# Factorial Foundations This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed. This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately. batch-20260926-factorial-foundations · 2026-09-26 · 100 questions This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt. ## 1. Exact Factorial Value Compute n! for the natural number n=0; give the answer as a decimal integer string. View answer and explanation1 0!=1, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0001/) Factorial · qf-factorial-exact-evaluation · candidate ## 2. Exact Factorial Value Compute n! for the natural number n=1; give the answer as a decimal integer string. View answer and explanation1 1!=1, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0002/) Factorial · qf-factorial-exact-evaluation · candidate ## 3. Exact Factorial Value Compute n! for the natural number n=2; give the answer as a decimal integer string. View answer and explanation2 2!=2, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0003/) Factorial · qf-factorial-exact-evaluation · candidate ## 4. Exact Factorial Value Compute n! for the natural number n=3; give the answer as a decimal integer string. View answer and explanation6 3!=6, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0004/) Factorial · qf-factorial-exact-evaluation · candidate ## 5. Exact Factorial Value Compute n! for the natural number n=4; give the answer as a decimal integer string. View answer and explanation24 4!=24, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0005/) Factorial · qf-factorial-exact-evaluation · candidate ## 6. Exact Factorial Value Compute n! for the natural number n=5; give the answer as a decimal integer string. View answer and explanation120 5!=120, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0006/) Factorial · qf-factorial-exact-evaluation · candidate ## 7. Exact Factorial Value Compute n! for the natural number n=6; give the answer as a decimal integer string. View answer and explanation720 6!=720, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0007/) Factorial · qf-factorial-exact-evaluation · candidate ## 8. Exact Factorial Value Compute n! for the natural number n=7; give the answer as a decimal integer string. View answer and explanation5040 7!=5040, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0008/) Factorial · qf-factorial-exact-evaluation · candidate ## 9. Exact Factorial Value Compute n! for the natural number n=8; give the answer as a decimal integer string. View answer and explanation40320 8!=40320, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0009/) Factorial · qf-factorial-exact-evaluation · candidate ## 10. Exact Factorial Value Compute n! for the natural number n=9; give the answer as a decimal integer string. View answer and explanation362880 9!=362880, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0010/) Factorial · qf-factorial-exact-evaluation · candidate ## 11. Exact Factorial Value Compute n! for the natural number n=10; give the answer as a decimal integer string. View answer and explanation3628800 10!=3628800, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0011/) Factorial · qf-factorial-exact-evaluation · candidate ## 12. Exact Factorial Value Compute n! for the natural number n=11; give the answer as a decimal integer string. View answer and explanation39916800 11!=39916800, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0012/) Factorial · qf-factorial-exact-evaluation · candidate ## 13. Exact Factorial Value Compute n! for the natural number n=12; give the answer as a decimal integer string. View answer and explanation479001600 12!=479001600, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0013/) Factorial · qf-factorial-exact-evaluation · candidate ## 14. Exact Factorial Value Compute n! for the natural number n=13; give the answer as a decimal integer string. View answer and explanation6227020800 13!=6227020800, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0014/) Factorial · qf-factorial-exact-evaluation · candidate ## 15. Exact Factorial Value Compute n! for the natural number n=14; give the answer as a decimal integer string. View answer and explanation87178291200 14!=87178291200, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0015/) Factorial · qf-factorial-exact-evaluation · candidate ## 16. Exact Factorial Value Compute n! for the natural number n=15; give the answer as a decimal integer string. View answer and explanation1307674368000 15!=1307674368000, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0016/) Factorial · qf-factorial-exact-evaluation · candidate ## 17. Exact Factorial Value Compute n! for the natural number n=16; give the answer as a decimal integer string. View answer and explanation20922789888000 16!=20922789888000, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0017/) Factorial · qf-factorial-exact-evaluation · candidate ## 18. Exact Factorial Value Compute n! for the natural number n=17; give the answer as a decimal integer string. View answer and explanation355687428096000 17!=355687428096000, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0018/) Factorial · qf-factorial-exact-evaluation · candidate ## 19. Exact Factorial Value Compute n! for the natural number n=18; give the answer as a decimal integer string. View answer and explanation6402373705728000 18!=6402373705728000, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0019/) Factorial · qf-factorial-exact-evaluation · candidate ## 20. Exact Factorial Value Compute n! for the natural number n=19; give the answer as a decimal integer string. View answer and explanation121645100408832000 19!=121645100408832000, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0020/) Factorial · qf-factorial-exact-evaluation · candidate ## 21. Exact Factorial Value Compute n! for the natural number n=20; give the answer as a decimal integer string. View answer and explanation2432902008176640000 20!=2432902008176640000, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0021/) Factorial · qf-factorial-exact-evaluation · candidate ## 22. Exact Factorial Value Compute n! for the natural number n=21; give the answer as a decimal integer string. View answer and explanation51090942171709440000 21!=51090942171709440000, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0022/) Factorial · qf-factorial-exact-evaluation · candidate ## 23. Exact Factorial Value Compute n! for the natural number n=22; give the answer as a decimal integer string. View answer and explanation1124000727777607680000 22!=1124000727777607680000, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0023/) Factorial · qf-factorial-exact-evaluation · candidate ## 24. Exact Factorial Value Compute n! for the natural number n=23; give the answer as a decimal integer string. View answer and explanation25852016738884976640000 23!=25852016738884976640000, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0024/) Factorial · qf-factorial-exact-evaluation · candidate ## 25. Exact Factorial Value Compute n! for the natural number n=24; give the answer as a decimal integer string. View answer and explanation620448401733239439360000 24!=620448401733239439360000, computed exactly with arbitrary-precision integer arithmetic. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-exact-evaluation-0025/) Factorial · qf-factorial-exact-evaluation · candidate ## 26. Successor Recurrence for Factorial Given 0!=1, use (n+1)!=(n+1)×n! to find 1!. View answer and explanation1 1×1=1, so 1!=1. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0001/) Factorial · qf-factorial-successor-recurrence · candidate ## 27. Successor Recurrence for Factorial Given 1!=1, use (n+1)!=(n+1)×n! to find 2!. View answer and explanation2 2×1=2, so 2!=2. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0002/) Factorial · qf-factorial-successor-recurrence · candidate ## 28. Successor Recurrence for Factorial Given 2!=2, use (n+1)!=(n+1)×n! to find 3!. View answer and explanation6 3×2=6, so 3!=6. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0003/) Factorial · qf-factorial-successor-recurrence · candidate ## 29. Successor Recurrence for Factorial Given 3!=6, use (n+1)!=(n+1)×n! to find 4!. View answer and explanation24 4×6=24, so 4!=24. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0004/) Factorial · qf-factorial-successor-recurrence · candidate ## 30. Successor Recurrence for Factorial Given 4!=24, use (n+1)!=(n+1)×n! to find 5!. View answer and explanation120 5×24=120, so 5!=120. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0005/) Factorial · qf-factorial-successor-recurrence · candidate ## 31. Successor Recurrence for Factorial Given 5!=120, use (n+1)!=(n+1)×n! to find 6!. View answer and explanation720 6×120=720, so 6!=720. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0006/) Factorial · qf-factorial-successor-recurrence · candidate ## 32. Successor Recurrence for Factorial Given 6!=720, use (n+1)!=(n+1)×n! to find 7!. View answer and explanation5040 7×720=5040, so 7!=5040. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0007/) Factorial · qf-factorial-successor-recurrence · candidate ## 33. Successor Recurrence for Factorial Given 7!=5040, use (n+1)!=(n+1)×n! to find 8!. View answer and explanation40320 8×5040=40320, so 8!=40320. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0008/) Factorial · qf-factorial-successor-recurrence · candidate ## 34. Successor Recurrence for Factorial Given 8!=40320, use (n+1)!=(n+1)×n! to find 9!. View answer and explanation362880 9×40320=362880, so 9!=362880. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0009/) Factorial · qf-factorial-successor-recurrence · candidate ## 35. Successor Recurrence for Factorial Given 9!=362880, use (n+1)!=(n+1)×n! to find 10!. View answer and explanation3628800 10×362880=3628800, so 10!=3628800. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0010/) Factorial · qf-factorial-successor-recurrence · candidate ## 36. Successor Recurrence for Factorial Given 10!=3628800, use (n+1)!=(n+1)×n! to find 11!. View answer and explanation39916800 11×3628800=39916800, so 11!=39916800. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0011/) Factorial · qf-factorial-successor-recurrence · candidate ## 37. Successor Recurrence for Factorial Given 11!=39916800, use (n+1)!=(n+1)×n! to find 12!. View answer and explanation479001600 12×39916800=479001600, so 12!=479001600. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0012/) Factorial · qf-factorial-successor-recurrence · candidate ## 38. Successor Recurrence for Factorial Given 12!=479001600, use (n+1)!=(n+1)×n! to find 13!. View answer and explanation6227020800 13×479001600=6227020800, so 13!=6227020800. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0013/) Factorial · qf-factorial-successor-recurrence · candidate ## 39. Successor Recurrence for Factorial Given 13!=6227020800, use (n+1)!=(n+1)×n! to find 14!. View answer and explanation87178291200 14×6227020800=87178291200, so 14!=87178291200. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0014/) Factorial · qf-factorial-successor-recurrence · candidate ## 40. Successor Recurrence for Factorial Given 14!=87178291200, use (n+1)!=(n+1)×n! to find 15!. View answer and explanation1307674368000 15×87178291200=1307674368000, so 15!=1307674368000. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0015/) Factorial · qf-factorial-successor-recurrence · candidate ## 41. Successor Recurrence for Factorial Given 15!=1307674368000, use (n+1)!=(n+1)×n! to find 16!. View answer and explanation20922789888000 16×1307674368000=20922789888000, so 16!=20922789888000. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0016/) Factorial · qf-factorial-successor-recurrence · candidate ## 42. Successor Recurrence for Factorial Given 16!=20922789888000, use (n+1)!=(n+1)×n! to find 17!. View answer and explanation355687428096000 17×20922789888000=355687428096000, so 17!=355687428096000. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0017/) Factorial · qf-factorial-successor-recurrence · candidate ## 43. Successor Recurrence for Factorial Given 17!=355687428096000, use (n+1)!=(n+1)×n! to find 18!. View answer and explanation6402373705728000 18×355687428096000=6402373705728000, so 18!=6402373705728000. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0018/) Factorial · qf-factorial-successor-recurrence · candidate ## 44. Successor Recurrence for Factorial Given 18!=6402373705728000, use (n+1)!=(n+1)×n! to find 19!. View answer and explanation121645100408832000 19×6402373705728000=121645100408832000, so 19!=121645100408832000. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0019/) Factorial · qf-factorial-successor-recurrence · candidate ## 45. Successor Recurrence for Factorial Given 19!=121645100408832000, use (n+1)!=(n+1)×n! to find 20!. View answer and explanation2432902008176640000 20×121645100408832000=2432902008176640000, so 20!=2432902008176640000. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0020/) Factorial · qf-factorial-successor-recurrence · candidate ## 46. Successor Recurrence for Factorial Given 20!=2432902008176640000, use (n+1)!=(n+1)×n! to find 21!. View answer and explanation51090942171709440000 21×2432902008176640000=51090942171709440000, so 21!=51090942171709440000. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0021/) Factorial · qf-factorial-successor-recurrence · candidate ## 47. Successor Recurrence for Factorial Given 21!=51090942171709440000, use (n+1)!=(n+1)×n! to find 22!. View answer and explanation1124000727777607680000 22×51090942171709440000=1124000727777607680000, so 22!=1124000727777607680000. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0022/) Factorial · qf-factorial-successor-recurrence · candidate ## 48. Successor Recurrence for Factorial Given 22!=1124000727777607680000, use (n+1)!=(n+1)×n! to find 23!. View answer and explanation25852016738884976640000 23×1124000727777607680000=25852016738884976640000, so 23!=25852016738884976640000. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0023/) Factorial · qf-factorial-successor-recurrence · candidate ## 49. Successor Recurrence for Factorial Given 23!=25852016738884976640000, use (n+1)!=(n+1)×n! to find 24!. View answer and explanation620448401733239439360000 24×25852016738884976640000=620448401733239439360000, so 24!=620448401733239439360000. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0024/) Factorial · qf-factorial-successor-recurrence · candidate ## 50. Successor Recurrence for Factorial Given 24!=620448401733239439360000, use (n+1)!=(n+1)×n! to find 25!. View answer and explanation15511210043330985984000000 25×620448401733239439360000=15511210043330985984000000, so 25!=15511210043330985984000000. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-successor-recurrence-0025/) Factorial · qf-factorial-successor-recurrence · candidate ## 51. Quotient of Consecutive Factorials Compute 1!/0!; the denominator is explicitly positive. View answer and explanation1 1/1=1, using 1!=1×0!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0001/) Factorial · qf-factorial-consecutive-quotient · candidate ## 52. Quotient of Consecutive Factorials Compute 2!/1!; the denominator is explicitly positive. View answer and explanation2 2/1=2, using 2!=2×1!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0002/) Factorial · qf-factorial-consecutive-quotient · candidate ## 53. Quotient of Consecutive Factorials Compute 3!/2!; the denominator is explicitly positive. View answer and explanation3 6/2=3, using 3!=3×2!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0003/) Factorial · qf-factorial-consecutive-quotient · candidate ## 54. Quotient of Consecutive Factorials Compute 4!/3!; the denominator is explicitly positive. View answer and explanation4 24/6=4, using 4!=4×3!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0004/) Factorial · qf-factorial-consecutive-quotient · candidate ## 55. Quotient of Consecutive Factorials Compute 5!/4!; the denominator is explicitly positive. View answer and explanation5 120/24=5, using 5!=5×4!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0005/) Factorial · qf-factorial-consecutive-quotient · candidate ## 56. Quotient of Consecutive Factorials Compute 6!/5!; the denominator is explicitly positive. View answer and explanation6 720/120=6, using 6!=6×5!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0006/) Factorial · qf-factorial-consecutive-quotient · candidate ## 57. Quotient of Consecutive Factorials Compute 7!/6!; the denominator is explicitly positive. View answer and explanation7 5040/720=7, using 7!=7×6!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0007/) Factorial · qf-factorial-consecutive-quotient · candidate ## 58. Quotient of Consecutive Factorials Compute 8!/7!; the denominator is explicitly positive. View answer and explanation8 40320/5040=8, using 8!=8×7!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0008/) Factorial · qf-factorial-consecutive-quotient · candidate ## 59. Quotient of Consecutive Factorials Compute 9!/8!; the denominator is explicitly positive. View answer and explanation9 362880/40320=9, using 9!=9×8!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0009/) Factorial · qf-factorial-consecutive-quotient · candidate ## 60. Quotient of Consecutive Factorials Compute 10!/9!; the denominator is explicitly positive. View answer and explanation10 3628800/362880=10, using 10!=10×9!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0010/) Factorial · qf-factorial-consecutive-quotient · candidate ## 61. Quotient of Consecutive Factorials Compute 11!/10!; the denominator is explicitly positive. View answer and explanation11 39916800/3628800=11, using 11!=11×10!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0011/) Factorial · qf-factorial-consecutive-quotient · candidate ## 62. Quotient of Consecutive Factorials Compute 12!/11!; the denominator is explicitly positive. View answer and explanation12 479001600/39916800=12, using 12!=12×11!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0012/) Factorial · qf-factorial-consecutive-quotient · candidate ## 63. Quotient of Consecutive Factorials Compute 13!/12!; the denominator is explicitly positive. View answer and explanation13 6227020800/479001600=13, using 13!=13×12!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0013/) Factorial · qf-factorial-consecutive-quotient · candidate ## 64. Quotient of Consecutive Factorials Compute 14!/13!; the denominator is explicitly positive. View answer and explanation14 87178291200/6227020800=14, using 14!=14×13!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0014/) Factorial · qf-factorial-consecutive-quotient · candidate ## 65. Quotient of Consecutive Factorials Compute 15!/14!; the denominator is explicitly positive. View answer and explanation15 1307674368000/87178291200=15, using 15!=15×14!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0015/) Factorial · qf-factorial-consecutive-quotient · candidate ## 66. Quotient of Consecutive Factorials Compute 16!/15!; the denominator is explicitly positive. View answer and explanation16 20922789888000/1307674368000=16, using 16!=16×15!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0016/) Factorial · qf-factorial-consecutive-quotient · candidate ## 67. Quotient of Consecutive Factorials Compute 17!/16!; the denominator is explicitly positive. View answer and explanation17 355687428096000/20922789888000=17, using 17!=17×16!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0017/) Factorial · qf-factorial-consecutive-quotient · candidate ## 68. Quotient of Consecutive Factorials Compute 18!/17!; the denominator is explicitly positive. View answer and explanation18 6402373705728000/355687428096000=18, using 18!=18×17!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0018/) Factorial · qf-factorial-consecutive-quotient · candidate ## 69. Quotient of Consecutive Factorials Compute 19!/18!; the denominator is explicitly positive. View answer and explanation19 121645100408832000/6402373705728000=19, using 19!=19×18!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0019/) Factorial · qf-factorial-consecutive-quotient · candidate ## 70. Quotient of Consecutive Factorials Compute 20!/19!; the denominator is explicitly positive. View answer and explanation20 2432902008176640000/121645100408832000=20, using 20!=20×19!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0020/) Factorial · qf-factorial-consecutive-quotient · candidate ## 71. Quotient of Consecutive Factorials Compute 21!/20!; the denominator is explicitly positive. View answer and explanation21 51090942171709440000/2432902008176640000=21, using 21!=21×20!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0021/) Factorial · qf-factorial-consecutive-quotient · candidate ## 72. Quotient of Consecutive Factorials Compute 22!/21!; the denominator is explicitly positive. View answer and explanation22 1124000727777607680000/51090942171709440000=22, using 22!=22×21!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0022/) Factorial · qf-factorial-consecutive-quotient · candidate ## 73. Quotient of Consecutive Factorials Compute 23!/22!; the denominator is explicitly positive. View answer and explanation23 25852016738884976640000/1124000727777607680000=23, using 23!=23×22!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0023/) Factorial · qf-factorial-consecutive-quotient · candidate ## 74. Quotient of Consecutive Factorials Compute 24!/23!; the denominator is explicitly positive. View answer and explanation24 620448401733239439360000/25852016738884976640000=24, using 24!=24×23!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0024/) Factorial · qf-factorial-consecutive-quotient · candidate ## 75. Quotient of Consecutive Factorials Compute 25!/24!; the denominator is explicitly positive. View answer and explanation25 15511210043330985984000000/620448401733239439360000=25, using 25!=25×24!. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-consecutive-quotient-0025/) Factorial · qf-factorial-consecutive-quotient · candidate ## 76. Missing Factor in the Successor Recurrence Fill the integer □ in 1!=□×0!, where 0!=1. View answer and explanation1 1= 1×1, so □=1. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0001/) Factorial · qf-factorial-missing-successor-factor · candidate ## 77. Missing Factor in the Successor Recurrence Fill the integer □ in 2!=□×1!, where 1!=1. View answer and explanation2 2= 2×1, so □=2. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0002/) Factorial · qf-factorial-missing-successor-factor · candidate ## 78. Missing Factor in the Successor Recurrence Fill the integer □ in 3!=□×2!, where 2!=2. View answer and explanation3 6= 3×2, so □=3. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0003/) Factorial · qf-factorial-missing-successor-factor · candidate ## 79. Missing Factor in the Successor Recurrence Fill the integer □ in 4!=□×3!, where 3!=6. View answer and explanation4 24= 4×6, so □=4. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0004/) Factorial · qf-factorial-missing-successor-factor · candidate ## 80. Missing Factor in the Successor Recurrence Fill the integer □ in 5!=□×4!, where 4!=24. View answer and explanation5 120= 5×24, so □=5. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0005/) Factorial · qf-factorial-missing-successor-factor · candidate ## 81. Missing Factor in the Successor Recurrence Fill the integer □ in 6!=□×5!, where 5!=120. View answer and explanation6 720= 6×120, so □=6. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0006/) Factorial · qf-factorial-missing-successor-factor · candidate ## 82. Missing Factor in the Successor Recurrence Fill the integer □ in 7!=□×6!, where 6!=720. View answer and explanation7 5040= 7×720, so □=7. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0007/) Factorial · qf-factorial-missing-successor-factor · candidate ## 83. Missing Factor in the Successor Recurrence Fill the integer □ in 8!=□×7!, where 7!=5040. View answer and explanation8 40320= 8×5040, so □=8. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0008/) Factorial · qf-factorial-missing-successor-factor · candidate ## 84. Missing Factor in the Successor Recurrence Fill the integer □ in 9!=□×8!, where 8!=40320. View answer and explanation9 362880= 9×40320, so □=9. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0009/) Factorial · qf-factorial-missing-successor-factor · candidate ## 85. Missing Factor in the Successor Recurrence Fill the integer □ in 10!=□×9!, where 9!=362880. View answer and explanation10 3628800= 10×362880, so □=10. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0010/) Factorial · qf-factorial-missing-successor-factor · candidate ## 86. Missing Factor in the Successor Recurrence Fill the integer □ in 11!=□×10!, where 10!=3628800. View answer and explanation11 39916800= 11×3628800, so □=11. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0011/) Factorial · qf-factorial-missing-successor-factor · candidate ## 87. Missing Factor in the Successor Recurrence Fill the integer □ in 12!=□×11!, where 11!=39916800. View answer and explanation12 479001600= 12×39916800, so □=12. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0012/) Factorial · qf-factorial-missing-successor-factor · candidate ## 88. Missing Factor in the Successor Recurrence Fill the integer □ in 13!=□×12!, where 12!=479001600. View answer and explanation13 6227020800= 13×479001600, so □=13. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0013/) Factorial · qf-factorial-missing-successor-factor · candidate ## 89. Missing Factor in the Successor Recurrence Fill the integer □ in 14!=□×13!, where 13!=6227020800. View answer and explanation14 87178291200= 14×6227020800, so □=14. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0014/) Factorial · qf-factorial-missing-successor-factor · candidate ## 90. Missing Factor in the Successor Recurrence Fill the integer □ in 15!=□×14!, where 14!=87178291200. View answer and explanation15 1307674368000= 15×87178291200, so □=15. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0015/) Factorial · qf-factorial-missing-successor-factor · candidate ## 91. Missing Factor in the Successor Recurrence Fill the integer □ in 16!=□×15!, where 15!=1307674368000. View answer and explanation16 20922789888000= 16×1307674368000, so □=16. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0016/) Factorial · qf-factorial-missing-successor-factor · candidate ## 92. Missing Factor in the Successor Recurrence Fill the integer □ in 17!=□×16!, where 16!=20922789888000. View answer and explanation17 355687428096000= 17×20922789888000, so □=17. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0017/) Factorial · qf-factorial-missing-successor-factor · candidate ## 93. Missing Factor in the Successor Recurrence Fill the integer □ in 18!=□×17!, where 17!=355687428096000. View answer and explanation18 6402373705728000= 18×355687428096000, so □=18. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0018/) Factorial · qf-factorial-missing-successor-factor · candidate ## 94. Missing Factor in the Successor Recurrence Fill the integer □ in 19!=□×18!, where 18!=6402373705728000. View answer and explanation19 121645100408832000= 19×6402373705728000, so □=19. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0019/) Factorial · qf-factorial-missing-successor-factor · candidate ## 95. Missing Factor in the Successor Recurrence Fill the integer □ in 20!=□×19!, where 19!=121645100408832000. View answer and explanation20 2432902008176640000= 20×121645100408832000, so □=20. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0020/) Factorial · qf-factorial-missing-successor-factor · candidate ## 96. Missing Factor in the Successor Recurrence Fill the integer □ in 21!=□×20!, where 20!=2432902008176640000. View answer and explanation21 51090942171709440000= 21×2432902008176640000, so □=21. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0021/) Factorial · qf-factorial-missing-successor-factor · candidate ## 97. Missing Factor in the Successor Recurrence Fill the integer □ in 22!=□×21!, where 21!=51090942171709440000. View answer and explanation22 1124000727777607680000= 22×51090942171709440000, so □=22. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0022/) Factorial · qf-factorial-missing-successor-factor · candidate ## 98. Missing Factor in the Successor Recurrence Fill the integer □ in 23!=□×22!, where 22!=1124000727777607680000. View answer and explanation23 25852016738884976640000= 23×1124000727777607680000, so □=23. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0023/) Factorial · qf-factorial-missing-successor-factor · candidate ## 99. Missing Factor in the Successor Recurrence Fill the integer □ in 24!=□×23!, where 23!=25852016738884976640000. View answer and explanation24 620448401733239439360000= 24×25852016738884976640000, so □=24. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0024/) Factorial · qf-factorial-missing-successor-factor · candidate ## 100. Missing Factor in the Successor Recurrence Fill the integer □ in 25!=□×24!, where 24!=620448401733239439360000. View answer and explanation25 15511210043330985984000000= 25×620448401733239439360000, so □=25. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260926-factorial-missing-successor-factor-0025/) Factorial · qf-factorial-missing-successor-factor · candidate [Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260926-factorial-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260926-factorial-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260926-factorial-foundations/validation-report.json)