# Question 9084: Positive-Modulus Guard before Natural Remainder Comparison

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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.

R0017-09084 · q-20261013-advance-20261018-r05-modular-congruence-0009



## Original question and givens

Fix modulus n=2, natural a=31, b=52. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.


- G001 — Fix modulus n=2, natural a=31, b=52. Give (n, a, b, n>0?, [a mod n,b mod n] or not-evaluated, congruent? or not-evaluated). This entry excludes zero-modulus conventions.





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





## Step-by-step derivation

S001 · impliesBefore

fixed natural modulus candidate=2

After

positive modulus=Yes

Conditions: none

Congruence compares remainders at one positive modulus,not numerical equality.Multiplicative preservation does not permit unconditional cancellation of a common factor.

Step identifiers and dependenciesRule: fixed_positive_modulus_scope · Uses: G001





S002 · impliesBefore

first remainder:natural dividend=31;positive divisor=2

After

31=15·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S003 · impliesBefore

second remainder:natural dividend=52;positive divisor=2

After

52=26·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S004 · impliesBefore

same positive modulus=2;remainders=[1, 0]

After

congruent=No

Conditions: none

The original a and b may differ;only their remainders at the same positive modulus must agree.

Step identifiers and dependenciesRule: same_modulus_remainder_equality · 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

(2, 31, 52, Yes, [1, 0], No)

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: 0703ceece6492ae730e729ea5173f16fbdf7fa2550c314bb0e094ab159404f3f
- candidate record SHA-256: 6244a2a3e2d5b3c7ed1cb57b4a08e2f815210bc28bf73d7c0b8fb98d224d6420
- semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
- question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261018-r05-modular-congruence-0009.json)