Question 6701: Adjacent Exact Factorials and Their Ratio

Original question and givens

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)).

• G001 — 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)).

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

N includes0;F(0)=1;F(j)=j·F(j−1)

exact integer products,not floating values or a permutation proof

actual factorial:start at F(0)

F(0)=1

true successor factorial := mul(1, 1)

true successor factorial = 1

successor factorial ratio := div(1, 1)

successor factorial ratio = 1

F(0)=1>0;F(1)=1

ratio=1

Reverse checks

• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute successor factorial ratio with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute true successor factorial with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(1, 1, 1)

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.