Question 21333: Prime Factors and a Complete Product Are Separate Validity Checks

Original question and givens

Use positive integer n=114>1 and candidate factors=[2, 3, 19] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).

• G001 — Use positive integer n=114>1 and candidate factors=[2, 3, 19] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).

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-unique-prime-factorization

Step-by-step derivation

source operation=upf-factorization-audit;original snapshot and mathematics retained

explicit ordered source output fields=[n, factors, all_factors_prime, factor_product, valid_factorization, certificate]

upf candidate factor index0:ORIGINAL candidate factor=2;ALL possible trial divisors2..floor(sqrt(v))=[]

ALL [divisor,remainder] rows=[];prime=Yes

upf candidate complete product at original factor0 := mul(1, 2)

upf candidate complete product at original factor0 = 2

upf candidate factor index1:ORIGINAL candidate factor=3;ALL possible trial divisors2..floor(sqrt(v))=[]

ALL [divisor,remainder] rows=[];prime=Yes

upf candidate complete product at original factor1 := mul(2, 3)

upf candidate complete product at original factor1 = 6

upf candidate factor index2:ORIGINAL candidate factor=19;ALL possible trial divisors2..floor(sqrt(v))=[2, 3, 4]

ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 3]];prime=Yes

upf candidate complete product at original factor2 := mul(6, 19)

upf candidate complete product at original factor2 = 114

upf candidate:ORIGINAL factors=[2, 3, 19];ALL primality flags=[Yes, Yes, Yes];original n=114

empty product convention1;actual complete product=114;all_prime=Yes;valid_prime_product=Yes

Reverse checks

• 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 upf candidate complete product at original factor2 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute upf candidate complete product at original factor1 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute upf candidate complete product at original factor0 with exact rational arithmetic, not rounded decimals.
• 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

(114, [2, 3, 19], Yes, 114, Yes, existence_requires_complete_prime_product)

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.