Question 9010: Additive Congruence Values, Premises, and Material Implication

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0017-09010 · q-20261013-advance-20261018-r05-modular-congruence-0060

Original question and givens

Use fixed positive modulus n=2, natural inputs a=10, b=12, c=21, d=25. 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=2, natural inputs a=10, b=12, c=21, d=25. 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

Step-by-step derivation

S001 · implies

Before

fixed natural modulus candidate=2

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=10;positive divisor=2

After

10=5·2+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

b remainder:natural dividend=12;positive divisor=2

After

12=6·2+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

c remainder:natural dividend=21;positive divisor=2

After

21=10·2+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, S003

S005 · implies

Before

d remainder:natural dividend=25;positive divisor=2

After

25=12·2+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, S004

S006 · calculation

Before

combined left value := sum(10, 21)

After

combined left value = 31

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 := sum(12, 25)

After

combined right value = 37

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=31;positive divisor=2

After

31=15·2+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=37;positive divisor=2

After

37=18·2+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=[0, 0];c,d remainders=[1, 1]

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(5, 6)

After

first 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, S010

S012 · calculation

Before

second quotient difference := sub(10, 12)

After

second quotient difference = -2

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

sum difference multiplier := sum(-1, -2)

After

sum difference multiplier = -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

combined difference := sub(31, 37)

After

combined difference = -6

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

modulus times difference multiplier := mul(2, -3)

After

modulus times difference multiplier = -6

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 · implies

Before

combined difference=-6;integer multiplier=-3;modulus=2

After

difference=n*multiplier=-6

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, S015

S017 · 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, 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

(2, [10, 12], [21, 25], 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: f0ac71426b4fdf9dfc1694d1a81509c7b0386705038994535b58d16fd45513bb
candidate record SHA-256: 82fd0ac246d6864fe14da32d9ea0a5962fefb99610f796e6934a7bea2e50162e
semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
Lean: NOT_RUN · human review: NOT_PERFORMED

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