# Question 6701: Adjacent Exact Factorials and Their Ratio

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0012-06701 · q-20261010-advance-20261014-r05-factorial-0001



## 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](https://ocme.evemisslab.com/en/mko/mko-factorial/)





## Step-by-step derivation

S001 · impliesBefore

N includes0;F(0)=1;F(j)=j·F(j−1)

After

exact integer products,not floating values or a permutation proof

Conditions: none

Use empty product1 and integer factor multiplication,retaining large values exactly.This finite recurrence audit is not a general proof of permutation counting.

Step identifiers and dependenciesRule: exact_factorial_protocol · Uses: G001





S002 · impliesBefore

actual factorial:start at F(0)

After

F(0)=1

Conditions: none

The product of zero factors is1,not0;this is the successor recurrence's base.

Step identifiers and dependenciesRule: factorial_empty_product_base · Uses: G001, S001





S003 · calculationBefore

true successor factorial := mul(1, 1)

After

true successor factorial = 1

Conditions: none

Compute F(n+1)=(n+1)F(n).

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

successor factorial ratio := div(1, 1)

After

successor factorial ratio = 1

Conditions: none

Since F(n)>0,division is legal and the exact ratio is n+1.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

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

After

ratio=1

Conditions: none

This includes n=0 with F(0)=F(1)=1,without0!=0 or floating approximations.

Step identifiers and dependenciesRule: positive_factorial_adjacent_ratio · Uses: G001, S004









## 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: ba307aaadfd397684db0e06fc777a6c14bf050ef64852e7559dc771ebf5651e1
- candidate record SHA-256: 0623e84c5b4381abcc3ca6e9b31a6e399d21ede6a46bccbb02480b66a31d5aff
- semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00
- question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r05-factorial-0001.json)