token_ids stringlengths 48 56.6k | text stringlengths 332 50.2k | stop_token_id int64 271 4.71k | stop_token_text stringclasses 8
values | stop_token_idx int64 10 13.1k |
|---|---|---|---|---|
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382 | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 47 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 176 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 183 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 268 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 3,593 | ).
| 365 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,939 | ?
| 374 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 535 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 666 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,447 | :
| 919 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,447 | :
| 943 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 271 | 993 | |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 271 | 1,030 | |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 271 | 1,057 | |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 271 | 1,083 | |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 692 | )
| 1,192 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 1,228 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 1,325 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 1,345 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 1,382 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,939 | ?
| 1,488 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,447 | :
| 1,492 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 1,521 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 1,877 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,939 | ?
| 1,931 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 2,059 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,939 | ?
| 2,129 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 2,354 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 3,593 | ).
| 2,392 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 2,569 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,939 | ?
| 2,640 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 2,700 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 2,932 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 3,113 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 3,231 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 3,384 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 3,538 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 3,579 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 3,594 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,447 | :
| 3,735 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 3,764 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 3,777 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 3,839 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 3,878 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 3,915 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 4,082 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 4,198 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,939 | ?
| 4,230 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,447 | :
| 4,247 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,447 | :
| 4,252 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 692 | )
| 4,277 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 692 | )
| 4,303 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 692 | )
| 4,327 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 692 | )
| 4,353 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 692 | )
| 4,379 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 692 | )
| 4,410 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 692 | )
| 4,441 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 692 | )
| 4,466 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 4,568 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 4,618 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 4,702 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 4,878 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 5,024 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 3,593 | ).
| 5,098 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 5,145 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,939 | ?
| 5,220 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 5,270 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 3,593 | ).
| 5,309 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 5,322 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 5,381 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 5,470 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 5,688 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,939 | ?
| 5,728 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 5,835 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,051 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,939 | ?
| 6,056 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,154 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,447 | :
| 6,362 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,371 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,381 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,395 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,407 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,428 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,496 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,519 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,534 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,566 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,572 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 1,447 | :
| 6,576 |
71486,11,773,358,1184,311,7071,700,1246,1657,24438,5109,1948,220,15,323,220,16,614,264,1985,315,279,63533,323,46912,320,9309,5326,304,15457,3793,8,6144,311,220,17,15,52962,13,6771,752,1191,553,8660,279,3491,382,5338,11,264,24438,1372,1948,220,15,323,220,16,646,387,5326,438,264,19419,1124,11520,37018,90,64,15170,65,1103... | <｜begin▁of▁sentence｜><｜User｜>Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product?
Please reason step by step, and put your final answer within \boxed{... | 382 | .
| 6,647 |
32313,11,1077,752,1430,311,7071,419,700,13,576,3405,374,10161,421,1667,279,17702,11331,564,4110,258,47311,311,44868,7437,2075,8,979,760,87,91,374,2686,1091,220,15,13,22,3059,304,458,1465,2686,1091,220,15,13,15,15,15,16,13,576,4226,1265,387,3007,476,3557,13,4710 | <｜begin▁of▁sentence｜><｜User｜>If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>
Okay, let me try to figure this out. The... | 4,710 | 61 | |
32313,11,1077,752,1430,311,7071,419,700,13,576,3405,374,10161,421,1667,279,17702,11331,564,4110,258,47311,311,44868,7437,2075,8,979,760,87,91,374,2686,1091,220,15,13,22,3059,304,458,1465,2686,1091,220,15,13,15,15,15,16,13,576,4226,1265,387,3007,476,3557,13,4710,5338,11,358,1184,311,6099,1246,11331,564,4110,258,4013,975... | <｜begin▁of▁sentence｜><｜User｜>If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>
Okay, let me try to figure this out. The... | 1,447 | :
| 106 |
32313,11,1077,752,1430,311,7071,419,700,13,576,3405,374,10161,421,1667,279,17702,11331,564,4110,258,47311,311,44868,7437,2075,8,979,760,87,91,374,2686,1091,220,15,13,22,3059,304,458,1465,2686,1091,220,15,13,15,15,15,16,13,576,4226,1265,387,3007,476,3557,13,4710,5338,11,358,1184,311,6099,1246,11331,564,4110,258,4013,975... | <｜begin▁of▁sentence｜><｜User｜>If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>
Okay, let me try to figure this out. The... | 4,710 | 191 | |
32313,11,1077,752,1430,311,7071,419,700,13,576,3405,374,10161,421,1667,279,17702,11331,564,4110,258,47311,311,44868,7437,2075,8,979,760,87,91,374,2686,1091,220,15,13,22,3059,304,458,1465,2686,1091,220,15,13,15,15,15,16,13,576,4226,1265,387,3007,476,3557,13,4710,5338,11,358,1184,311,6099,1246,11331,564,4110,258,4013,975... | <｜begin▁of▁sentence｜><｜User｜>If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>
Okay, let me try to figure this out. The... | 1,447 | :
| 212 |
32313,11,1077,752,1430,311,7071,419,700,13,576,3405,374,10161,421,1667,279,17702,11331,564,4110,258,47311,311,44868,7437,2075,8,979,760,87,91,374,2686,1091,220,15,13,22,3059,304,458,1465,2686,1091,220,15,13,15,15,15,16,13,576,4226,1265,387,3007,476,3557,13,4710,5338,11,358,1184,311,6099,1246,11331,564,4110,258,4013,975... | <｜begin▁of▁sentence｜><｜User｜>If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>
Okay, let me try to figure this out. The... | 271 | 235 | |
32313,11,1077,752,1430,311,7071,419,700,13,576,3405,374,10161,421,1667,279,17702,11331,564,4110,258,47311,311,44868,7437,2075,8,979,760,87,91,374,2686,1091,220,15,13,22,3059,304,458,1465,2686,1091,220,15,13,15,15,15,16,13,576,4226,1265,387,3007,476,3557,13,4710,5338,11,358,1184,311,6099,1246,11331,564,4110,258,4013,975... | <｜begin▁of▁sentence｜><｜User｜>If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>
Okay, let me try to figure this out. The... | 4,710 | 322 | |
32313,11,1077,752,1430,311,7071,419,700,13,576,3405,374,10161,421,1667,279,17702,11331,564,4110,258,47311,311,44868,7437,2075,8,979,760,87,91,374,2686,1091,220,15,13,22,3059,304,458,1465,2686,1091,220,15,13,15,15,15,16,13,576,4226,1265,387,3007,476,3557,13,4710,5338,11,358,1184,311,6099,1246,11331,564,4110,258,4013,975... | <｜begin▁of▁sentence｜><｜User｜>If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>
Okay, let me try to figure this out. The... | 1,447 | :
| 333 |
32313,11,1077,752,1430,311,7071,419,700,13,576,3405,374,10161,421,1667,279,17702,11331,564,4110,258,47311,311,44868,7437,2075,8,979,760,87,91,374,2686,1091,220,15,13,22,3059,304,458,1465,2686,1091,220,15,13,15,15,15,16,13,576,4226,1265,387,3007,476,3557,13,4710,5338,11,358,1184,311,6099,1246,11331,564,4110,258,4013,975... | <｜begin▁of▁sentence｜><｜User｜>If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>
Okay, let me try to figure this out. The... | 692 | )
| 340 |
32313,11,1077,752,1430,311,7071,419,700,13,576,3405,374,10161,421,1667,279,17702,11331,564,4110,258,47311,311,44868,7437,2075,8,979,760,87,91,374,2686,1091,220,15,13,22,3059,304,458,1465,2686,1091,220,15,13,15,15,15,16,13,576,4226,1265,387,3007,476,3557,13,4710,5338,11,358,1184,311,6099,1246,11331,564,4110,258,4013,975... | <｜begin▁of▁sentence｜><｜User｜>If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>
Okay, let me try to figure this out. The... | 692 | )
| 347 |
32313,11,1077,752,1430,311,7071,419,700,13,576,3405,374,10161,421,1667,279,17702,11331,564,4110,258,47311,311,44868,7437,2075,8,979,760,87,91,374,2686,1091,220,15,13,22,3059,304,458,1465,2686,1091,220,15,13,15,15,15,16,13,576,4226,1265,387,3007,476,3557,13,4710,5338,11,358,1184,311,6099,1246,11331,564,4110,258,4013,975... | <｜begin▁of▁sentence｜><｜User｜>If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>
Okay, let me try to figure this out. The... | 692 | )
| 354 |
32313,11,1077,752,1430,311,7071,419,700,13,576,3405,374,10161,421,1667,279,17702,11331,564,4110,258,47311,311,44868,7437,2075,8,979,760,87,91,374,2686,1091,220,15,13,22,3059,304,458,1465,2686,1091,220,15,13,15,15,15,16,13,576,4226,1265,387,3007,476,3557,13,4710,5338,11,358,1184,311,6099,1246,11331,564,4110,258,4013,975... | <｜begin▁of▁sentence｜><｜User｜>If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.
Please reason step by step, and put your final answer within \boxed{}.<｜Assistant｜><think>
Okay, let me try to figure this out. The... | 692 | )
| 360 |
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