Question 31813: Quotient of Consecutive Factorials

Original question and givens

Compute 13!/12!; the denominator is explicitly positive.

• G001 — Compute 13!/12!; the denominator is explicitly positive.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-factorial

Step-by-step derivation

source roles = [qf-factorial-consecutive-quotient, no-domain-field, consecutive_quotient, no-form-field]

derived = [[exact_integer_STRING]]

Factorial requested n ORIGINAL factorial factor2 := mul(1, 2)

Factorial requested n ORIGINAL factorial factor2 = 2

Factorial requested n ORIGINAL factorial factor3 := mul(2, 3)

Factorial requested n ORIGINAL factorial factor3 = 6

Factorial requested n ORIGINAL factorial factor4 := mul(6, 4)

Factorial requested n ORIGINAL factorial factor4 = 24

Factorial requested n ORIGINAL factorial factor5 := mul(24, 5)

Factorial requested n ORIGINAL factorial factor5 = 120

Factorial requested n ORIGINAL factorial factor6 := mul(120, 6)

Factorial requested n ORIGINAL factorial factor6 = 720

Factorial requested n ORIGINAL factorial factor7 := mul(720, 7)

Factorial requested n ORIGINAL factorial factor7 = 5040

Factorial requested n ORIGINAL factorial factor8 := mul(5040, 8)

Factorial requested n ORIGINAL factorial factor8 = 40320

Factorial requested n ORIGINAL factorial factor9 := mul(40320, 9)

Factorial requested n ORIGINAL factorial factor9 = 362880

Factorial requested n ORIGINAL factorial factor10 := mul(362880, 10)

Factorial requested n ORIGINAL factorial factor10 = 3628800

Factorial requested n ORIGINAL factorial factor11 := mul(3628800, 11)

Factorial requested n ORIGINAL factorial factor11 = 39916800

Factorial requested n ORIGINAL factorial factor12 := mul(39916800, 12)

Factorial requested n ORIGINAL factorial factor12 = 479001600

Factorial requested n ORIGINAL factorial factor13 := mul(479001600, 13)

Factorial requested n ORIGINAL factorial factor13 = 6227020800

source roles = [Factorial requested n, 13]

derived = [6227020800]

source roles = [13, 6227020800]

derived = [6227020800]

Factorial previous n minus1 ORIGINAL factorial factor2 := mul(1, 2)

Factorial previous n minus1 ORIGINAL factorial factor2 = 2

Factorial previous n minus1 ORIGINAL factorial factor3 := mul(2, 3)

Factorial previous n minus1 ORIGINAL factorial factor3 = 6

Factorial previous n minus1 ORIGINAL factorial factor4 := mul(6, 4)

Factorial previous n minus1 ORIGINAL factorial factor4 = 24

Factorial previous n minus1 ORIGINAL factorial factor5 := mul(24, 5)

Factorial previous n minus1 ORIGINAL factorial factor5 = 120

Factorial previous n minus1 ORIGINAL factorial factor6 := mul(120, 6)

Factorial previous n minus1 ORIGINAL factorial factor6 = 720

Factorial previous n minus1 ORIGINAL factorial factor7 := mul(720, 7)

Factorial previous n minus1 ORIGINAL factorial factor7 = 5040

Factorial previous n minus1 ORIGINAL factorial factor8 := mul(5040, 8)

Factorial previous n minus1 ORIGINAL factorial factor8 = 40320

Factorial previous n minus1 ORIGINAL factorial factor9 := mul(40320, 9)

Factorial previous n minus1 ORIGINAL factorial factor9 = 362880

Factorial previous n minus1 ORIGINAL factorial factor10 := mul(362880, 10)

Factorial previous n minus1 ORIGINAL factorial factor10 = 3628800

Factorial previous n minus1 ORIGINAL factorial factor11 := mul(3628800, 11)

Factorial previous n minus1 ORIGINAL factorial factor11 = 39916800

Factorial previous n minus1 ORIGINAL factorial factor12 := mul(39916800, 12)

Factorial previous n minus1 ORIGINAL factorial factor12 = 479001600

source roles = [Factorial previous n minus1, 12]

derived = [479001600]

Factorial ORIGINAL n factorial over positive previous factorial := div(6227020800, 479001600)

Factorial ORIGINAL n factorial over positive previous factorial = 13

Factorial quotient reverse ORIGINAL previous factorial times n := mul(479001600, 13)

Factorial quotient reverse ORIGINAL previous factorial times n = 6227020800

source roles = [13, 6227020800, 479001600]

derived = [13, 6227020800]

Reverse checks

• S030 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S029 · premise_and_result_audit — Recompute Factorial quotient reverse ORIGINAL previous factorial times n with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Factorial ORIGINAL n factorial over positive previous factorial with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S026 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor12 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor11 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor10 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor9 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor8 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor7 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S014 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S013 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor13 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor12 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor11 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor10 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor9 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor8 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor7 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.

Result

13

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.