Question 9060: Multiplicative Congruence Values, Premises, and Material Implication

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], product congruence?, premises imply conclusion?). Do not cancel factors without additional conditions.

• 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], 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.

• mko-modular-congruence

Step-by-step derivation

fixed natural modulus candidate=2

positive modulus=Yes

a remainder:natural dividend=10;positive divisor=2

10=5·2+0

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

12=6·2+0

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

21=10·2+1

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

25=12·2+1

combined left value := mul(10, 21)

combined left value = 210

combined right value := mul(12, 25)

combined right value = 300

combined left remainder:natural dividend=210;positive divisor=2

210=105·2+0

combined right remainder:natural dividend=300;positive divisor=2

300=150·2+0

a,b remainders=[0, 0];c,d remainders=[1, 1]

both premises=Yes;conclusion=Yes

first quotient difference := sub(5, 6)

first quotient difference = -1

second quotient difference := sub(10, 12)

second quotient difference = -2

product difference first term := mul(10, -2)

product difference first term = -20

product difference second term := mul(25, -1)

product difference second term = -25

product difference multiplier := sum(-20, -25)

product difference multiplier = -45

combined difference := sub(210, 300)

combined difference = -90

modulus times difference multiplier := mul(2, -45)

modulus times difference multiplier = -90

combined difference=-90;integer multiplier=-45;modulus=2

difference=n*multiplier=-90

premises=Yes;conclusion=Yes

material implication=Yes

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

(2, [10, 12], [21, 25], 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.