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Question 8866: Natural Divisibility and an Explicit Multiplier Witness

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0016-08866 · q-20261013-advance-20261018-r02-natural-divisibility-0016

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=15, b=240. 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=15, b=240. 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: natural_divisibility_zero_scope · Uses: G001

S002 · implies

Before

given divisibility:divide natural 240 by positive 15

After

240=16·15+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 dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S001

S003 · implies

Before

given divisibility:a=15;b=240;quotient=16;remainder=0

After

divides=Yes;witness/status=16

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 dependencies

Rule: 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

(15, 240, Yes, 16)

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: 1a299e0b3b192303fadbc66f5e5bd7ca5e0769c39cb059b65c2629d6f42b9792
  • candidate record SHA-256: 7650f43b865496944b72df99b44903bfb824f6444c2d82eac0ccdf2736dcc8b6
  • semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2
  • question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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