Question 31806: Quotient of Consecutive Factorials

Original question and givens

Compute 6!/5!; the denominator is explicitly positive.

• G001 — Compute 6!/5!; 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

source roles = [Factorial requested n, 6]

derived = [720]

source roles = [6, 720]

derived = [720]

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

source roles = [Factorial previous n minus1, 5]

derived = [120]

Factorial ORIGINAL n factorial over positive previous factorial := div(720, 120)

Factorial ORIGINAL n factorial over positive previous factorial = 6

Factorial quotient reverse ORIGINAL previous factorial times n := mul(120, 6)

Factorial quotient reverse ORIGINAL previous factorial times n = 720

source roles = [6, 720, 120]

derived = [6, 720]

Reverse checks

• S016 · 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.
• S015 · premise_and_result_audit — Recompute Factorial quotient reverse ORIGINAL previous factorial times n with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Factorial ORIGINAL n factorial over positive previous factorial with exact rational arithmetic, not rounded decimals.
• S013 · 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.
• S012 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Factorial previous n minus1 ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
• S008 · 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.
• S007 · 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.
• 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

6

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.