# Question 8805: Complete Positive Divisor Inventory Including the Zero Case

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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.

R0016-08805 · q-20261013-advance-20261018-r02-natural-divisibility-0030



## Original question and givens

Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=14. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.


- G001 — Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=14. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.





## 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-natural-divisibility](https://ocme.evemisslab.com/en/mko/mko-natural-divisibility/)





## Step-by-step derivation

S001 · impliesBefore

natural numbers include0;b is divisible by a iff b=a*k for some natural k

After

0 divides only0;every natural divides0

Conditions: none

Divisibility is an existential multiplier relation without presuming a positive or nonzero divisor.Every natural,including0,divides0.A positive-divisor inventory separately excludes0.

Step identifiers and dependenciesRule: natural_divisibility_zero_scope · Uses: G001





S002 · impliesBefore

positive divisor candidate1:divide natural 14 by positive 1

After

14=14·1+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

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





S003 · impliesBefore

positive divisor candidate1:a=1;b=14;quotient=14;remainder=0

After

divides=Yes;witness/status=14

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

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





S004 · impliesBefore

positive divisor candidate2:divide natural 14 by positive 2

After

14=7·2+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

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





S005 · impliesBefore

positive divisor candidate2:a=2;b=14;quotient=7;remainder=0

After

divides=Yes;witness/status=7

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

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





S006 · impliesBefore

positive divisor candidate3:divide natural 14 by positive 3

After

14=4·3+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

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





S007 · impliesBefore

positive divisor candidate3:a=3;b=14;quotient=4;remainder=2

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

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





S008 · impliesBefore

positive divisor candidate4:divide natural 14 by positive 4

After

14=3·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

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





S009 · impliesBefore

positive divisor candidate4:a=4;b=14;quotient=3;remainder=2

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S008





S010 · impliesBefore

positive divisor candidate5:divide natural 14 by positive 5

After

14=2·5+4

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S009





S011 · impliesBefore

positive divisor candidate5:a=5;b=14;quotient=2;remainder=4

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S010





S012 · impliesBefore

positive divisor candidate6:divide natural 14 by positive 6

After

14=2·6+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S011





S013 · impliesBefore

positive divisor candidate6:a=6;b=14;quotient=2;remainder=2

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S012





S014 · impliesBefore

positive divisor candidate7:divide natural 14 by positive 7

After

14=2·7+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S013





S015 · impliesBefore

positive divisor candidate7:a=7;b=14;quotient=2;remainder=0

After

divides=Yes;witness/status=2

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S014





S016 · impliesBefore

positive divisor candidate8:divide natural 14 by positive 8

After

14=1·8+6

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S015





S017 · impliesBefore

positive divisor candidate8:a=8;b=14;quotient=1;remainder=6

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S016





S018 · impliesBefore

positive divisor candidate9:divide natural 14 by positive 9

After

14=1·9+5

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S017





S019 · impliesBefore

positive divisor candidate9:a=9;b=14;quotient=1;remainder=5

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S018





S020 · impliesBefore

positive divisor candidate10:divide natural 14 by positive 10

After

14=1·10+4

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S019





S021 · impliesBefore

positive divisor candidate10:a=10;b=14;quotient=1;remainder=4

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S020





S022 · impliesBefore

positive divisor candidate11:divide natural 14 by positive 11

After

14=1·11+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S021





S023 · impliesBefore

positive divisor candidate11:a=11;b=14;quotient=1;remainder=3

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S022





S024 · impliesBefore

positive divisor candidate12:divide natural 14 by positive 12

After

14=1·12+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S023





S025 · impliesBefore

positive divisor candidate12:a=12;b=14;quotient=1;remainder=2

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S024





S026 · impliesBefore

positive divisor candidate13:divide natural 14 by positive 13

After

14=1·13+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S025





S027 · impliesBefore

positive divisor candidate13:a=13;b=14;quotient=1;remainder=1

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S026





S028 · impliesBefore

positive divisor candidate14:divide natural 14 by positive 14

After

14=1·14+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S027





S029 · impliesBefore

positive divisor candidate14:a=14;b=14;quotient=1;remainder=0

After

divides=Yes;witness/status=1

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S028





S030 · impliesBefore

positive b=14;complete candidate interval=[1,14]

After

positive divisors=[1, 2, 7, 14];count=4

Conditions: none

If positive d divides positive b,then b=d*k with k≥1,so d≤b.Thus1 through b is the complete candidate domain,not a sample of some factors.

Step identifiers and dependenciesRule: complete_positive_divisor_bound · Uses: G001, S029









## Reverse checks


- S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — 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

(14, [1, 2, 7, 14], 4)

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: 9e78a38295ac06f298dbdbdf3f50636fec9aaf086767b8a240decfc61ce81369
- candidate record SHA-256: 0e994ce955ebf52ffea40d8a8ef00f86065a5dbab096d79fa3d31410c220bf42
- semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2
- question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1
- Lean: NOT_RUN · human review: NOT_PERFORMED

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