# Question 8801: Complete Positive Divisor Inventory Including the Zero Case [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-0026) 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. R0016-08801 · q-20261013-advance-20261018-r02-natural-divisibility-0026 ## 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=0. 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=0. 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 b=0;positive candidate d is arbitrary After d*0=0 for every positive d;positive divisors=all positive natural numbers;count=infinite Conditions: none Every positive d has multiplier0,so the positive-divisor set is all positive naturals and is infinite.This is not a finite-window sample,and divisor0 is not a positive divisor. Step identifiers and dependenciesRule: zero_all_positive_divisors_unbounded · Uses: G001, S001 ## Reverse checks - 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 (0, all-positive-naturals, infinite) 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: 7ac5eb8a1d773434df82883bc195b25e920be10716de9865c2f9488e68b65ea0 - candidate record SHA-256: 0fd0fa9a609007684a5f71bd073e6bc84c3074583876ba3798324cdc11c4a671 - 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-0026.json)