Question 9061: Multiplicative Congruence Values, Premises, and Material Implication

Original question and givens

Use fixed positive modulus n=3, natural inputs a=11, b=17, c=23, d=32. 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=3, natural inputs a=11, b=17, c=23, d=32. 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=3

positive modulus=Yes

a remainder:natural dividend=11;positive divisor=3

11=3·3+2

b remainder:natural dividend=17;positive divisor=3

17=5·3+2

c remainder:natural dividend=23;positive divisor=3

23=7·3+2

d remainder:natural dividend=32;positive divisor=3

32=10·3+2

combined left value := mul(11, 23)

combined left value = 253

combined right value := mul(17, 32)

combined right value = 544

combined left remainder:natural dividend=253;positive divisor=3

253=84·3+1

combined right remainder:natural dividend=544;positive divisor=3

544=181·3+1

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

both premises=Yes;conclusion=Yes

first quotient difference := sub(3, 5)

first quotient difference = -2

second quotient difference := sub(7, 10)

second quotient difference = -3

product difference first term := mul(11, -3)

product difference first term = -33

product difference second term := mul(32, -2)

product difference second term = -64

product difference multiplier := sum(-33, -64)

product difference multiplier = -97

combined difference := sub(253, 544)

combined difference = -291

modulus times difference multiplier := mul(3, -97)

modulus times difference multiplier = -291

combined difference=-291;integer multiplier=-97;modulus=3

difference=n*multiplier=-291

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

(3, [11, 17], [23, 32], 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.