# Question 8826: Compose Two Given Natural Divisibility Witnesses [Return to catalogue page 45](https://ocme.evemisslab.com/en/questions/page/45/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261018-r02-natural-divisibility-foundations/#q-20261013-advance-20261018-r02-natural-divisibility-0076) 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. R0016-08826 · q-20261013-advance-20261018-r02-natural-divisibility-0076 ## 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=0, k=0, l=0; 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=0, k=0, l=0; 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](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 · calculationBefore first multiple b=a*k := mul(0, 0) After first multiple b=a*k = 0 Conditions: none Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore second multiple c=b*l := mul(0, 0) After second multiple c=b*l = 0 Conditions: none Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore composed multiplier k*l := mul(0, 0) After composed multiplier k*l = 0 Conditions: none Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore a times composed multiplier := mul(0, 0) After a times composed multiplier = 0 Conditions: none Evaluate exactly with this step's source operands and operation order.Check a denominator is nonzero before division. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore composed divisibility:a=0;b=0 After 0 divides b=Yes Conditions: none Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero. Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S005 S007 · impliesBefore b=a*k=0;c=b*l=0;k*l=0 After c=a*(k*l)=0;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 dependenciesRule: divisibility_compose_given_witnesses · Uses: G001, S006 ## Reverse checks - 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 (0, 0, 0, 0, 0, 0, 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: 0b5db2de0b76485c80f54ca8f23b270357e3eef1e3bae24212f09908d5916bf6 - candidate record SHA-256: 7dbc4bfe4f5cd09c01d751137e8113f8c19d1f15a438ba0efd414da0f442d06d - 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-0076.json)