# Question 8854: Natural Divisibility and an Explicit Multiplier Witness

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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.

R0016-08854 · q-20261013-advance-20261018-r02-natural-divisibility-0004



## 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. a=3, b=12. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.


- 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. a=3, b=12. Give (a, b, a divides b?, natural witness k for b=a*k or does-not-exist). For 0 divides 0 choose k=0, without claiming uniqueness.





## 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

given divisibility:divide natural 12 by positive 3

After

12=4·3+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

given divisibility:a=3;b=12;quotient=4;remainder=0

After

divides=Yes;witness/status=4

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









## Reverse checks


- 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

(3, 12, Yes, 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: 9ebbfd111de201827384dd33683f563ea68e047727f8969b3c5013e0cdc48869
- candidate record SHA-256: 615c1395f3c68b443efb100b51654fe39876787013857307eb9c3a8e971ee154
- semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2
- question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261018-r02-natural-divisibility-0004.json)