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R0016-08806 · q-20261013-advance-20261018-r02-natural-divisibility-0031
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=15. 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=15. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.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
positive divisor candidate1:divide natural 15 by positive 1
After
15=15·1+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, S001
Before
positive divisor candidate1:a=1;b=15;quotient=15;remainder=0
After
divides=Yes;witness/status=15
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, S002
Before
positive divisor candidate2:divide natural 15 by positive 2
After
15=7·2+1
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, S003
Before
positive divisor candidate2:a=2;b=15;quotient=7;remainder=1
After
divides=No;witness/status=does not exist
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, S004
Before
positive divisor candidate3:divide natural 15 by positive 3
After
15=5·3+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
positive divisor candidate3:a=3;b=15;quotient=5;remainder=0
After
divides=Yes;witness/status=5
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
positive divisor candidate4:divide natural 15 by positive 4
After
15=3·4+3
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, S007
Before
positive divisor candidate4:a=4;b=15;quotient=3;remainder=3
After
divides=No;witness/status=does not exist
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, S008
Before
positive divisor candidate5:divide natural 15 by positive 5
After
15=3·5+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, S009
Before
positive divisor candidate5:a=5;b=15;quotient=3;remainder=0
After
divides=Yes;witness/status=3
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, S010
Before
positive divisor candidate6:divide natural 15 by positive 6
After
15=2·6+3
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, S011
Before
positive divisor candidate6:a=6;b=15;quotient=2;remainder=3
After
divides=No;witness/status=does not exist
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, S012
Before
positive divisor candidate7:divide natural 15 by positive 7
After
15=2·7+1
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, S013
Before
positive divisor candidate7:a=7;b=15;quotient=2;remainder=1
After
divides=No;witness/status=does not exist
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, S014
Before
positive divisor candidate8:divide natural 15 by positive 8
After
15=1·8+7
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, S015
Before
positive divisor candidate8:a=8;b=15;quotient=1;remainder=7
After
divides=No;witness/status=does not exist
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, S016
Before
positive divisor candidate9:divide natural 15 by positive 9
After
15=1·9+6
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, S017
Before
positive divisor candidate9:a=9;b=15;quotient=1;remainder=6
After
divides=No;witness/status=does not exist
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, S018
Before
positive divisor candidate10:divide natural 15 by positive 10
After
15=1·10+5
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, S019
Before
positive divisor candidate10:a=10;b=15;quotient=1;remainder=5
After
divides=No;witness/status=does not exist
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, S020
Before
positive divisor candidate11:divide natural 15 by positive 11
After
15=1·11+4
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, S021
Before
positive divisor candidate11:a=11;b=15;quotient=1;remainder=4
After
divides=No;witness/status=does not exist
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, S022
Before
positive divisor candidate12:divide natural 15 by positive 12
After
15=1·12+3
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, S023
Before
positive divisor candidate12:a=12;b=15;quotient=1;remainder=3
After
divides=No;witness/status=does not exist
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, S024
Before
positive divisor candidate13:divide natural 15 by positive 13
After
15=1·13+2
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, S025
Before
positive divisor candidate13:a=13;b=15;quotient=1;remainder=2
After
divides=No;witness/status=does not exist
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, S026
Before
positive divisor candidate14:divide natural 15 by positive 14
After
15=1·14+1
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, S027
Before
positive divisor candidate14:a=14;b=15;quotient=1;remainder=1
After
divides=No;witness/status=does not exist
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, S028
Before
positive divisor candidate15:divide natural 15 by positive 15
After
15=1·15+0
Conditions: none
A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.
Rule: natural_euclidean_division_witness · Uses: G001, S029
Before
positive divisor candidate15:a=15;b=15;quotient=1;remainder=0
After
divides=Yes;witness/status=1
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, S030
Before
positive b=15;complete candidate interval=[1,15]
After
positive divisors=[1, 3, 5, 15];count=4
Conditions: none
If positive d divides positive b,then b=d*k with k≥1,so d≤b.Thus1 through b is the complete candidate domain,not a sample of some factors.
Rule: complete_positive_divisor_bound · Uses: G001, S031
S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(15, [1, 3, 5, 15], 4)
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: 780eb58931acb54512340328c4cfceb4a5a4405d1507dea06db21bed308edaadcandidate record SHA-256: e173bd7220dcc7d7a1a248f0bd59a1dafc545045835fb04919cbd2ba05d338f6semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1Lean: NOT_RUN · human review: NOT_PERFORMED