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Question 9087: 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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← Question derivation index · Original question batch

R0017-09087 · q-20261013-advance-20261018-r05-modular-congruence-0012

Original question and givens

Fix modulus n=5, natural a=40, b=67. 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=5, natural a=40, b=67. 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=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 dependencies

Rule: fixed_positive_modulus_scope · Uses: G001

S002 · implies

Before

first remainder:natural dividend=40;positive divisor=5

After

40=8·5+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, S001

S003 · implies

Before

second remainder:natural dividend=67;positive divisor=5

After

67=13·5+2

Conditions: none

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

Step identifiers and dependencies

Rule: positive_euclidean_division · Uses: G001, S002

S004 · implies

Before

same positive modulus=5;remainders=[0, 2]

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

(5, 40, 67, Yes, [0, 2], 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: 1e48a6327566f1d9e60dcb17e4e8a82f6410f66b9ed5f22373a9a31fa5e769e5
  • candidate record SHA-256: 770111d69aea854863ea21e359907104cd740574400ac589c88b1bbaa3e49964
  • semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
  • question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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