# Question 6420: Rational Division Eligibility and Multiplication Back

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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.

R0011-06420 · q-20261009-advance-20261013-r03-rational-number-0070



## Original question and givens

Exact rational representations have nonzero denominators; a=20/-3, b=21/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.


- G001 — Exact rational representations have nonzero denominators; a=20/-3, b=21/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.





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





## Step-by-step derivation

S001 · impliesBefore

raw a=[20, -3];raw b=[21, -4]

After

a=-20/3;b=-21/4

Conditions: none

Original denominators need only be nonzero;negative denominators are valid.Normalize to reduced positive-denominator fractions for exact arithmetic while retaining the original representations and sources.

Step identifiers and dependenciesRule: nonzero_signed_rational_representation · Uses: G001





S002 · calculationBefore

guarded rational quotient := div(-20/3, -21/4)

After

guarded rational quotient = 80/63

Conditions: none

Use the exact quotient only after checking b≠0.

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





S003 · calculationBefore

rational reverse multiplication := mul(80/63, -21/4)

After

rational reverse multiplication = -20/3

Conditions: none

Multiply back by the same divisor to recover a.

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





S004 · impliesBefore

b=-21/4≠0;quotient=80/63

After

quotient·b=-20/3=a

Conditions: none

This division/reverse-multiplication chain requires a nonzero divisor;it is not a formal proof of the entire rational-number structure.

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









## Reverse checks


- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute rational reverse multiplication with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute guarded rational quotient with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(Yes, 80/63, -20/3)

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: eb0a517f101fe03ff4ecb9f99af5a9a15dee389a5b159d5e2232f0edf6bf05c1
- candidate record SHA-256: 803836b0a218908cd40b758fd5f21376930f090eb0fffdfda468925722804b32
- semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80
- question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261009-advance-20261013-r03-rational-number-0070.json)