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R0016-08831 · q-20261013-advance-20261018-r02-natural-divisibility-0081
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=5, k=0, l=1; 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=5, k=0, l=1; define b=a*k and c=b*l. Give (a, b, c, k, l, k*l, a divides c?).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.
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.
Rule: natural_divisibility_zero_scope · Uses: G001
Before
first multiple b=a*k := mul(5, 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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
second multiple c=b*l := mul(0, 1)
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
composed multiplier k*l := mul(0, 1)
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
a times composed multiplier := mul(5, 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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
composed divisibility:divide natural 0 by positive 5
After
0=0·5+0
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S005
Before
composed divisibility:a=5;b=0;quotient=0;remainder=0
After
divides=Yes;witness/status=0
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.
Rule: positive_divisor_remainder_criterion · Uses: G001, S006
Before
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.
Rule: divisibility_compose_given_witnesses · Uses: G001, S007
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.(5, 0, 0, 0, 1, 0, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 1136183f6ae57705a626fc506a60cf2ffed4cede6f1a1a251235688672fdbfd8candidate record SHA-256: 5c3c8d3f0282303769049e449f2f651b182f165acaf106d5ad52872d128a5506semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1Lean: NOT_RUN · human review: NOT_PERFORMED