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R0016-08809 · q-20261013-advance-20261018-r02-natural-divisibility-0034
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=18. 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=18. 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 18 by positive 1
After
18=18·1+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=18;quotient=18;remainder=0
After
divides=Yes;witness/status=18
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 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.
Rule: natural_euclidean_division_witness · Uses: G001, S003
Before
positive divisor candidate2: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.
Rule: positive_divisor_remainder_criterion · Uses: G001, S004
Before
positive divisor candidate3:divide natural 18 by positive 3
After
18=6·3+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=18;quotient=6;remainder=0
After
divides=Yes;witness/status=6
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 18 by positive 4
After
18=4·4+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, S007
Before
positive divisor candidate4:a=4;b=18;quotient=4;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, S008
Before
positive divisor candidate5:divide natural 18 by positive 5
After
18=3·5+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, S009
Before
positive divisor candidate5:a=5;b=18;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, S010
Before
positive divisor candidate6:divide natural 18 by positive 6
After
18=3·6+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, S011
Before
positive divisor candidate6:a=6;b=18;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, S012
Before
positive divisor candidate7:divide natural 18 by positive 7
After
18=2·7+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, S013
Before
positive divisor candidate7:a=7;b=18;quotient=2;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, S014
Before
positive divisor candidate8:divide natural 18 by positive 8
After
18=2·8+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, S015
Before
positive divisor candidate8:a=8;b=18;quotient=2;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, S016
Before
positive divisor candidate9:divide natural 18 by positive 9
After
18=2·9+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, S017
Before
positive divisor candidate9:a=9;b=18;quotient=2;remainder=0
After
divides=Yes;witness/status=2
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 18 by positive 10
After
18=1·10+8
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=18;quotient=1;remainder=8
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 18 by positive 11
After
18=1·11+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, S021
Before
positive divisor candidate11:a=11;b=18;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, S022
Before
positive divisor candidate12:divide natural 18 by positive 12
After
18=1·12+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, S023
Before
positive divisor candidate12:a=12;b=18;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, S024
Before
positive divisor candidate13:divide natural 18 by positive 13
After
18=1·13+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, S025
Before
positive divisor candidate13:a=13;b=18;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, S026
Before
positive divisor candidate14:divide natural 18 by positive 14
After
18=1·14+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, S027
Before
positive divisor candidate14:a=14;b=18;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, S028
Before
positive divisor candidate15:divide natural 18 by positive 15
After
18=1·15+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, S029
Before
positive divisor candidate15:a=15;b=18;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, S030
Before
positive divisor candidate16:divide natural 18 by positive 16
After
18=1·16+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, S031
Before
positive divisor candidate16:a=16;b=18;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, S032
Before
positive divisor candidate17:divide natural 18 by positive 17
After
18=1·17+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, S033
Before
positive divisor candidate17:a=17;b=18;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, S034
Before
positive divisor candidate18:divide natural 18 by positive 18
After
18=1·18+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, S035
Before
positive divisor candidate18:a=18;b=18;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, S036
Before
positive b=18;complete candidate interval=[1,18]
After
positive divisors=[1, 2, 3, 6, 9, 18];count=6
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, S037
S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(18, [1, 2, 3, 6, 9, 18], 6)
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.
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