id
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26
62
setting_key
stringclasses
41 values
messages
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stringclasses
1 value
omega_config
stringclasses
41 values
omega_split
stringclasses
2 values
numbertheory_qr_sum_test_out_032
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_033
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_034
numbertheory_qr_sum
[ { "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 reversing its digits are divisible by S. Let Q and R be the quotient and remainder, respectively, when M is divided by 1000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
432
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_035
numbertheory_qr_sum
[ { "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 reversing its digits are divisible by S. Let Q and R be the quotient and remainder, respectively, when M is divided by 100. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
96
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_036
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_037
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_038
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100005
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_039
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_040
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_041
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_042
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_043
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_044
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_045
numbertheory_qr_sum
[ { "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 reversing its digits are divisible by S. Let Q and R be the quotient and remainder, respectively, when M is divided by 100. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
96
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_046
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_047
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_048
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_049
numbertheory_qr_sum
[ { "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 reversing its digits are divisible by S. Let Q and R be the quotient and remainder, respectively, when M is divided by 100. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
96
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_050
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_051
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_052
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_053
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_054
numbertheory_qr_sum
[ { "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 reversing its digits are divisible by S. Let Q and R be the quotient and remainder, respectively, when M is divided by 10000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
8856
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_055
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_056
numbertheory_qr_sum
[ { "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 reversing its digits are divisible by S. Let Q and R be the quotient and remainder, respectively, when M is divided by 1000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
1008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_057
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100004
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_058
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_059
numbertheory_qr_sum
[ { "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 reversing its digits are divisible by S. Let Q and R be the quotient and remainder, respectively, when M is divided by 1000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
963
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_060
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_061
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_062
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_063
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_064
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_065
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_066
numbertheory_qr_sum
[ { "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 reversing its digits are divisible by S. Let Q and R be the quotient and remainder, respectively, when M is divided by 10000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
10008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_067
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_068
numbertheory_qr_sum
[ { "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 reversing its digits are divisible by S. Let Q and R be the quotient and remainder, respectively, when M is divided by 100. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
72
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_069
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_070
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_071
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_072
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_073
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_074
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100000
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_075
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_076
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_077
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_078
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_079
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_080
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_081
numbertheory_qr_sum
[ { "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 reversing its digits are divisible by S. Let Q and R be the quotient and remainder, respectively, when M is divided by 10000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" } ]
8856
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_082
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_083
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_084
numbertheory_qr_sum
[ { "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{Your answer}", "role": "user" } ]
100008
math
numbertheory_qr_sum
test_out
numbertheory_qr_sum_test_out_085
numbertheory_qr_sum
[ { "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
numbertheory_qr_sum
test_out