Question 8834: Compose Two Given Natural Divisibility Witnesses

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0016-08834 · q-20261013-advance-20261018-r02-natural-divisibility-0084

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=2, k=3, l=3; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).

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=2, k=3, l=3; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).

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

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 · calculation

Before

first multiple b=a*k := mul(2, 3)

After

first multiple b=a*k = 6

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

second multiple c=b*l := mul(6, 3)

After

second multiple c=b*l = 18

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

composed multiplier k*l := mul(3, 3)

After

composed multiplier k*l = 9

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

a times composed multiplier := mul(2, 9)

After

a times composed multiplier = 18

Conditions: none

Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

composed divisibility:divide natural 18 by positive 2

After

18=9·2+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 dependencies

Rule: natural_euclidean_division_witness · Uses: G001, S005

S007 · implies

Before

composed divisibility:a=2;b=18;quotient=9;remainder=0

After

divides=Yes;witness/status=9

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, S006

S008 · implies

Before

b=a*k=6;c=b*l=18;k*l=9

After

c=a*(k*l)=18;a divides c=Yes

Conditions: none

Compose the two supplied witnesses by multiplication and associativity to get c=a*(k*l).Zero multipliers and a=0 require neither cancellation nor division.

Step identifiers and dependencies

Rule: divisibility_compose_given_witnesses · 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 — Recompute a times composed multiplier with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute composed multiplier k*l with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute second multiple c=b*l with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute first multiple b=a*k with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(2, 6, 18, 3, 3, 9, Yes)

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: d1d5eef4871970714710ab0497d7a4b4fd4b2a40d61811634ff55bb75994d026
candidate record SHA-256: 9695a3712b0cfc9e3ba0041f4800b34148f7ad4c6073e2584c06f19a8de991fc
semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2
question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1
Lean: NOT_RUN · human review: NOT_PERFORMED

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