# Exact Factorial Windows and Candidate Audits

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-20261010-advance-20261014-r05-factorial-foundations · 2026-10-10 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=0. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(1, 1, 1)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (1, 1, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0001/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 2. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=1. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(1, 2, 2)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (1, 2, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0002/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 3. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=2. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(2, 6, 3)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (2, 6, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0003/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 4. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=3. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(6, 24, 4)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (6, 24, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0004/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 5. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=4. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(24, 120, 5)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (24, 120, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0005/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 6. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=5. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(120, 720, 6)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (120, 720, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0006/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 7. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=6. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(720, 5040, 7)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (720, 5040, 7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0007/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 8. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=7. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(5040, 40320, 8)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (5040, 40320, 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0008/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 9. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=8. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(40320, 362880, 9)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (40320, 362880, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0009/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 10. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=9. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(362880, 3628800, 10)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (362880, 3628800, 10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0010/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 11. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=10. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(3628800, 39916800, 11)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (3628800, 39916800, 11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0011/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 12. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=11. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(39916800, 479001600, 12)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (39916800, 479001600, 12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0012/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 13. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=12. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(479001600, 6227020800, 13)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (479001600, 6227020800, 13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0013/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 14. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=13. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(6227020800, 87178291200, 14)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (6227020800, 87178291200, 14).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0014/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 15. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=14. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(87178291200, 1307674368000, 15)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (87178291200, 1307674368000, 15).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0015/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 16. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=15. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(1307674368000, 20922789888000, 16)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (1307674368000, 20922789888000, 16).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0016/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 17. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=16. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(20922789888000, 355687428096000, 17)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (20922789888000, 355687428096000, 17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0017/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 18. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=17. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(355687428096000, 6402373705728000, 18)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (355687428096000, 6402373705728000, 18).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0018/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 19. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=18. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(6402373705728000, 121645100408832000, 19)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (6402373705728000, 121645100408832000, 19).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0019/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 20. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=19. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(121645100408832000, 2432902008176640000, 20)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (121645100408832000, 2432902008176640000, 20).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0020/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 21. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=20. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(2432902008176640000, 51090942171709440000, 21)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (2432902008176640000, 51090942171709440000, 21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0021/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 22. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=21. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(51090942171709440000, 1124000727777607680000, 22)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (51090942171709440000, 1124000727777607680000, 22).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0022/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 23. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=22. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(1124000727777607680000, 25852016738884976640000, 23)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (1124000727777607680000, 25852016738884976640000, 23).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0023/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 24. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=23. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(25852016738884976640000, 620448401733239439360000, 24)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (25852016738884976640000, 620448401733239439360000, 24).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0024/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 25. Adjacent Exact Factorials and Their Ratio

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=24. Give decimal integers (F(n), F(n+1), F(n+1)/F(n)).

View answer and explanation(620448401733239439360000, 15511210043330985984000000, 25)

Compute both factorials by integer products, then divide exactly; these finite values are not a general permutation-counting proof. The exact result, retaining the requested order, is (620448401733239439360000, 15511210043330985984000000, 25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0025/)

Factorial · qf-density14-factorial-adjacent-ratio · candidate





## 26. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=0≤hi=0≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(1, [], Yes)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (1, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0026/)

Factorial · qf-density14-factorial-window-product · candidate





## 27. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=0≤hi=1≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(1, [1], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (1, [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0027/)

Factorial · qf-density14-factorial-window-product · candidate





## 28. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=0≤hi=20≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(2432902008176640000, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (2432902008176640000, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0028/)

Factorial · qf-density14-factorial-window-product · candidate





## 29. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=1≤hi=3≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(6, [2, 3], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (6, [2, 3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0029/)

Factorial · qf-density14-factorial-window-product · candidate





## 30. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=2≤hi=5≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(60, [3, 4, 5], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (60, [3, 4, 5], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0030/)

Factorial · qf-density14-factorial-window-product · candidate





## 31. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=3≤hi=7≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(840, [4, 5, 6, 7], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (840, [4, 5, 6, 7], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0031/)

Factorial · qf-density14-factorial-window-product · candidate





## 32. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=4≤hi=6≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(30, [5, 6], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (30, [5, 6], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0032/)

Factorial · qf-density14-factorial-window-product · candidate





## 33. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=5≤hi=8≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(336, [6, 7, 8], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (336, [6, 7, 8], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0033/)

Factorial · qf-density14-factorial-window-product · candidate





## 34. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=6≤hi=10≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(5040, [7, 8, 9, 10], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (5040, [7, 8, 9, 10], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0034/)

Factorial · qf-density14-factorial-window-product · candidate





## 35. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=7≤hi=9≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(72, [8, 9], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (72, [8, 9], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0035/)

Factorial · qf-density14-factorial-window-product · candidate





## 36. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=8≤hi=11≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(990, [9, 10, 11], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (990, [9, 10, 11], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0036/)

Factorial · qf-density14-factorial-window-product · candidate





## 37. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=9≤hi=13≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(17160, [10, 11, 12, 13], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (17160, [10, 11, 12, 13], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0037/)

Factorial · qf-density14-factorial-window-product · candidate





## 38. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=10≤hi=12≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(132, [11, 12], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (132, [11, 12], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0038/)

Factorial · qf-density14-factorial-window-product · candidate





## 39. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=11≤hi=14≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(2184, [12, 13, 14], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (2184, [12, 13, 14], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0039/)

Factorial · qf-density14-factorial-window-product · candidate





## 40. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=12≤hi=16≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(43680, [13, 14, 15, 16], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (43680, [13, 14, 15, 16], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0040/)

Factorial · qf-density14-factorial-window-product · candidate





## 41. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=13≤hi=15≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(210, [14, 15], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (210, [14, 15], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0041/)

Factorial · qf-density14-factorial-window-product · candidate





## 42. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=14≤hi=17≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(4080, [15, 16, 17], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (4080, [15, 16, 17], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0042/)

Factorial · qf-density14-factorial-window-product · candidate





## 43. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=15≤hi=19≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(93024, [16, 17, 18, 19], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (93024, [16, 17, 18, 19], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0043/)

Factorial · qf-density14-factorial-window-product · candidate





## 44. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=16≤hi=18≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(306, [17, 18], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (306, [17, 18], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0044/)

Factorial · qf-density14-factorial-window-product · candidate





## 45. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=17≤hi=20≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(6840, [18, 19, 20], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (6840, [18, 19, 20], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0045/)

Factorial · qf-density14-factorial-window-product · candidate





## 46. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=18≤hi=22≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(175560, [19, 20, 21, 22], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (175560, [19, 20, 21, 22], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0046/)

Factorial · qf-density14-factorial-window-product · candidate





## 47. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=19≤hi=21≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(420, [20, 21], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (420, [20, 21], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0047/)

Factorial · qf-density14-factorial-window-product · candidate





## 48. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=20≤hi=23≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(10626, [21, 22, 23], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (10626, [21, 22, 23], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0048/)

Factorial · qf-density14-factorial-window-product · candidate





## 49. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=21≤hi=25≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(303600, [22, 23, 24, 25], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (303600, [22, 23, 24, 25], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0049/)

Factorial · qf-density14-factorial-window-product · candidate





## 50. Factorial Window Product and Empty-Window Eligibility

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=22≤hi=24≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).

View answer and explanation(552, [23, 24], No)

Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (552, [23, 24], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0050/)

Factorial · qf-density14-factorial-window-product · candidate





## 51. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=0; candidate decimal integer strings u=1, v=1. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(1, 1, 0, 0, Yes)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (1, 1, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0051/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 52. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=3; candidate decimal integer strings u=12, v=48. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(6, 24, 6, 0, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (6, 24, 6, 0, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0052/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 53. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=2; candidate decimal integer strings u=2, v=7. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(2, 6, 0, 1, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (2, 6, 0, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0053/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 54. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=3; candidate decimal integer strings u=6, v=24. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(6, 24, 0, 0, Yes)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (6, 24, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0054/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 55. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=4; candidate decimal integer strings u=25, v=125. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(24, 120, 1, 0, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (24, 120, 1, 0, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0055/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 56. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=5; candidate decimal integer strings u=120, v=721. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(120, 720, 0, 1, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (120, 720, 0, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0056/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 57. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=6; candidate decimal integer strings u=720, v=5040. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(720, 5040, 0, 0, Yes)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (720, 5040, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0057/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 58. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=7; candidate decimal integer strings u=5041, v=40328. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(5040, 40320, 1, 0, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (5040, 40320, 1, 0, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0058/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 59. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=8; candidate decimal integer strings u=40320, v=362881. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(40320, 362880, 0, 1, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (40320, 362880, 0, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0059/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 60. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=9; candidate decimal integer strings u=362880, v=3628800. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(362880, 3628800, 0, 0, Yes)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (362880, 3628800, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0060/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 61. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=10; candidate decimal integer strings u=3628801, v=39916811. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(3628800, 39916800, 1, 0, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (3628800, 39916800, 1, 0, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0061/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 62. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=11; candidate decimal integer strings u=39916800, v=479001601. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(39916800, 479001600, 0, 1, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (39916800, 479001600, 0, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0062/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 63. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=12; candidate decimal integer strings u=479001600, v=6227020800. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(479001600, 6227020800, 0, 0, Yes)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (479001600, 6227020800, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0063/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 64. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=13; candidate decimal integer strings u=6227020801, v=87178291214. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(6227020800, 87178291200, 1, 0, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (6227020800, 87178291200, 1, 0, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0064/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 65. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=14; candidate decimal integer strings u=87178291200, v=1307674368001. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(87178291200, 1307674368000, 0, 1, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (87178291200, 1307674368000, 0, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0065/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 66. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=15; candidate decimal integer strings u=1307674368000, v=20922789888000. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(1307674368000, 20922789888000, 0, 0, Yes)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (1307674368000, 20922789888000, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0066/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 67. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=16; candidate decimal integer strings u=20922789888001, v=355687428096017. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(20922789888000, 355687428096000, 1, 0, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (20922789888000, 355687428096000, 1, 0, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0067/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 68. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=17; candidate decimal integer strings u=355687428096000, v=6402373705728001. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(355687428096000, 6402373705728000, 0, 1, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (355687428096000, 6402373705728000, 0, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0068/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 69. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=18; candidate decimal integer strings u=6402373705728000, v=121645100408832000. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(6402373705728000, 121645100408832000, 0, 0, Yes)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (6402373705728000, 121645100408832000, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0069/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 70. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=19; candidate decimal integer strings u=121645100408832001, v=2432902008176640020. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(121645100408832000, 2432902008176640000, 1, 0, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (121645100408832000, 2432902008176640000, 1, 0, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0070/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 71. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=20; candidate decimal integer strings u=2432902008176640000, v=51090942171709440001. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(2432902008176640000, 51090942171709440000, 0, 1, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (2432902008176640000, 51090942171709440000, 0, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0071/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 72. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=21; candidate decimal integer strings u=51090942171709440000, v=1124000727777607680000. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(51090942171709440000, 1124000727777607680000, 0, 0, Yes)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (51090942171709440000, 1124000727777607680000, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0072/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 73. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=22; candidate decimal integer strings u=1124000727777607680001, v=25852016738884976640023. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(1124000727777607680000, 25852016738884976640000, 1, 0, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (1124000727777607680000, 25852016738884976640000, 1, 0, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0073/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 74. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=23; candidate decimal integer strings u=25852016738884976640000, v=620448401733239439360001. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(25852016738884976640000, 620448401733239439360000, 0, 1, No)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (25852016738884976640000, 620448401733239439360000, 0, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0074/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 75. Reported Factorials versus Exact Values and Recurrence

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=24; candidate decimal integer strings u=620448401733239439360000, v=15511210043330985984000000. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).

View answer and explanation(620448401733239439360000, 15511210043330985984000000, 0, 0, Yes)

Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (620448401733239439360000, 15511210043330985984000000, 0, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0075/)

Factorial · qf-density14-factorial-reported-successor-audit · candidate





## 76. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=1, k=1∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(1, 1, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (1, 1, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0076/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 77. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=19, k=19∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(121645100408832000, 6402373705728000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (121645100408832000, 6402373705728000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0077/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 78. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=4, k=3∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(24, 8, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (24, 8, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0078/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 79. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=5, k=4∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(120, 30, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (120, 30, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0079/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 80. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=6, k=5∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(720, 144, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (720, 144, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0080/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 81. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=7, k=6∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(5040, 840, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (5040, 840, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0081/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 82. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=8, k=7∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(40320, 5760, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (40320, 5760, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0082/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 83. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=9, k=8∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(362880, 45360, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (362880, 45360, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0083/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 84. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=10, k=9∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(3628800, 403200, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (3628800, 403200, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0084/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 85. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=11, k=10∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(39916800, 3991680, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (39916800, 3991680, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0085/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 86. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=12, k=11∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(479001600, 43545600, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (479001600, 43545600, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0086/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 87. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=13, k=12∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(6227020800, 518918400, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (6227020800, 518918400, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0087/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 88. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=14, k=13∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(87178291200, 6706022400, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (87178291200, 6706022400, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0088/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 89. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=15, k=14∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(1307674368000, 93405312000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (1307674368000, 93405312000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0089/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 90. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=16, k=15∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(20922789888000, 1394852659200, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (20922789888000, 1394852659200, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0090/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 91. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=17, k=16∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(355687428096000, 22230464256000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (355687428096000, 22230464256000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0091/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 92. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=18, k=17∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(6402373705728000, 376610217984000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (6402373705728000, 376610217984000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0092/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 93. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=19, k=18∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(121645100408832000, 6758061133824000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (121645100408832000, 6758061133824000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0093/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 94. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=20, k=19∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(2432902008176640000, 128047474114560000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (2432902008176640000, 128047474114560000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0094/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 95. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=21, k=20∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(51090942171709440000, 2554547108585472000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (51090942171709440000, 2554547108585472000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0095/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 96. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=22, k=21∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(1124000727777607680000, 53523844179886080000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (1124000727777607680000, 53523844179886080000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0096/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 97. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=23, k=22∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(25852016738884976640000, 1175091669949317120000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (25852016738884976640000, 1175091669949317120000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0097/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 98. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=24, k=23∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(620448401733239439360000, 26976017466662584320000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (620448401733239439360000, 26976017466662584320000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0098/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 99. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=25, k=24∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(15511210043330985984000000, 646300418472124416000000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (15511210043330985984000000, 646300418472124416000000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0099/)

Factorial · qf-density14-factorial-single-factor-omission · candidate





## 100. Omitting One Existing Factor and Multiplying Back

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=26, k=25∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).

View answer and explanation(403291461126605635584000000, 16131658445064225423360000, 0)

Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (403291461126605635584000000, 16131658445064225423360000, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r05-factorial-0100/)

Factorial · qf-density14-factorial-single-factor-omission · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261010-advance-20261014-r05-factorial-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261010-advance-20261014-r05-factorial-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261010-advance-20261014-r05-factorial-foundations/validation-report.json)