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Question 9053: Multiplicative Congruence Values, Premises, and Material Implication

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

R0017-09053 · q-20261013-advance-20261018-r05-modular-congruence-0078

Original question and givens

Use fixed positive modulus n=4, natural inputs a=3, b=15, c=7, d=11. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.

  • G001 — Use fixed positive modulus n=4, natural inputs a=3, b=15, c=7, d=11. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a*c) mod n,(b*d) mod n], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.

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=4

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

a remainder:natural dividend=3;positive divisor=4

After

3=0·4+3

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

b remainder:natural dividend=15;positive divisor=4

After

15=3·4+3

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

c remainder:natural dividend=7;positive divisor=4

After

7=1·4+3

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, S003

S005 · implies

Before

d remainder:natural dividend=11;positive divisor=4

After

11=2·4+3

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, S004

S006 · calculation

Before

combined left value := mul(3, 7)

After

combined left value = 21

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

combined right value := mul(15, 11)

After

combined right value = 165

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

combined left remainder:natural dividend=21;positive divisor=4

After

21=5·4+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, S007

S009 · implies

Before

combined right remainder:natural dividend=165;positive divisor=4

After

165=41·4+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, S008

S010 · implies

Before

a,b remainders=[3, 3];c,d remainders=[3, 3]

After

both premises=Yes;conclusion=Yes

Conditions: none

Audit both premises first,then independently compute the sum/product remainders.A failed premise does not force a false conclusion.

Step identifiers and dependencies

Rule: modular_premises_and_conclusion_separate · Uses: G001, S009

S011 · calculation

Before

first quotient difference := sub(0, 3)

After

first quotient difference = -3

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

second quotient difference := sub(1, 2)

After

second quotient difference = -1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

product difference first term := mul(3, -1)

After

product difference first term = -3

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

product difference second term := mul(11, -3)

After

product difference second term = -33

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

product difference multiplier := sum(-3, -33)

After

product difference multiplier = -36

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

combined difference := sub(21, 165)

After

combined difference = -144

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

modulus times difference multiplier := mul(4, -36)

After

modulus times difference multiplier = -144

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

combined difference=-144;integer multiplier=-36;modulus=4

After

difference=n*multiplier=-144

Conditions: none

Equal remainders give a−b=n(qa−qb) and c−d=n(qc−qd).Add directly,or use ac−bd=a(c−d)+d(a−b).The multiplier may be a negative integer;no factor cancellation is needed.

Step identifiers and dependencies

Rule: applicable_modular_preservation_witness · Uses: G001, S017

S019 · implies

Before

premises=Yes;conclusion=Yes

After

material implication=Yes

Conditions: none

A false premise makes the material implication true by its truth definition,not by supplying applicable preservation premises.

Step identifiers and dependencies

Rule: modular_implication_not_applicable_premise · Uses: G001, S018

Reverse checks

  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Recompute modulus times difference multiplier with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute combined difference with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute product difference multiplier with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute product difference second term with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute product difference first term with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute second quotient difference with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute first quotient difference with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Recompute combined right value with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute combined left value with exact rational arithmetic, not rounded decimals.
  • 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 — 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

(4, [3, 15], [7, 11], Yes, [1, 1], Yes, Yes)

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: 2216e88a87470f76988269d73b4d2ef3d5374fe87c2da2523110b7bbea3701e8
  • candidate record SHA-256: 2474b9dc0f28d30caec60785f7c85bfb6fd663f350f7b501605b264d3ca123d4
  • semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
  • question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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