# Question 9116: Complete Positive Divisors and Prime Qualification

[Return to catalogue page 46](https://ocme.evemisslab.com/en/questions/page/46/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261018-r04-prime-number-foundations/#q-20261013-advance-20261018-r04-prime-number-0016)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0017-09116 · q-20261013-advance-20261018-r04-prime-number-0016



## Original question and givens

Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=15. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).


- G001 — Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. p=15. Give (p, positive divisors in increasing order or all-positive-naturals for p=0, p≥2?, prime?).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-prime-number](https://ocme.evemisslab.com/en/mko/mko-prime-number/)





## Step-by-step derivation

S001 · impliesBefore

natural-number prime protocol;0 and1 are not prime

After

a failed candidate witness alone is not a primality proof

Conditions: none

Naturals include0.Primality retains both n≥2 and a complete factor bound.A failed single factor candidate only invalidates that witness;it does not certify primality.

Step identifiers and dependenciesRule: prime_complete_evidence_scope · Uses: G001





S002 · impliesBefore

positive factor pair candidate1:natural dividend=15;positive divisor=1

After

15=15·1+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S001





S003 · impliesBefore

candidate d=1;quotient=15;remainder=0

After

factor pair=[1, 15]

Conditions: none

Only a zero remainder adds d and n/d;deduplicate a square-root pair.Every factor pair has an endpoint≤sqrt(n),so pairing covers all positive divisors.

Step identifiers and dependenciesRule: complete_positive_factor_pair_inventory · Uses: G001, S002





S004 · impliesBefore

positive factor pair candidate2:natural dividend=15;positive divisor=2

After

15=7·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S003





S005 · impliesBefore

candidate d=2;quotient=7;remainder=1

After

factor pair=none

Conditions: none

Only a zero remainder adds d and n/d;deduplicate a square-root pair.Every factor pair has an endpoint≤sqrt(n),so pairing covers all positive divisors.

Step identifiers and dependenciesRule: complete_positive_factor_pair_inventory · Uses: G001, S004





S006 · impliesBefore

positive factor pair candidate3:natural dividend=15;positive divisor=3

After

15=5·3+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S005





S007 · impliesBefore

candidate d=3;quotient=5;remainder=0

After

factor pair=[3, 5]

Conditions: none

Only a zero remainder adds d and n/d;deduplicate a square-root pair.Every factor pair has an endpoint≤sqrt(n),so pairing covers all positive divisors.

Step identifiers and dependenciesRule: complete_positive_factor_pair_inventory · Uses: G001, S006





S008 · impliesBefore

all positive divisors=[1, 3, 5, 15];n=15

After

n>=2=Yes;exactly1 and n=No;prime=No

Conditions: none

Primality requires n≥2 and the complete positive-divisor set exactly{1,n}.At n=1 there is only one divisor1,not two copies in[1,1].

Step identifiers and dependenciesRule: prime_positive_divisors_with_lower_bound · Uses: G001, S007









## Reverse checks


- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(15, [1, 3, 5, 15], Yes, No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: d87e2ddb52b24eaaebd600727689ae6c2e9a5ef7303616ff9caed238caefb725
- candidate record SHA-256: e5b9f2cef92bba3b6e93f64a2cac8c022041b831c489ebbf745e653a58427401
- semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
- question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261018-r04-prime-number-0016.json)