Question 9071: Multiplicative Congruence Values, Premises, and Material Implication

Original question and givens

Use fixed positive modulus n=4, natural inputs a=21, b=33, c=43, 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=4, natural inputs a=21, b=33, c=43, 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=4

positive modulus=Yes

a remainder:natural dividend=21;positive divisor=4

21=5·4+1

b remainder:natural dividend=33;positive divisor=4

33=8·4+1

c remainder:natural dividend=43;positive divisor=4

43=10·4+3

d remainder:natural dividend=47;positive divisor=4

47=11·4+3

combined left value := mul(21, 43)

combined left value = 903

combined right value := mul(33, 47)

combined right value = 1551

combined left remainder:natural dividend=903;positive divisor=4

903=225·4+3

combined right remainder:natural dividend=1551;positive divisor=4

1551=387·4+3

a,b remainders=[1, 1];c,d remainders=[3, 3]

both premises=Yes;conclusion=Yes

first quotient difference := sub(5, 8)

first quotient difference = -3

second quotient difference := sub(10, 11)

second quotient difference = -1

product difference first term := mul(21, -1)

product difference first term = -21

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

product difference second term = -141

product difference multiplier := sum(-21, -141)

product difference multiplier = -162

combined difference := sub(903, 1551)

combined difference = -648

modulus times difference multiplier := mul(4, -162)

modulus times difference multiplier = -648

combined difference=-648;integer multiplier=-162;modulus=4

difference=n*multiplier=-648

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

(4, [21, 33], [43, 47], Yes, [3, 3], 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.