Question 9068: Multiplicative Congruence Values, Premises, and Material Implication

Original question and givens

Use fixed positive modulus n=10, natural inputs a=18, b=48, c=37, d=47. 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=10, natural inputs a=18, b=48, c=37, d=47. 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=10

positive modulus=Yes

a remainder:natural dividend=18;positive divisor=10

18=1·10+8

b remainder:natural dividend=48;positive divisor=10

48=4·10+8

c remainder:natural dividend=37;positive divisor=10

37=3·10+7

d remainder:natural dividend=47;positive divisor=10

47=4·10+7

combined left value := mul(18, 37)

combined left value = 666

combined right value := mul(48, 47)

combined right value = 2256

combined left remainder:natural dividend=666;positive divisor=10

666=66·10+6

combined right remainder:natural dividend=2256;positive divisor=10

2256=225·10+6

a,b remainders=[8, 8];c,d remainders=[7, 7]

both premises=Yes;conclusion=Yes

first quotient difference := sub(1, 4)

first quotient difference = -3

second quotient difference := sub(3, 4)

second quotient difference = -1

product difference first term := mul(18, -1)

product difference first term = -18

product difference second term := mul(47, -3)

product difference second term = -141

product difference multiplier := sum(-18, -141)

product difference multiplier = -159

combined difference := sub(666, 2256)

combined difference = -1590

modulus times difference multiplier := mul(10, -159)

modulus times difference multiplier = -1590

combined difference=-1590;integer multiplier=-159;modulus=10

difference=n*multiplier=-1590

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

(10, [18, 48], [37, 47], Yes, [6, 6], 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.