# Question 9021: Additive Congruence Values, Premises, and Material Implication

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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.

R0017-09021 · q-20261013-advance-20261018-r05-modular-congruence-0071



## Original question and givens

Use fixed positive modulus n=4, natural inputs a=21, b=33, c=43, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).


- G001 — Use fixed positive modulus n=4, natural inputs a=21, b=33, c=43, d=47. Give (n, [a,b], [c,d], both a≡b and c≡d?, [(a+c) mod n,(b+d) mod n], sum congruence?, premises imply conclusion?).





## 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=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 dependenciesRule: fixed_positive_modulus_scope · Uses: G001





S002 · impliesBefore

a 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 dependenciesRule: positive_euclidean_division · Uses: G001, S001





S003 · impliesBefore

b remainder:natural dividend=33;positive divisor=4

After

33=8·4+1

Conditions: none

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

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





S004 · impliesBefore

c remainder:natural dividend=43;positive divisor=4

After

43=10·4+3

Conditions: none

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

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S003





S005 · impliesBefore

d remainder:natural dividend=47;positive divisor=4

After

47=11·4+3

Conditions: none

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

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S004





S006 · calculationBefore

combined left value := sum(21, 43)

After

combined left value = 64

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, S005





S007 · calculationBefore

combined right value := sum(33, 47)

After

combined right value = 80

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, S006





S008 · impliesBefore

combined left remainder:natural dividend=64;positive divisor=4

After

64=16·4+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, S007





S009 · impliesBefore

combined right remainder:natural dividend=80;positive divisor=4

After

80=20·4+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, S008





S010 · impliesBefore

a,b remainders=[1, 1];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 dependenciesRule: modular_premises_and_conclusion_separate · Uses: G001, S009





S011 · calculationBefore

first quotient difference := sub(5, 8)

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

second quotient difference := sub(10, 11)

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

sum difference multiplier := sum(-3, -1)

After

sum difference multiplier = -4

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, S012





S014 · calculationBefore

combined difference := sub(64, 80)

After

combined difference = -16

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, S013





S015 · calculationBefore

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

After

modulus times difference multiplier = -16

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, S014





S016 · impliesBefore

combined difference=-16;integer multiplier=-4;modulus=4

After

difference=n*multiplier=-16

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 dependenciesRule: applicable_modular_preservation_witness · Uses: G001, S015





S017 · impliesBefore

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 dependenciesRule: modular_implication_not_applicable_premise · Uses: G001, S016









## Reverse checks


- S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S015 · premise_and_result_audit — Recompute modulus times difference multiplier with exact rational arithmetic, not rounded decimals.
- S014 · premise_and_result_audit — Recompute combined difference with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Recompute sum difference multiplier 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, [21, 33], [43, 47], Yes, [0, 0], 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: be4a71ec63d39d9169a00bb65aa206ba3e6754f80a1c6f1180835117e7125137
- candidate record SHA-256: 9287f241cb06b8f8ea96fb09ed5a508cffdf7f44d7693d0bb6bf9b95d5f80dad
- 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-0071.json)