Question 9063: Multiplicative Congruence Values, Premises, and Material Implication

Original question and givens

Use fixed positive modulus n=5, natural inputs a=13, b=18, c=27, d=37. 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=5, natural inputs a=13, b=18, c=27, d=37. 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=5

positive modulus=Yes

a remainder:natural dividend=13;positive divisor=5

13=2·5+3

b remainder:natural dividend=18;positive divisor=5

18=3·5+3

c remainder:natural dividend=27;positive divisor=5

27=5·5+2

d remainder:natural dividend=37;positive divisor=5

37=7·5+2

combined left value := mul(13, 27)

combined left value = 351

combined right value := mul(18, 37)

combined right value = 666

combined left remainder:natural dividend=351;positive divisor=5

351=70·5+1

combined right remainder:natural dividend=666;positive divisor=5

666=133·5+1

a,b remainders=[3, 3];c,d remainders=[2, 2]

both premises=Yes;conclusion=Yes

first quotient difference := sub(2, 3)

first quotient difference = -1

second quotient difference := sub(5, 7)

second quotient difference = -2

product difference first term := mul(13, -2)

product difference first term = -26

product difference second term := mul(37, -1)

product difference second term = -37

product difference multiplier := sum(-26, -37)

product difference multiplier = -63

combined difference := sub(351, 666)

combined difference = -315

modulus times difference multiplier := mul(5, -63)

modulus times difference multiplier = -315

combined difference=-315;integer multiplier=-63;modulus=5

difference=n*multiplier=-315

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

(5, [13, 18], [27, 37], 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.