# Question 9047: A Natural Congruence Class in an Inclusive Window

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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.

R0017-09047 · q-20261013-advance-20261018-r05-modular-congruence-0047



## Original question and givens

Fix positive modulus n=5, natural a=64, bound=46. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).


- G001 — Fix positive modulus n=5, natural a=64, bound=46. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod 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-modular-congruence](https://ocme.evemisslab.com/en/mko/mko-modular-congruence/)





## Step-by-step derivation

S001 · impliesBefore

fixed natural modulus candidate=5

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

representative remainder:natural dividend=64;positive divisor=5

After

64=12·5+4

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 · calculationBefore

window bound minus residue := sub(46, 4)

After

window bound minus residue = 42

Conditions: none

Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.

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





S004 · impliesBefore

last progression index:natural dividend=42;positive divisor=5

After

42=8·5+2

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, S003





S005 · impliesBefore

residue=4;modulus=5;0<=k<=8

After

all class members=[4, 9, 14, 19, 24, 29, 34, 39, 44]

Conditions: none

Euclidean uniqueness gives x≡a exactly when x=r+k*n.Natural x and0≤r<n force k≥0;the bound gives k≤floor((bound−r)/n).Retain the full inclusive progression,not a sample.

Step identifiers and dependenciesRule: complete_residue_progression · 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute window bound minus residue 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

(5, 64, 46, 4, [4, 9, 14, 19, 24, 29, 34, 39, 44])

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: d1477118cdaa56f262ee7b74d32f27d8e8a85d912a7036096d2f85934b3c1e47
- candidate record SHA-256: 855296eaed66f40485f0d560f22c691d4df650bdd82a7a849718f628977dfbc0
- 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-0047.json)