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 20051 | math |
numbertheory_lte_qr_train_0009 | numbertheory_lte_qr | null | [
{
"role": "user",
"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}"
}
] | 1 | math |
numbertheory_lte_qr_train_0010 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^2. Find the least positive integer m such that m^2+(-1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0011 | numbertheory_lte_qr | null | [
{
"role": "user",
"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{Y... | 15 | math |
numbertheory_lte_qr_train_0012 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^6. 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{Y... | 508 | math |
numbertheory_lte_qr_train_0013 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^3, and that n^5+(1) is also divisible by p^5. Find the smallest positive integer m such that m^5+(1) is divisible by p^3.\n\nPresent the answer in LaTex format: \\box... | 31 | math |
numbertheory_lte_qr_train_0014 | numbertheory_lte_qr | null | [
{
"role": "user",
"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. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 17 | math |
numbertheory_lte_qr_train_0015 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 3). 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}"
}
] | 6 | math |
numbertheory_lte_qr_train_0016 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4, 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^4.\n\nPresent the answer in LaTex format: \\box... | 146991 | math |
numbertheory_lte_qr_train_0017 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4, 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^4.\n\nPresent the answer in LaTex format: \\box... | 47 | math |
numbertheory_lte_qr_train_0018 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^3, and that n^6+(0) is also divisible by p^6. Find the smallest positive integer m such that m^6+(0) is divisible by p^3.\n\nPresent the answer in LaTex format: \\box... | 2 | math |
numbertheory_lte_qr_train_0019 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 4). For the least p where n^2=-1 (mod p^2) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
numbertheory_lte_qr_train_0020 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^6. Find the least positive integer m such that m^3+(-2) is divisible by p^6.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 5303 | math |
numbertheory_lte_qr_train_0021 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=2. How many solutions m in [1, 2, ..., 2^1] satisfy m^2+1=0 (mod 2^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0022 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=13. How many solutions m in [1, 2, ..., 13^1] satisfy m^1+1=0 (mod 13^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0023 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^2, and that n^6+(-1) is also divisible by p^4. Find the smallest positive integer m such that m^6+(-1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\... | 1 | math |
numbertheory_lte_qr_train_0024 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4, and n^3+(-2) is also a multiple of p^5. Find the largest positive integer m such that m^3+(-2) is a multiple of p^4.\n\nPresent the answer in LaTex format: \\boxe... | 2178 | math |
numbertheory_lte_qr_train_0025 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^3, and n^6+(0) is also a multiple of p^6. Find the largest positive integer m such that m^6+(0) is a multiple of p^3.\n\nPresent the answer in LaTex format: \\boxed{Y... | 62 | math |
numbertheory_lte_qr_train_0026 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^5, and that n^2+(1) is also divisible by p^6. Find the smallest positive integer m such that m^2+(1) is divisible by p^5.\n\nPresent the answer in LaTex format: \\box... | 1068 | math |
numbertheory_lte_qr_train_0027 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^5. Find the smallest positive integer m such that m^5+(-2) is divisible by p^3.\n\nPresent the answer in LaTex format: \\... | 23 | math |
numbertheory_lte_qr_train_0028 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4. 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{Y... | 443 | math |
numbertheory_lte_qr_train_0029 | numbertheory_lte_qr | null | [
{
"role": "user",
"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{Y... | 2057 | math |
numbertheory_lte_qr_train_0030 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=5. How many solutions m in [1, 2, ..., 5^3] satisfy m^3+1=0 (mod 5^3)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0031 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^2, and n^3+(0) is also a multiple of p^6. Find the largest positive integer m such that m^3+(0) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxed{Y... | 60 | math |
numbertheory_lte_qr_train_0032 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=7. How many solutions m in [1, 2, ..., 7^2] satisfy m^1+1=0 (mod 7^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0033 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0034 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=41. How many solutions m in [1, 2, ..., 41^1] satisfy m^3+1=0 (mod 41^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0035 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^3, and n^3+(2) is also a multiple of p^5. Find the largest positive integer m such that m^3+(2) is a multiple of p^3.\n\nPresent the answer in LaTex format: \\boxed{Y... | 947 | math |
numbertheory_lte_qr_train_0036 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^5, and that n^2+(-1) is also divisible by p^6. Find the smallest positive integer m such that m^2+(-1) is divisible by p^5.\n\nPresent the answer in LaTex format: \\... | 1 | math |
numbertheory_lte_qr_train_0037 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 5). 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}"
}
] | 14640 | math |
numbertheory_lte_qr_train_0038 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^2, 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^2.\n\nPresent the answer in LaTex format: \\boxed{Y... | 28 | math |
numbertheory_lte_qr_train_0039 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=41. How many solutions m in [1, 2, ..., 41^2] satisfy m^4+1=0 (mod 41^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
numbertheory_lte_qr_train_0040 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4. Find the least positive integer m such that m^3+(-1) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0041 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^3, and that n^6+(2) is also divisible by p^5. Find the smallest positive integer m such that m^6+(2) is divisible by p^3.\n\nPresent the answer in LaTex format: \\box... | 60003 | math |
numbertheory_lte_qr_train_0042 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^6. 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{Y... | 56 | math |
numbertheory_lte_qr_train_0043 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^2. Find the least positive integer m such that m^2+(1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
numbertheory_lte_qr_train_0044 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4. Find the least positive integer m such that m^2+(-1) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0045 | numbertheory_lte_qr | null | [
{
"role": "user",
"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. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0046 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^3, and n^3+(2) is also a multiple of p^4. Find the largest positive integer m such that m^3+(2) is a multiple of p^3.\n\nPresent the answer in LaTex format: \\boxed{Y... | 322 | math |
numbertheory_lte_qr_train_0047 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4. Find the smallest positive integer m such that m^3+(-2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\... | 303 | math |
numbertheory_lte_qr_train_0048 | numbertheory_lte_qr | null | [
{
"role": "user",
"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, 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^4.\n\nPresent the answer in LaTex format: \\box... | 10322 | math |
numbertheory_lte_qr_train_0049 | numbertheory_lte_qr | null | [
{
"role": "user",
"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{Y... | 31 | math |
numbertheory_lte_qr_train_0050 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 2). For the least p where n^1=-1 (mod p^3) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 26 | math |
numbertheory_lte_qr_train_0051 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=23. How many solutions m in [1, 2, ..., 23^4] satisfy m^3+1=0 (mod 23^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0052 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^5. Find the largest positive integer m such that m^5+(-1) is a multiple of p^3.\n\nPresent the answer in LaTex format: \\boxe... | 1 | math |
numbertheory_lte_qr_train_0053 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^6. Find the least positive integer m such that m^6+(2) is divisible by p^6.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 584201 | math |
numbertheory_lte_qr_train_0054 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4. Find the smallest positive integer m such that m^4+(2) is divisible by p^3.\n\nPresent the answer in LaTex format: \\box... | 34 | math |
numbertheory_lte_qr_train_0055 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=19. How many solutions m in [1, 2, ..., 19^4] satisfy m^1+1=0 (mod 19^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0056 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 3). For the least p where n^2=-1 (mod p^2) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 70 | math |
numbertheory_lte_qr_train_0057 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be the smallest prime where n^3+-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}"
}
] | 1 | math |
numbertheory_lte_qr_train_0058 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be the smallest prime where n^6+-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}"
}
] | 1 | math |
numbertheory_lte_qr_train_0059 | numbertheory_lte_qr | null | [
{
"role": "user",
"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{Y... | 4 | math |
numbertheory_lte_qr_train_0060 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^3, 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^3.\n\nPresent the answer in LaTex format: \\... | 1 | math |
numbertheory_lte_qr_train_0061 | numbertheory_lte_qr | null | [
{
"role": "user",
"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. Find the least positive integer m such that m^4+(2) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 47 | math |
numbertheory_lte_qr_train_0062 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4, 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^4.\n\nPresent the answer in LaTex format: \\... | 4681 | math |
numbertheory_lte_qr_train_0063 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=13. How many solutions m in [1, 2, ..., 13^4] satisfy m^3+1=0 (mod 13^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
numbertheory_lte_qr_train_0064 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4. Find the smallest positive integer m such that m^3+(-1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\... | 1 | math |
numbertheory_lte_qr_train_0065 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4, and that n^2+(-2) is also divisible by p^6. Find the smallest positive integer m such that m^2+(-2) is divisible by p^4.\n\nPresent the answer in LaTex format: \\... | 38181 | math |
numbertheory_lte_qr_train_0066 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^5. Find the smallest positive integer m such that m^4+(2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\box... | 47 | math |
numbertheory_lte_qr_train_0067 | numbertheory_lte_qr | null | [
{
"role": "user",
"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 the least positive integer m such that m^3+(-2) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 53 | math |
numbertheory_lte_qr_train_0068 | numbertheory_lte_qr | null | [
{
"role": "user",
"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}"
}
] | 1 | math |
numbertheory_lte_qr_train_0069 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 2). 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}"
}
] | 8 | math |
numbertheory_lte_qr_train_0070 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=5. How many solutions m in [1, 2, ..., 5^1] satisfy m^1+1=0 (mod 5^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0071 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=13. How many solutions m in [1, 2, ..., 13^1] satisfy m^3+1=0 (mod 13^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
numbertheory_lte_qr_train_0072 | numbertheory_lte_qr | null | [
{
"role": "user",
"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{Y... | 63 | math |
numbertheory_lte_qr_train_0073 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^2, 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^2.\n\nPresent the answer in LaTex format: \\... | 1 | math |
numbertheory_lte_qr_train_0074 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4. Find the smallest positive integer m such that m^6+(2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\box... | 1439 | math |
numbertheory_lte_qr_train_0075 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be the smallest prime where n^6+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}"
}
] | 57 | math |
numbertheory_lte_qr_train_0076 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=5. How many solutions m in [1, 2, ..., 5^1] satisfy m^2+1=0 (mod 5^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0077 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^5. Find the smallest positive integer m such that m^6+(-1) is divisible by p^3.\n\nPresent the answer in LaTex format: \\... | 1 | math |
numbertheory_lte_qr_train_0078 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^2, 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^2.\n\nPresent the answer in LaTex format: \\... | 1 | math |
numbertheory_lte_qr_train_0079 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=37. How many solutions m in [1, 2, ..., 37^2] satisfy m^1+1=0 (mod 37^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0080 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^5. Find the smallest positive integer m such that m^6+(0) is divisible by p^2.\n\nPresent the answer in LaTex format: \\box... | 2 | math |
numbertheory_lte_qr_train_0081 | numbertheory_lte_qr | null | [
{
"role": "user",
"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{Y... | 443 | math |
numbertheory_lte_qr_train_0082 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=17. How many solutions m in [1, 2, ..., 17^1] satisfy m^1+1=0 (mod 17^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0083 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 6). For the least p where n^5=-1 (mod p^2) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 48 | math |
numbertheory_lte_qr_train_0084 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^3. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0085 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^5. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
numbertheory_lte_qr_train_0086 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^5+(0) is a multiple of p^4, and n^5+(0) is also a multiple of p^5. Find the largest positive integer m such that m^5+(0) is a multiple of p^4.\n\nPresent the answer in LaTex format: \\boxed{Y... | 30 | math |
numbertheory_lte_qr_train_0087 | numbertheory_lte_qr | null | [
{
"role": "user",
"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. Find the least 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 |
numbertheory_lte_qr_train_0088 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 2). For the least p where n^5=-1 (mod p^4) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 80 | math |
numbertheory_lte_qr_train_0089 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^2. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 5 | math |
numbertheory_lte_qr_train_0090 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=19. How many solutions m in [1, 2, ..., 19^2] satisfy m^1+1=0 (mod 19^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0091 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^2+(1) is a multiple of p^2, and n^2+(1) is also a multiple of p^3. Find the largest positive integer m such that m^2+(1) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxed{Y... | 68 | math |
numbertheory_lte_qr_train_0092 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^4. 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{Y... | 12 | math |
numbertheory_lte_qr_train_0093 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^2+(-1) is a multiple of p^5, and n^2+(-1) is also a multiple of p^6. Find the largest positive integer m such that m^2+(-1) is a multiple of p^5.\n\nPresent the answer in LaTex format: \\boxe... | 63 | math |
numbertheory_lte_qr_train_0094 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^5. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 5 | math |
numbertheory_lte_qr_train_0095 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=29. How many solutions m in [1, 2, ..., 29^2] satisfy m^2+1=0 (mod 29^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0096 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^6. Find the least positive integer m such that m^3+(2) is divisible by p^6.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 10322 | math |
numbertheory_lte_qr_train_0097 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be the smallest prime where n^3+-1=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}"
}
] | 1 | math |
numbertheory_lte_qr_train_0098 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^2, and n^6+(0) is also a multiple of p^4. Find the largest positive integer m such that m^6+(0) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxed{Y... | 14 | math |
numbertheory_lte_qr_train_0099 | numbertheory_lte_qr | null | [
{
"role": "user",
"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. Find the least positive integer m such that m^5+(0) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 13