# Question 21327: Prime Factors and a Complete Product Are Separate Validity Checks [Return to catalogue page 107](https://ocme.evemisslab.com/en/questions/page/107/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations/#q-20261014-advance-20261108-r06-unique-prime-factorization-0002) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0041-21327 · q-20261014-advance-20261108-r06-unique-prime-factorization-0002 ## Original question and givens Use positive integer n=18>1 and candidate factors=[2, 3, 3] 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=18>1 and candidate factors=[2, 3, 3] 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](https://ocme.evemisslab.com/en/mko/mko-unique-prime-factorization/) ## Step-by-step derivation S001 · impliesBefore source operation=upf-factorization-audit;original snapshot and mathematics retained After explicit ordered source output fields=[n, factors, all_factors_prime, factor_product, valid_factorization, certificate] Conditions: none Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore upf candidate factor index0:ORIGINAL candidate factor=2;ALL possible trial divisors2..floor(sqrt(v))=[] After ALL [divisor,remainder] rows=[];prime=Yes Conditions: none A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation. Step identifiers and dependenciesRule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S001 S003 · calculationBefore upf candidate complete product at original factor0 := mul(1, 2) After upf candidate complete product at original factor0 = 2 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore upf candidate factor index1:ORIGINAL candidate factor=3;ALL possible trial divisors2..floor(sqrt(v))=[] After ALL [divisor,remainder] rows=[];prime=Yes Conditions: none A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation. Step identifiers and dependenciesRule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S003 S005 · calculationBefore upf candidate complete product at original factor1 := mul(2, 3) After upf candidate complete product at original factor1 = 6 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore upf candidate factor index2:ORIGINAL candidate factor=3;ALL possible trial divisors2..floor(sqrt(v))=[] After ALL [divisor,remainder] rows=[];prime=Yes Conditions: none A composite integer greater than1 has a divisor at most its square root.Complete nondivisibility over that interval certifies primality,including the empty interval for2.Matching a product cannot replace this obligation. Step identifiers and dependenciesRule: upf_every_trial_divisor_primality_not_product_only · Uses: G001, S005 S007 · calculationBefore upf candidate complete product at original factor2 := mul(6, 3) After upf candidate complete product at original factor2 = 18 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore upf candidate:ORIGINAL factors=[2, 3, 3];ALL primality flags=[Yes, Yes, Yes];original n=18 After empty product convention1;actual complete product=18;all_prime=Yes;valid_prime_product=Yes Conditions: none Keep original factor order.An empty list vacuously has all-prime entries,but its product1 differs from n>1.A matching composite product is not a prime factorization.The SAFE guard concerns source projection,not theorem truth. Step identifiers and dependenciesRule: upf_candidate_prime_and_complete_product_separate · Uses: G001, S007 ## 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 (18, [2, 3, 3], Yes, 18, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 26c029015237b7b4d1cdea625d54fef4641c6bb81ca026ee6ef4824a7d9143c4 - candidate record SHA-256: 00d582258490344a68aef4bf18d3881e13e3cacbcb4c7a27ea2d86caca34e549 - semantic review receipt SHA-256: 8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a - question-first witness SHA-256: ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0002.json)