messages listlengths 1 1 | ground_truth stringlengths 2 6 | dataset stringclasses 1
value |
|---|---|---|
[
{
"content": "let N be the maximum 3-digit positive integer with the property that whenever one of its digits changes to 4, the resulting number is a multiple of 4. Let S be the LCM of the digits of N. let M be the largest 4-digit positive integer with the property that both the number and the number formed by ... | 432 | MATH |
[
{
"content": "let N be the tenth smallest 6-digit positive integer with the property that the sum of the digits of n is divisible by 11. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100109 | MATH |
[
{
"content": "let N be the biggest 6-digit positive integer with the property that whenever two of its digits changes to 9, the resulting number is a multiple of 11. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Y... | 100008 | MATH |
[
{
"content": "let N be the greatest 6-digit positive integer with the property that both the number and the number formed by reversing its digits are divisible by 6. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Y... | 100002 | MATH |
[
{
"content": "let N be the maximum 2-digit positive integer with the property that whenever one of its digits changes to 2, the resulting number is a multiple of 2. Let S be the LCM of the digits of N. let M be the largest 3-digit positive integer with the property that both the number and the number formed by ... | 90 | MATH |
[
{
"content": "let N be the maximum 3-digit positive integer with the property that whenever one of its digits changes to 4, the resulting number is a multiple of 2. Let S be the LCM of the digits of N. let M be the largest 4-digit positive integer with the property that both the number and the number formed by ... | 432 | MATH |
[
{
"content": "let N be the maximum 6-digit positive integer with the property that whenever one of its digits changes to 9, the resulting number is a multiple of 7. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the maximum 3-digit positive integer with the property that whenever one of its digits changes to 2, the resulting number is a multiple of 3. Let S be the LCM of the digits of N. let M be the largest 4-digit positive integer with the property that both the number and the number formed by ... | 896 | MATH |
[
{
"content": "let N be the maximum 6-digit positive integer with the property that whenever one of its digits changes to 6, the resulting number is a multiple of 6. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100005 | MATH |
[
{
"content": "let N be the maximum 6-digit positive integer with the property that whenever one of its digits changes to 5, the resulting number is a multiple of 5. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100004 | MATH |
[
{
"content": "let N be the maximum 3-digit positive integer with the property that whenever one of its digits changes to 4, the resulting number is a multiple of 3. Let S be the LCM of the digits of N. let M be the largest 4-digit positive integer with the property that both the number and the number formed by ... | 963 | MATH |
[
{
"content": "let N be the biggest 6-digit positive integer with the property that whenever two of its digits changes to 2, the resulting number is a multiple of 7. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the third greatest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 12. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999804 | MATH |
[
{
"content": "let N be the third smallest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 4. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100008 | MATH |
[
{
"content": "let N be the biggest 6-digit positive integer with the property that whenever two of its digits changes to 9, the resulting number is a multiple of 9. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the maximum 2-digit positive integer with the property that whenever one of its digits changes to 2, the resulting number is a multiple of 3. Let S be the LCM of the digits of N. let M be the largest 3-digit positive integer with the property that both the number and the number formed by ... | 68 | MATH |
[
{
"content": "let N be the tenth greatest 6-digit positive integer with the property that the sum of the digits of n is divisible by 11. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999890 | MATH |
[
{
"content": "let N be the maximum 6-digit positive integer with the property that whenever one of its digits changes to 2, the resulting number is a multiple of 7. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the tenth smallest 6-digit positive integer with the property that the sum of the digits of n is divisible by 12. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100128 | MATH |
[
{
"content": "let N be the third smallest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 6. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100020 | MATH |
[
{
"content": "let N be the third greatest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 3. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999993 | MATH |
[
{
"content": "let N be the maximum 2-digit positive integer with the property that whenever one of its digits changes to 3, the resulting number is a multiple of 3. Let S be the LCM of the digits of N. let M be the largest 3-digit positive integer with the property that both the number and the number formed by ... | 108 | MATH |
[
{
"content": "let N be the maximum 4-digit positive integer with the property that whenever one of its digits changes to 4, the resulting number is a multiple of 4. Let S be the LCM of the digits of N. let M be the largest 5-digit positive integer with the property that both the number and the number formed by ... | 8856 | MATH |
[
{
"content": "let N be the maximum 6-digit positive integer with the property that whenever one of its digits changes to 9, the resulting number is a multiple of 9. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the tenth smallest 6-digit positive integer with the property that the sum of the digits of n is divisible by 5. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100045 | MATH |
[
{
"content": "let N be the tenth greatest 6-digit positive integer with the property that the sum of the digits of n is divisible by 4. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999962 | MATH |
[
{
"content": "let N be the third greatest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 10. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999700 | MATH |
[
{
"content": "let N be the greatest 6-digit positive integer with the property that both the number and the number formed by reversing its digits are divisible by 7. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Y... | 100008 | MATH |
[
{
"content": "let N be the third smallest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 7. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 111181 | MATH |
[
{
"content": "let N be the least 6-digit positive integer with the property that whenever two of its digits changes to 9, the resulting number is a multiple of 11. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{You... | 100008 | MATH |
[
{
"content": "let N be the maximum 2-digit positive integer with the property that whenever one of its digits changes to 4, the resulting number is a multiple of 2. Let S be the LCM of the digits of N. let M be the largest 3-digit positive integer with the property that both the number and the number formed by ... | 90 | MATH |
[
{
"content": "let N be the tenth greatest 6-digit positive integer with the property that the sum of the digits of n is divisible by 8. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999922 | MATH |
[
{
"content": "let N be the biggest 6-digit positive integer with the property that whenever two of its digits changes to 9, the resulting number is a multiple of 3. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the third greatest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 11. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 777777 | MATH |
[
{
"content": "let N be the maximum 3-digit positive integer with the property that whenever one of its digits changes to 2, the resulting number is a multiple of 2. Let S be the LCM of the digits of N. let M be the largest 4-digit positive integer with the property that both the number and the number formed by ... | 432 | MATH |
[
{
"content": "let N be the maximum 2-digit positive integer with the property that whenever one of its digits changes to 4, the resulting number is a multiple of 3. Let S be the LCM of the digits of N. let M be the largest 3-digit positive integer with the property that both the number and the number formed by ... | 96 | MATH |
[
{
"content": "let N be the maximum 6-digit positive integer with the property that whenever one of its digits changes to 9, the resulting number is a multiple of 11. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Y... | 100008 | MATH |
[
{
"content": "let N be the third greatest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 9. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999981 | MATH |
[
{
"content": "let N be the biggest 6-digit positive integer with the property that whenever two of its digits changes to 6, the resulting number is a multiple of 6. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100005 | MATH |
[
{
"content": "let N be the third greatest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 6. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999960 | MATH |
[
{
"content": "let N be the biggest 6-digit positive integer with the property that whenever two of its digits changes to 9, the resulting number is a multiple of 7. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the third smallest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 5. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100050 | MATH |
[
{
"content": "let N be the third greatest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 4. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999880 | MATH |
[
{
"content": "let N be the third greatest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 5. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999905 | MATH |
[
{
"content": "let N be the tenth smallest 6-digit positive integer with the property that the sum of the digits of n is divisible by 9. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100089 | MATH |
[
{
"content": "let N be the maximum 2-digit positive integer with the property that whenever one of its digits changes to 4, the resulting number is a multiple of 4. Let S be the LCM of the digits of N. let M be the largest 3-digit positive integer with the property that both the number and the number formed by ... | 96 | MATH |
[
{
"content": "let N be the greatest 6-digit positive integer with the property that both the number and the number formed by reversing its digits are divisible by 13. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{... | 100008 | MATH |
[
{
"content": "let N be the biggest 6-digit positive integer with the property that whenever two of its digits changes to 6, the resulting number is a multiple of 3. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the tenth smallest 6-digit positive integer with the property that the sum of the digits of n is divisible by 10. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100090 | MATH |
[
{
"content": "let N be the maximum 2-digit positive integer with the property that whenever one of its digits changes to 1, the resulting number is a multiple of 3. Let S be the LCM of the digits of N. let M be the largest 3-digit positive integer with the property that both the number and the number formed by ... | 96 | MATH |
[
{
"content": "let N be the tenth greatest 6-digit positive integer with the property that the sum of the digits of n is divisible by 7. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999933 | MATH |
[
{
"content": "let N be the third smallest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 9. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100026 | MATH |
[
{
"content": "let N be the greatest 6-digit positive integer with the property that both the number and the number formed by reversing its digits are divisible by 11. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{... | 100008 | MATH |
[
{
"content": "let N be the greatest 6-digit positive integer with the property that both the number and the number formed by reversing its digits are divisible by 9. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Y... | 100008 | MATH |
[
{
"content": "let N be the maximum 4-digit positive integer with the property that whenever one of its digits changes to 4, the resulting number is a multiple of 2. Let S be the LCM of the digits of N. let M be the largest 5-digit positive integer with the property that both the number and the number formed by ... | 8856 | MATH |
[
{
"content": "let N be the maximum 6-digit positive integer with the property that whenever one of its digits changes to 6, the resulting number is a multiple of 3. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the maximum 3-digit positive integer with the property that whenever one of its digits changes to 3, the resulting number is a multiple of 3. Let S be the LCM of the digits of N. let M be the largest 4-digit positive integer with the property that both the number and the number formed by ... | 1008 | MATH |
[
{
"content": "let N be the biggest 6-digit positive integer with the property that whenever two of its digits changes to 5, the resulting number is a multiple of 5. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100004 | MATH |
[
{
"content": "let N be the tenth smallest 6-digit positive integer with the property that the sum of the digits of n is divisible by 6. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100056 | MATH |
[
{
"content": "let N be the maximum 3-digit positive integer with the property that whenever one of its digits changes to 1, the resulting number is a multiple of 3. Let S be the LCM of the digits of N. let M be the largest 4-digit positive integer with the property that both the number and the number formed by ... | 963 | MATH |
[
{
"content": "let N be the tenth greatest 6-digit positive integer with the property that the sum of the digits of n is divisible by 5. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999954 | MATH |
[
{
"content": "let N be the tenth greatest 6-digit positive integer with the property that the sum of the digits of n is divisible by 10. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999904 | MATH |
[
{
"content": "let N be the third greatest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 7. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999929 | MATH |
[
{
"content": "let N be the tenth smallest 6-digit positive integer with the property that the sum of the digits of n is divisible by 7. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100067 | MATH |
[
{
"content": "let N be the tenth smallest 6-digit positive integer with the property that the sum of the digits of n is divisible by 3. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100029 | MATH |
[
{
"content": "let N be the maximum 6-digit positive integer with the property that whenever one of its digits changes to 3, the resulting number is a multiple of 3. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the maximum 4-digit positive integer with the property that whenever one of its digits changes to 3, the resulting number is a multiple of 3. Let S be the LCM of the digits of N. let M be the largest 5-digit positive integer with the property that both the number and the number formed by ... | 10008 | MATH |
[
{
"content": "let N be the tenth smallest 6-digit positive integer with the property that the sum of the digits of n is divisible by 4. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100038 | MATH |
[
{
"content": "let N be the maximum 2-digit positive integer with the property that whenever one of its digits changes to 2, the resulting number is a multiple of 4. Let S be the LCM of the digits of N. let M be the largest 3-digit positive integer with the property that both the number and the number formed by ... | 72 | MATH |
[
{
"content": "let N be the maximum 6-digit positive integer with the property that whenever one of its digits changes to 9, the resulting number is a multiple of 3. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the third smallest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 10. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100200 | MATH |
[
{
"content": "let N be the tenth smallest 6-digit positive integer with the property that the sum of the digits of n is divisible by 8. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100078 | MATH |
[
{
"content": "let N be the third smallest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 3. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100008 | MATH |
[
{
"content": "let N be the third greatest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 8. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 998808 | MATH |
[
{
"content": "let N be the greatest 6-digit positive integer with the property that both the number and the number formed by reversing its digits are divisible by 5. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Y... | 100000 | MATH |
[
{
"content": "let N be the tenth greatest 6-digit positive integer with the property that the sum of the digits of n is divisible by 12. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999885 | MATH |
[
{
"content": "let N be the tenth greatest 6-digit positive integer with the property that the sum of the digits of n is divisible by 3. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999972 | MATH |
[
{
"content": "let N be the third smallest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 11. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 333333 | MATH |
[
{
"content": "let N be the greatest 6-digit positive integer with the property that both the number and the number formed by reversing its digits are divisible by 3. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Y... | 100008 | MATH |
[
{
"content": "let N be the smallest 6-digit positive integer with the property that whenever one of its digits changes to 9, the resulting number is a multiple of 11. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{... | 100008 | MATH |
[
{
"content": "let N be the tenth greatest 6-digit positive integer with the property that the sum of the digits of n is divisible by 6. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999942 | MATH |
[
{
"content": "let N be the maximum 4-digit positive integer with the property that whenever one of its digits changes to 2, the resulting number is a multiple of 2. Let S be the LCM of the digits of N. let M be the largest 5-digit positive integer with the property that both the number and the number formed by ... | 8856 | MATH |
[
{
"content": "let N be the third smallest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 12. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100080 | MATH |
[
{
"content": "let N be the third smallest 6-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 8. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100080 | MATH |
[
{
"content": "let N be the biggest 6-digit positive integer with the property that whenever two of its digits changes to 3, the resulting number is a multiple of 3. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 100008 | MATH |
[
{
"content": "let N be the tenth greatest 6-digit positive integer with the property that the sum of the digits of n is divisible by 9. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 999918 | MATH |
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