# Question 4605: Inverse Candidate and Multiplicative Defect

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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.

R0008-04605 · q-20261008-advance-20261011-r02-arithmetic-operations-0055



## Original question and givens

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=5/2, candidate r=1/6. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.


- G001 — All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=5/2, candidate r=1/6. Give (is r a multiplicative inverse of a?, ar−1). 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-arithmetic-operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/)





## Step-by-step derivation

S001 · impliesBefore

exact arithmetic in Q;N={0,1,2,...} when membership is asked

After

do not truncate subtraction or rational division

Conditions: none

Compute exact results in Q before testing N-membership;do not clamp negative differences to0 or replace rational division by integer division.Division by0 remains undefined.

Step identifiers and dependenciesRule: exact_rational_arithmetic_scope · Uses: G001





S002 · calculationBefore

candidate inverse product := mul(5/2, 1/6)

After

candidate inverse product = 5/12

Conditions: none

A multiplicative inverse must multiply the original number to1.

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





S003 · calculationBefore

inverse signed defect := sub(5/12, 1)

After

inverse signed defect = -7/12

Conditions: none

Compute the signed defect ar−1 in that order.

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





S004 · impliesBefore

a=5/2;r=1/6;ar=5/12

After

inverse valid=No;ar−1=-7/12

Conditions: none

For a=0 every candidate product is0,so no inverse exists;even with nonzero a,the candidate must actually produce1.

Step identifiers and dependenciesRule: multiplicative_inverse_zero_boundary · 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 inverse signed defect with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute candidate inverse product 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

(No, -7/12)

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: cb7d23b502ef6b8368e8677d432377a0b9c7628b11ff9b2a3bb7ff32c56748a2
- candidate record SHA-256: 9426511ae8b263c07607954b22ed7b9d7fd0c58ab1fd3d16e1bf58d96c31777f
- semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289
- question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0055.json)