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Question 9085: Positive-Modulus Guard before Natural Remainder Comparison

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0017-09085 · q-20261013-advance-20261018-r05-modular-congruence-0010

Original question and givens

Fix modulus n=3, natural a=34, b=57. 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=3, natural a=34, b=57. 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.

Step-by-step derivation

S001 · implies

Before

fixed natural modulus candidate=3

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 dependencies

Rule: fixed_positive_modulus_scope · Uses: G001

S002 · implies

Before

first remainder:natural dividend=34;positive divisor=3

After

34=11·3+1

Conditions: none

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

Step identifiers and dependencies

Rule: positive_euclidean_division · Uses: G001, S001

S003 · implies

Before

second remainder:natural dividend=57;positive divisor=3

After

57=19·3+0

Conditions: none

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

Step identifiers and dependencies

Rule: positive_euclidean_division · Uses: G001, S002

S004 · implies

Before

same positive modulus=3;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 dependencies

Rule: 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

(3, 34, 57, 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: 8c5d2b4f4872eddf88e1b0b34a7cad6c8071508fb011595b9c62436ff07476fb
  • candidate record SHA-256: dc26607ac97c81dfbfac8c6c467db828b9677ffec8f1dd9a0255f1cb7f9dab7d
  • semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
  • question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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