# Valid Prime Products, Multiplicities, Euclid Witnesses and Sign Units

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=12>1 and candidate factors=[2, 3, 2] 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).

View answer and explanation(12, [2, 3, 2], Yes, 12, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (12, [2, 3, 2], Yes, 12, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0001/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 2. Prime Factors and a Complete Product Are Separate Validity Checks

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).

View answer and explanation(18, [2, 3, 3], Yes, 18, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (18, [2, 3, 3], Yes, 18, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0002/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 3. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=30>1 and candidate factors=[2, 3, 5] 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).

View answer and explanation(30, [2, 3, 5], Yes, 30, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (30, [2, 3, 5], Yes, 30, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0003/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 4. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=42>1 and candidate factors=[2, 3, 7] 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).

View answer and explanation(42, [2, 3, 7], Yes, 42, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (42, [2, 3, 7], Yes, 42, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0004/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 5. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=66>1 and candidate factors=[2, 3, 11] 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).

View answer and explanation(66, [2, 3, 11], Yes, 66, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (66, [2, 3, 11], Yes, 66, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0005/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 6. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=78>1 and candidate factors=[2, 3, 13] 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).

View answer and explanation(78, [2, 3, 13], Yes, 78, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (78, [2, 3, 13], Yes, 78, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0006/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 7. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=102>1 and candidate factors=[2, 3, 17] 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).

View answer and explanation(102, [2, 3, 17], Yes, 102, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (102, [2, 3, 17], Yes, 102, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0007/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 8. Prime Factors and a Complete Product Are Separate Validity Checks

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).

View answer and explanation(114, [2, 3, 19], Yes, 114, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (114, [2, 3, 19], Yes, 114, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0008/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 9. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=138>1 and candidate factors=[2, 3, 23] 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).

View answer and explanation(138, [2, 3, 23], Yes, 138, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (138, [2, 3, 23], Yes, 138, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0009/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 10. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=174>1 and candidate factors=[2, 3, 29] 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).

View answer and explanation(174, [2, 3, 29], Yes, 174, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (174, [2, 3, 29], Yes, 174, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0010/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 11. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=186>1 and candidate factors=[2, 3, 31] 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).

View answer and explanation(186, [2, 3, 31], Yes, 186, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (186, [2, 3, 31], Yes, 186, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0011/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 12. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=222>1 and candidate factors=[2, 3, 37] 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).

View answer and explanation(222, [2, 3, 37], Yes, 222, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (222, [2, 3, 37], Yes, 222, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0012/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 13. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=246>1 and candidate factors=[2, 3, 41] 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).

View answer and explanation(246, [2, 3, 41], Yes, 246, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (246, [2, 3, 41], Yes, 246, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0013/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 14. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=258>1 and candidate factors=[2, 3, 43] 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).

View answer and explanation(258, [2, 3, 43], Yes, 258, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (258, [2, 3, 43], Yes, 258, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0014/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 15. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=282>1 and candidate factors=[2, 3, 47] 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).

View answer and explanation(282, [2, 3, 47], Yes, 282, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (282, [2, 3, 47], Yes, 282, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0015/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 16. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=318>1 and candidate factors=[2, 3, 53] 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).

View answer and explanation(318, [2, 3, 53], Yes, 318, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (318, [2, 3, 53], Yes, 318, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0016/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 17. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=354>1 and candidate factors=[2, 3, 59] 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).

View answer and explanation(354, [2, 3, 59], Yes, 354, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (354, [2, 3, 59], Yes, 354, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0017/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 18. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=366>1 and candidate factors=[2, 3, 61] 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).

View answer and explanation(366, [2, 3, 61], Yes, 366, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (366, [2, 3, 61], Yes, 366, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0018/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 19. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=402>1 and candidate factors=[2, 3, 67] 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).

View answer and explanation(402, [2, 3, 67], Yes, 402, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (402, [2, 3, 67], Yes, 402, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0019/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 20. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=426>1 and candidate factors=[2, 3, 71] 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).

View answer and explanation(426, [2, 3, 71], Yes, 426, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (426, [2, 3, 71], Yes, 426, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0020/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 21. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=438>1 and candidate factors=[2, 3, 73] 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).

View answer and explanation(438, [2, 3, 73], Yes, 438, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (438, [2, 3, 73], Yes, 438, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0021/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 22. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=474>1 and candidate factors=[2, 3, 79] 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).

View answer and explanation(474, [2, 3, 79], Yes, 474, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (474, [2, 3, 79], Yes, 474, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0022/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 23. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=498>1 and candidate factors=[2, 3, 83] 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).

View answer and explanation(498, [2, 3, 83], Yes, 498, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (498, [2, 3, 83], Yes, 498, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0023/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 24. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=534>1 and candidate factors=[2, 3, 89] 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).

View answer and explanation(534, [2, 3, 89], Yes, 534, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (534, [2, 3, 89], Yes, 534, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0024/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 25. Prime Factors and a Complete Product Are Separate Validity Checks

Use positive integer n=582>1 and candidate factors=[2, 3, 97] 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).

View answer and explanation(582, [2, 3, 97], Yes, 582, Yes, existence_requires_complete_prime_product)

Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (582, [2, 3, 97], Yes, 582, Yes, existence_requires_complete_prime_product).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0025/)

Unique prime factorization of integers · qf-mod39-upf-factorization-audit · candidate





## 26. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=12>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(12, [2, 2, 3], [[2, 2], [3, 1]], 12, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (12, [2, 2, 3], [[2, 2], [3, 1]], 12, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0026/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 27. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=18>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(18, [2, 3, 3], [[2, 1], [3, 2]], 18, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (18, [2, 3, 3], [[2, 1], [3, 2]], 18, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0027/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 28. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=30>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(30, [2, 3, 5], [[2, 1], [3, 1], [5, 1]], 30, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (30, [2, 3, 5], [[2, 1], [3, 1], [5, 1]], 30, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0028/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 29. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=42>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(42, [2, 3, 7], [[2, 1], [3, 1], [7, 1]], 42, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (42, [2, 3, 7], [[2, 1], [3, 1], [7, 1]], 42, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0029/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 30. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=66>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(66, [2, 3, 11], [[2, 1], [3, 1], [11, 1]], 66, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (66, [2, 3, 11], [[2, 1], [3, 1], [11, 1]], 66, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0030/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 31. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=78>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(78, [2, 3, 13], [[2, 1], [3, 1], [13, 1]], 78, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (78, [2, 3, 13], [[2, 1], [3, 1], [13, 1]], 78, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0031/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 32. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=102>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(102, [2, 3, 17], [[2, 1], [3, 1], [17, 1]], 102, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (102, [2, 3, 17], [[2, 1], [3, 1], [17, 1]], 102, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0032/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 33. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=114>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(114, [2, 3, 19], [[2, 1], [3, 1], [19, 1]], 114, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (114, [2, 3, 19], [[2, 1], [3, 1], [19, 1]], 114, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0033/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 34. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=138>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(138, [2, 3, 23], [[2, 1], [3, 1], [23, 1]], 138, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (138, [2, 3, 23], [[2, 1], [3, 1], [23, 1]], 138, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0034/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 35. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=174>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(174, [2, 3, 29], [[2, 1], [3, 1], [29, 1]], 174, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (174, [2, 3, 29], [[2, 1], [3, 1], [29, 1]], 174, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0035/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 36. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=186>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(186, [2, 3, 31], [[2, 1], [3, 1], [31, 1]], 186, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (186, [2, 3, 31], [[2, 1], [3, 1], [31, 1]], 186, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0036/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 37. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=222>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(222, [2, 3, 37], [[2, 1], [3, 1], [37, 1]], 222, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (222, [2, 3, 37], [[2, 1], [3, 1], [37, 1]], 222, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0037/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 38. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=246>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(246, [2, 3, 41], [[2, 1], [3, 1], [41, 1]], 246, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (246, [2, 3, 41], [[2, 1], [3, 1], [41, 1]], 246, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0038/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 39. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=258>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(258, [2, 3, 43], [[2, 1], [3, 1], [43, 1]], 258, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (258, [2, 3, 43], [[2, 1], [3, 1], [43, 1]], 258, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0039/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 40. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=282>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(282, [2, 3, 47], [[2, 1], [3, 1], [47, 1]], 282, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (282, [2, 3, 47], [[2, 1], [3, 1], [47, 1]], 282, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0040/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 41. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=318>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(318, [2, 3, 53], [[2, 1], [3, 1], [53, 1]], 318, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (318, [2, 3, 53], [[2, 1], [3, 1], [53, 1]], 318, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0041/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 42. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=354>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0042/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 43. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=366>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(366, [2, 3, 61], [[2, 1], [3, 1], [61, 1]], 366, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (366, [2, 3, 61], [[2, 1], [3, 1], [61, 1]], 366, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0043/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 44. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=402>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(402, [2, 3, 67], [[2, 1], [3, 1], [67, 1]], 402, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (402, [2, 3, 67], [[2, 1], [3, 1], [67, 1]], 402, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0044/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 45. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=426>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(426, [2, 3, 71], [[2, 1], [3, 1], [71, 1]], 426, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (426, [2, 3, 71], [[2, 1], [3, 1], [71, 1]], 426, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0045/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 46. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=438>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(438, [2, 3, 73], [[2, 1], [3, 1], [73, 1]], 438, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (438, [2, 3, 73], [[2, 1], [3, 1], [73, 1]], 438, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0046/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 47. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=474>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(474, [2, 3, 79], [[2, 1], [3, 1], [79, 1]], 474, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (474, [2, 3, 79], [[2, 1], [3, 1], [79, 1]], 474, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0047/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 48. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=498>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(498, [2, 3, 83], [[2, 1], [3, 1], [83, 1]], 498, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (498, [2, 3, 83], [[2, 1], [3, 1], [83, 1]], 498, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0048/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 49. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=534>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(534, [2, 3, 89], [[2, 1], [3, 1], [89, 1]], 534, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (534, [2, 3, 89], [[2, 1], [3, 1], [89, 1]], 534, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0049/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 50. Complete Trial Division Produces the Canonical Prime Multiset

Use positive integer n=582>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).

View answer and explanation(582, [2, 3, 97], [[2, 1], [3, 1], [97, 1]], 582, Yes, complete_trial_division_range)

Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (582, [2, 3, 97], [[2, 1], [3, 1], [97, 1]], 582, Yes, complete_trial_division_range).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0050/)

Unique prime factorization of integers · qf-mod39-upf-canonical-factorization · candidate





## 51. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=12>1, candidate factors_left=[2, 3, 2] and factors_right=[2, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 2, 3], [2, 2, 3], [[2, Yes, 0, 2, Yes], [2, Yes, 0, 2, Yes], [3, Yes, 2, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 2, 3], [2, 2, 3], [[2, Yes, 0, 2, Yes], [2, Yes, 0, 2, Yes], [3, Yes, 2, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0051/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 52. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=18>1, candidate factors_left=[2, 3, 3] and factors_right=[3, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 3], [2, 3, 3], [[2, Yes, 1, 2, Yes], [3, Yes, 0, 3, Yes], [3, Yes, 0, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 3], [2, 3, 3], [[2, Yes, 1, 2, Yes], [3, Yes, 0, 3, Yes], [3, Yes, 0, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0052/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 53. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=30>1, candidate factors_left=[2, 3, 5] and factors_right=[5, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 5], [2, 3, 5], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [5, Yes, 0, 5, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 5], [2, 3, 5], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [5, Yes, 0, 5, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0053/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 54. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=42>1, candidate factors_left=[2, 3, 7] and factors_right=[7, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 7], [2, 3, 7], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [7, Yes, 0, 7, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 7], [2, 3, 7], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [7, Yes, 0, 7, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0054/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 55. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=66>1, candidate factors_left=[2, 3, 11] and factors_right=[11, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 11], [2, 3, 11], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [11, Yes, 0, 11, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 11], [2, 3, 11], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [11, Yes, 0, 11, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0055/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 56. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=78>1, candidate factors_left=[2, 3, 13] and factors_right=[13, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 13], [2, 3, 13], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [13, Yes, 0, 13, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 13], [2, 3, 13], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [13, Yes, 0, 13, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0056/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 57. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=102>1, candidate factors_left=[2, 3, 17] and factors_right=[17, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 17], [2, 3, 17], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [17, Yes, 0, 17, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 17], [2, 3, 17], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [17, Yes, 0, 17, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0057/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 58. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=114>1, candidate factors_left=[2, 3, 19] and factors_right=[19, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 19], [2, 3, 19], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [19, Yes, 0, 19, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 19], [2, 3, 19], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [19, Yes, 0, 19, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0058/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 59. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=138>1, candidate factors_left=[2, 3, 23] and factors_right=[23, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 23], [2, 3, 23], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [23, Yes, 0, 23, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 23], [2, 3, 23], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [23, Yes, 0, 23, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0059/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 60. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=174>1, candidate factors_left=[2, 3, 29] and factors_right=[29, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 29], [2, 3, 29], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [29, Yes, 0, 29, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 29], [2, 3, 29], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [29, Yes, 0, 29, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0060/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 61. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=186>1, candidate factors_left=[2, 3, 31] and factors_right=[31, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 31], [2, 3, 31], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [31, Yes, 0, 31, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 31], [2, 3, 31], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [31, Yes, 0, 31, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0061/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 62. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=222>1, candidate factors_left=[2, 3, 37] and factors_right=[37, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 37], [2, 3, 37], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [37, Yes, 0, 37, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 37], [2, 3, 37], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [37, Yes, 0, 37, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0062/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 63. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=246>1, candidate factors_left=[2, 3, 41] and factors_right=[41, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 41], [2, 3, 41], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [41, Yes, 0, 41, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 41], [2, 3, 41], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [41, Yes, 0, 41, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0063/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 64. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=258>1, candidate factors_left=[2, 3, 43] and factors_right=[43, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 43], [2, 3, 43], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [43, Yes, 0, 43, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 43], [2, 3, 43], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [43, Yes, 0, 43, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0064/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 65. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=282>1, candidate factors_left=[2, 3, 47] and factors_right=[47, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 47], [2, 3, 47], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [47, Yes, 0, 47, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 47], [2, 3, 47], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [47, Yes, 0, 47, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0065/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 66. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=318>1, candidate factors_left=[2, 3, 53] and factors_right=[53, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 53], [2, 3, 53], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [53, Yes, 0, 53, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 53], [2, 3, 53], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [53, Yes, 0, 53, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0066/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 67. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=354>1, candidate factors_left=[2, 3, 59] and factors_right=[59, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 59], [2, 3, 59], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [59, Yes, 0, 59, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 59], [2, 3, 59], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [59, Yes, 0, 59, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0067/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 68. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=366>1, candidate factors_left=[2, 3, 61] and factors_right=[61, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 61], [2, 3, 61], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [61, Yes, 0, 61, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 61], [2, 3, 61], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [61, Yes, 0, 61, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0068/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 69. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=402>1, candidate factors_left=[2, 3, 67] and factors_right=[67, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 67], [2, 3, 67], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [67, Yes, 0, 67, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 67], [2, 3, 67], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [67, Yes, 0, 67, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0069/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 70. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=426>1, candidate factors_left=[2, 3, 71] and factors_right=[71, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 71], [2, 3, 71], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [71, Yes, 0, 71, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 71], [2, 3, 71], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [71, Yes, 0, 71, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0070/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 71. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=438>1, candidate factors_left=[2, 3, 73] and factors_right=[73, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 73], [2, 3, 73], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [73, Yes, 0, 73, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 73], [2, 3, 73], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [73, Yes, 0, 73, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0071/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 72. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=474>1, candidate factors_left=[2, 3, 79] and factors_right=[79, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 79], [2, 3, 79], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [79, Yes, 0, 79, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 79], [2, 3, 79], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [79, Yes, 0, 79, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0072/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 73. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=498>1, candidate factors_left=[2, 3, 83] and factors_right=[83, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 83], [2, 3, 83], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [83, Yes, 0, 83, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 83], [2, 3, 83], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [83, Yes, 0, 83, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0073/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 74. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=534>1, candidate factors_left=[2, 3, 89] and factors_right=[89, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 89], [2, 3, 89], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [89, Yes, 0, 89, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 89], [2, 3, 89], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [89, Yes, 0, 89, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0074/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 75. Only Two Valid Prime Products Can Supply This Uniqueness Certificate

Use positive integer n=582>1, candidate factors_left=[2, 3, 97] and factors_right=[97, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).

View answer and explanation(Yes, Yes, [2, 3, 97], [2, 3, 97], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [97, Yes, 0, 97, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)

Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 97], [2, 3, 97], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [97, Yes, 0, 97, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0075/)

Unique prime factorization of integers · qf-mod39-upf-uniqueness-audit · candidate





## 76. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=0. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(0, No, null, [], zero_has_no_prime_factorization)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (0, No, null, [], zero_has_no_prime_factorization).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0076/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 77. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=1. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(1, No, 1, [], units_have_empty_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (1, No, 1, [], units_have_empty_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0077/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 78. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-1. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-1, No, -1, [], units_have_empty_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-1, No, -1, [], units_have_empty_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0078/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 79. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-42. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-42, No, -1, [2, 3, 7], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-42, No, -1, [2, 3, 7], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0079/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 80. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-66. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-66, No, -1, [2, 3, 11], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-66, No, -1, [2, 3, 11], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0080/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 81. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-78. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-78, No, -1, [2, 3, 13], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-78, No, -1, [2, 3, 13], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0081/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 82. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-102. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-102, No, -1, [2, 3, 17], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-102, No, -1, [2, 3, 17], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0082/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 83. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-114. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-114, No, -1, [2, 3, 19], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-114, No, -1, [2, 3, 19], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0083/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 84. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-138. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-138, No, -1, [2, 3, 23], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-138, No, -1, [2, 3, 23], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0084/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 85. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-174. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-174, No, -1, [2, 3, 29], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-174, No, -1, [2, 3, 29], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0085/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 86. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-186. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-186, No, -1, [2, 3, 31], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-186, No, -1, [2, 3, 31], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0086/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 87. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-222. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-222, No, -1, [2, 3, 37], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-222, No, -1, [2, 3, 37], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0087/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 88. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-246. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-246, No, -1, [2, 3, 41], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-246, No, -1, [2, 3, 41], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0088/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 89. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-258. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-258, No, -1, [2, 3, 43], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-258, No, -1, [2, 3, 43], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0089/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 90. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-282. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-282, No, -1, [2, 3, 47], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-282, No, -1, [2, 3, 47], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0090/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 91. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-318. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-318, No, -1, [2, 3, 53], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-318, No, -1, [2, 3, 53], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0091/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 92. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-354. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-354, No, -1, [2, 3, 59], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-354, No, -1, [2, 3, 59], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0092/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 93. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-366. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-366, No, -1, [2, 3, 61], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-366, No, -1, [2, 3, 61], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0093/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 94. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-402. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-402, No, -1, [2, 3, 67], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-402, No, -1, [2, 3, 67], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0094/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 95. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-426. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-426, No, -1, [2, 3, 71], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-426, No, -1, [2, 3, 71], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0095/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 96. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-438. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-438, No, -1, [2, 3, 73], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-438, No, -1, [2, 3, 73], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0096/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 97. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-474. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-474, No, -1, [2, 3, 79], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-474, No, -1, [2, 3, 79], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0097/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 98. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-498. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-498, No, -1, [2, 3, 83], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-498, No, -1, [2, 3, 83], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0098/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 99. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-534. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-534, No, -1, [2, 3, 89], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-534, No, -1, [2, 3, 89], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0099/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate





## 100. Separate Zero, Sign Units and the Positive Prime Multiset

Use original bounded integer n=-582. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).

This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization.

View answer and explanation(-582, No, -1, [2, 3, 97], integer_sign_separated_from_positive_prime_multiset)

Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-582, No, -1, [2, 3, 97], integer_sign_separated_from_positive_prime_multiset).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r06-unique-prime-factorization-0100/)

Unique prime factorization of integers · qf-mod39-upf-unit-sign-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations/validation-report.json)