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_0200 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 7). 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}"
}
] | 24388 | math |
numbertheory_lte_qr_train_0201 | 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^5, 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^5.\n\nPresent the answer in LaTex format: \\boxe... | 1 | math |
numbertheory_lte_qr_train_0202 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=2. How many solutions m in [1, 2, ..., 2^4] satisfy m^3+1=0 (mod 2^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0203 | 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^4, 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^4.\n\nPresent the answer in LaTex format: \\boxe... | 1 | math |
numbertheory_lte_qr_train_0204 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 6). 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}"
}
] | 342 | math |
numbertheory_lte_qr_train_0205 | 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^3. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 17 | math |
numbertheory_lte_qr_train_0206 | 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^5, 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^5.\n\nPresent the answer in LaTex format: \\box... | 2 | math |
numbertheory_lte_qr_train_0207 | 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. Find the least positive integer m such that m^4+(-2) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 121 | math |
numbertheory_lte_qr_train_0208 | 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^3, 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^3.\n\nPresent the answer in LaTex format: \\boxed{Y... | 63 | math |
numbertheory_lte_qr_train_0209 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 4). 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}"
}
] | 624 | math |
numbertheory_lte_qr_train_0210 | 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^2, and n^5+(0) is also a multiple of p^4. Find the largest positive integer m such that m^5+(0) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxed{Y... | 14 | math |
numbertheory_lte_qr_train_0211 | 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^3. Find the least positive integer m such that m^4+(1) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 399 | math |
numbertheory_lte_qr_train_0212 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 8). 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}"
}
] | 83520 | math |
numbertheory_lte_qr_train_0213 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=5. How many solutions m in [1, 2, ..., 5^2] satisfy m^2+1=0 (mod 5^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0214 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 7). 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}"
}
] | 840 | math |
numbertheory_lte_qr_train_0215 | 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^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{Y... | 221 | math |
numbertheory_lte_qr_train_0216 | 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, and that n^2+(0) is also divisible by p^6. Find the smallest positive integer m such that m^2+(0) is divisible by p^5.\n\nPresent the answer in LaTex format: \\box... | 8 | math |
numbertheory_lte_qr_train_0217 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=53. How many solutions m in [1, 2, ..., 53^2] satisfy m^3+1=0 (mod 53^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0218 | 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 p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0219 | 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^5. Find the least positive integer m such that m^5+(1) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 31 | math |
numbertheory_lte_qr_train_0220 | 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^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}"
}
] | 196 | math |
numbertheory_lte_qr_train_0221 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be the smallest prime where n^5+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}"
}
] | 7 | math |
numbertheory_lte_qr_train_0222 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=43. How many solutions m in [1, 2, ..., 43^4] satisfy m^1+1=0 (mod 43^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0223 | 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+(0) is divisible by p^3, and that n^4+(0) is also divisible by p^4. Find the smallest positive integer m such that m^4+(0) is divisible by p^3.\n\nPresent the answer in LaTex format: \\box... | 2 | math |
numbertheory_lte_qr_train_0224 | 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^6. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 17 | math |
numbertheory_lte_qr_train_0225 | 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^6. Find the smallest positive integer m such that m^4+(-2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\... | 21488 | math |
numbertheory_lte_qr_train_0226 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be the smallest prime where n^7+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}"
}
] | 3 | math |
numbertheory_lte_qr_train_0227 | 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^6. Find the least positive integer m such that m^4+(-1) is divisible by p^6.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0228 | 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^4. Find the smallest positive integer m such that m^6+(-2) is divisible by p^3.\n\nPresent the answer in LaTex format: \\... | 7203 | math |
numbertheory_lte_qr_train_0229 | 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^2. Find the least positive integer m such that m^2+(-2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 10 | math |
numbertheory_lte_qr_train_0230 | 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^4, and n^5+(1) is also a multiple of p^6. Find the largest positive integer m such that m^5+(1) is a multiple of p^4.\n\nPresent the answer in LaTex format: \\boxed{Y... | 63 | math |
numbertheory_lte_qr_train_0231 | 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_0232 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=37. How many solutions m in [1, 2, ..., 37^3] satisfy m^2+1=0 (mod 37^3)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0233 | 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^3. Find the smallest positive integer m such that m^3+(2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\box... | 72 | math |
numbertheory_lte_qr_train_0234 | 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^5. Find the smallest positive integer m such that m^3+(1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\box... | 31 | math |
numbertheory_lte_qr_train_0235 | 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^3. 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... | 2623 | math |
numbertheory_lte_qr_train_0236 | 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. 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}"
}
] | 303 | math |
numbertheory_lte_qr_train_0237 | 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: \\boxe... | 79468 | math |
numbertheory_lte_qr_train_0238 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 2). For the least p where n^6=-1 (mod p^3) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 57 | math |
numbertheory_lte_qr_train_0239 | 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 the least positive integer m such that m^2+(-1) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0240 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=31. How many solutions m in [1, 2, ..., 31^2] satisfy m^3+1=0 (mod 31^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
numbertheory_lte_qr_train_0241 | 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^2. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0242 | 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: \\box... | 7 | math |
numbertheory_lte_qr_train_0243 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 8). 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}"
}
] | 4912 | math |
numbertheory_lte_qr_train_0244 | 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^6. Find the smallest positive integer m such that m^3+(2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\box... | 10322 | math |
numbertheory_lte_qr_train_0245 | 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^6. Find the smallest positive integer m such that m^5+(1) is divisible by p^3.\n\nPresent the answer in LaTex format: \\box... | 63 | math |
numbertheory_lte_qr_train_0246 | 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^4. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 11 | math |
numbertheory_lte_qr_train_0247 | 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^6. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0248 | 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. Find the least 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 |
numbertheory_lte_qr_train_0249 | 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: \\box... | 182 | math |
numbertheory_lte_qr_train_0250 | 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^2, 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^2.\n\nPresent the answer in LaTex format: \\... | 38181 | math |
numbertheory_lte_qr_train_0251 | 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^6. Find the largest positive integer m such that m^5+(-2) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxe... | 23 | math |
numbertheory_lte_qr_train_0252 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 6). 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}"
}
] | 19 | math |
numbertheory_lte_qr_train_0253 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=3. How many solutions m in [1, 2, ..., 3^2] satisfy m^1+1=0 (mod 3^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0254 | 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^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: \\box... | 10322 | math |
numbertheory_lte_qr_train_0255 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^2, and n^4+(-2) is also a multiple of p^6. Find the largest positive integer m such that m^4+(-2) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxe... | 96161 | math |
numbertheory_lte_qr_train_0256 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 8). For the least p where n^3=-1 (mod p^2) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 288 | math |
numbertheory_lte_qr_train_0257 | 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^3, and n^3+(-1) is also a multiple of p^5. Find the largest positive integer m such that m^3+(-1) is a multiple of p^3.\n\nPresent the answer in LaTex format: \\boxe... | 1 | math |
numbertheory_lte_qr_train_0258 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 8). 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}"
}
] | 288 | math |
numbertheory_lte_qr_train_0259 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=53. How many solutions m in [1, 2, ..., 53^4] satisfy m^3+1=0 (mod 53^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0260 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 6). 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}"
}
] | 1048 | math |
numbertheory_lte_qr_train_0261 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=13. How many solutions m in [1, 2, ..., 13^1] satisfy m^2+1=0 (mod 13^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0262 | 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^3, and that n^5+(0) is also divisible by p^4. Find the smallest positive integer m such that m^5+(0) is divisible by p^3.\n\nPresent the answer in LaTex format: \\box... | 2 | math |
numbertheory_lte_qr_train_0263 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 8). 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}"
}
] | 1985 | math |
numbertheory_lte_qr_train_0264 | 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^4, and that n^2+(0) is also divisible by p^5. Find the smallest positive integer m such that m^2+(0) is divisible by p^4.\n\nPresent the answer in LaTex format: \\box... | 8 | math |
numbertheory_lte_qr_train_0265 | 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^2, 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^2.\n\nPresent the answer in LaTex format: \\boxe... | 1 | math |
numbertheory_lte_qr_train_0266 | 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^3. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0267 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be the smallest prime where n^4+-2=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}"
}
] | 121 | math |
numbertheory_lte_qr_train_0268 | 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^5. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0269 | 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^3, 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^3.\n\nPresent the answer in LaTex format: \\box... | 146991 | math |
numbertheory_lte_qr_train_0270 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=11. How many solutions m in [1, 2, ..., 11^1] satisfy m^3+1=0 (mod 11^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0271 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=2. How many solutions m in [1, 2, ..., 2^4] satisfy m^1+1=0 (mod 2^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0272 | 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^3, 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^3.\n\nPresent the answer in LaTex format: \\boxe... | 12240 | math |
numbertheory_lte_qr_train_0273 | 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. Find the least positive integer m such that m^4+(2) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0274 | 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^2, and that n^5+(-1) is also divisible by p^4. Find the smallest positive integer m such that m^5+(-1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\... | 1 | math |
numbertheory_lte_qr_train_0275 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=3. How many solutions m in [1, 2, ..., 3^4] satisfy m^1+1=0 (mod 3^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0276 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 3). For the least p where n^4=-1 (mod p^2) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1032 | math |
numbertheory_lte_qr_train_0277 | 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+(0) is divisible by p^6. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | math |
numbertheory_lte_qr_train_0278 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=41. How many solutions m in [1, 2, ..., 41^3] satisfy m^1+1=0 (mod 41^3)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0279 | 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: \\boxe... | 2178 | math |
numbertheory_lte_qr_train_0280 | 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^6. 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... | 5303 | math |
numbertheory_lte_qr_train_0281 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=7. How many solutions m in [1, 2, ..., 7^3] satisfy m^1+1=0 (mod 7^3)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0282 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "let p=13. How many solutions m in [1, 2, ..., 13^3] satisfy m^3+1=0 (mod 13^3)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
numbertheory_lte_qr_train_0283 | 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^6. Find the largest positive integer m such that m^2+(-1) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxe... | 63 | math |
numbertheory_lte_qr_train_0284 | 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^3. 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... | 6 | math |
numbertheory_lte_qr_train_0285 | 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^4. 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... | 58 | math |
numbertheory_lte_qr_train_0286 | numbertheory_lte_qr | null | [
{
"role": "user",
"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^2, and n^4+(2) is also a multiple of p^4. Find the largest positive integer m such that m^4+(2) is a multiple of p^2.\n\nPresent the answer in LaTex format: \\boxed{Y... | 47 | math |
numbertheory_lte_qr_train_0287 | 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^2, and that n^4+(1) is also divisible by p^4. Find the smallest positive integer m such that m^4+(1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\box... | 20051 | math |
numbertheory_lte_qr_train_0288 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 7). For the least p where n^4=-1 (mod p^3) has a solution, find the minimal m for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 576595 | math |
numbertheory_lte_qr_train_0289 | 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^5. Find p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 11 | math |
numbertheory_lte_qr_train_0290 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 4). 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}"
}
] | 124 | math |
numbertheory_lte_qr_train_0291 | 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^5, 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^5.\n\nPresent the answer in LaTex format: \\... | 23 | math |
numbertheory_lte_qr_train_0292 | 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^4, and n^2+(1) is also a multiple of p^5. Find the largest positive integer m such that m^2+(1) is a multiple of p^4.\n\nPresent the answer in LaTex format: \\boxed{Y... | 2057 | math |
numbertheory_lte_qr_train_0293 | 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+(0) is divisible by p^4. Find the least positive integer m such that m^3+(0) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
numbertheory_lte_qr_train_0294 | 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, and that n^2+(-1) is also divisible by p^4. Find the smallest positive integer m such that m^2+(-1) is divisible by p^3.\n\nPresent the answer in LaTex format: \\... | 1 | math |
numbertheory_lte_qr_train_0295 | 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 the least positive integer m such that m^6+(-1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
numbertheory_lte_qr_train_0296 | 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_0297 | 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: \\box... | 31 | math |
numbertheory_lte_qr_train_0298 | numbertheory_lte_qr | null | [
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 2). 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}"
}
] | 182 | math |
numbertheory_lte_qr_train_0299 | 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. Find the least positive integer m such that m^3+(-1) is divisible by p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | math |
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