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R0016-08779 · q-20261013-advance-20261018-r02-natural-divisibility-0054
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=3, b=6, bound=33. Give (a, b, bound, all common multiples in [0,bound] in increasing order).
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=3, b=6, bound=33. Give (a, b, bound, all common multiples in [0,bound] in increasing order).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
window0 first divisor:divide natural 0 by positive 3
After
0=0·3+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, S001
Before
window0 first divisor:a=3;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, S002
Before
window0 second divisor:divide natural 0 by positive 6
After
0=0·6+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, S003
Before
window0 second divisor:a=6;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, S004
Before
candidate=0;a divides=Yes;b divides=Yes
After
common multiple=Yes
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S005
Before
window1 first divisor:divide natural 1 by positive 3
After
1=0·3+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, S006
Before
window1 first divisor:a=3;b=1;quotient=0;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, S007
Before
window1 second divisor:divide natural 1 by positive 6
After
1=0·6+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, S008
Before
window1 second divisor:a=6;b=1;quotient=0;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, S009
Before
candidate=1;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S010
Before
window2 first divisor:divide natural 2 by positive 3
After
2=0·3+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, S011
Before
window2 first divisor:a=3;b=2;quotient=0;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, S012
Before
window2 second divisor:divide natural 2 by positive 6
After
2=0·6+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, S013
Before
window2 second divisor:a=6;b=2;quotient=0;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, S014
Before
candidate=2;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S015
Before
window3 first divisor:divide natural 3 by positive 3
After
3=1·3+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, S016
Before
window3 first divisor:a=3;b=3;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, S017
Before
window3 second divisor:divide natural 3 by positive 6
After
3=0·6+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, S018
Before
window3 second divisor:a=6;b=3;quotient=0;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, S019
Before
candidate=3;a divides=Yes;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S020
Before
window4 first divisor:divide natural 4 by positive 3
After
4=1·3+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, S021
Before
window4 first divisor:a=3;b=4;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, S022
Before
window4 second divisor:divide natural 4 by positive 6
After
4=0·6+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, S023
Before
window4 second divisor:a=6;b=4;quotient=0;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, S024
Before
candidate=4;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S025
Before
window5 first divisor:divide natural 5 by positive 3
After
5=1·3+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, S026
Before
window5 first divisor:a=3;b=5;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, S027
Before
window5 second divisor:divide natural 5 by positive 6
After
5=0·6+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, S028
Before
window5 second divisor:a=6;b=5;quotient=0;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, S029
Before
candidate=5;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S030
Before
window6 first divisor:divide natural 6 by positive 3
After
6=2·3+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, S031
Before
window6 first divisor:a=3;b=6;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, S032
Before
window6 second divisor:divide natural 6 by positive 6
After
6=1·6+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, S033
Before
window6 second divisor:a=6;b=6;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, S034
Before
candidate=6;a divides=Yes;b divides=Yes
After
common multiple=Yes
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S035
Before
window7 first divisor:divide natural 7 by positive 3
After
7=2·3+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, S036
Before
window7 first divisor:a=3;b=7;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, S037
Before
window7 second divisor:divide natural 7 by positive 6
After
7=1·6+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, S038
Before
window7 second divisor:a=6;b=7;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, S039
Before
candidate=7;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S040
Before
window8 first divisor:divide natural 8 by positive 3
After
8=2·3+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, S041
Before
window8 first divisor:a=3;b=8;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, S042
Before
window8 second divisor:divide natural 8 by positive 6
After
8=1·6+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, S043
Before
window8 second divisor:a=6;b=8;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, S044
Before
candidate=8;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S045
Before
window9 first divisor:divide natural 9 by positive 3
After
9=3·3+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, S046
Before
window9 first divisor:a=3;b=9;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, S047
Before
window9 second divisor:divide natural 9 by positive 6
After
9=1·6+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, S048
Before
window9 second divisor:a=6;b=9;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, S049
Before
candidate=9;a divides=Yes;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S050
Before
window10 first divisor:divide natural 10 by positive 3
After
10=3·3+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, S051
Before
window10 first divisor:a=3;b=10;quotient=3;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, S052
Before
window10 second divisor:divide natural 10 by positive 6
After
10=1·6+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, S053
Before
window10 second divisor:a=6;b=10;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, S054
Before
candidate=10;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S055
Before
window11 first divisor:divide natural 11 by positive 3
After
11=3·3+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, S056
Before
window11 first divisor:a=3;b=11;quotient=3;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, S057
Before
window11 second divisor:divide natural 11 by positive 6
After
11=1·6+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, S058
Before
window11 second divisor:a=6;b=11;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, S059
Before
candidate=11;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S060
Before
window12 first divisor:divide natural 12 by positive 3
After
12=4·3+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, S061
Before
window12 first divisor:a=3;b=12;quotient=4;remainder=0
After
divides=Yes;witness/status=4
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, S062
Before
window12 second divisor:divide natural 12 by positive 6
After
12=2·6+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, S063
Before
window12 second divisor:a=6;b=12;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, S064
Before
candidate=12;a divides=Yes;b divides=Yes
After
common multiple=Yes
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S065
Before
window13 first divisor:divide natural 13 by positive 3
After
13=4·3+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, S066
Before
window13 first divisor:a=3;b=13;quotient=4;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, S067
Before
window13 second divisor:divide natural 13 by positive 6
After
13=2·6+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, S068
Before
window13 second divisor:a=6;b=13;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, S069
Before
candidate=13;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S070
Before
window14 first divisor:divide natural 14 by positive 3
After
14=4·3+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, S071
Before
window14 first divisor:a=3;b=14;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, S072
Before
window14 second divisor:divide natural 14 by positive 6
After
14=2·6+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, S073
Before
window14 second divisor:a=6;b=14;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, S074
Before
candidate=14;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S075
Before
window15 first divisor: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, S076
Before
window15 first divisor: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, S077
Before
window15 second divisor: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, S078
Before
window15 second divisor: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, S079
Before
candidate=15;a divides=Yes;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S080
Before
window16 first divisor:divide natural 16 by positive 3
After
16=5·3+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, S081
Before
window16 first divisor:a=3;b=16;quotient=5;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, S082
Before
window16 second divisor:divide natural 16 by positive 6
After
16=2·6+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, S083
Before
window16 second divisor:a=6;b=16;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, S084
Before
candidate=16;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S085
Before
window17 first divisor:divide natural 17 by positive 3
After
17=5·3+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, S086
Before
window17 first divisor:a=3;b=17;quotient=5;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, S087
Before
window17 second divisor:divide natural 17 by positive 6
After
17=2·6+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, S088
Before
window17 second divisor:a=6;b=17;quotient=2;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, S089
Before
candidate=17;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S090
Before
window18 first divisor: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, S091
Before
window18 first divisor: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, S092
Before
window18 second divisor: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, S093
Before
window18 second divisor: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, S094
Before
candidate=18;a divides=Yes;b divides=Yes
After
common multiple=Yes
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S095
Before
window19 first divisor:divide natural 19 by positive 3
After
19=6·3+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, S096
Before
window19 first divisor:a=3;b=19;quotient=6;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, S097
Before
window19 second divisor:divide natural 19 by positive 6
After
19=3·6+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, S098
Before
window19 second divisor:a=6;b=19;quotient=3;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, S099
Before
candidate=19;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S100
Before
window20 first divisor:divide natural 20 by positive 3
After
20=6·3+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, S101
Before
window20 first divisor:a=3;b=20;quotient=6;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, S102
Before
window20 second divisor:divide natural 20 by positive 6
After
20=3·6+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, S103
Before
window20 second divisor:a=6;b=20;quotient=3;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, S104
Before
candidate=20;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S105
Before
window21 first divisor:divide natural 21 by positive 3
After
21=7·3+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, S106
Before
window21 first divisor:a=3;b=21;quotient=7;remainder=0
After
divides=Yes;witness/status=7
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, S107
Before
window21 second divisor:divide natural 21 by positive 6
After
21=3·6+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, S108
Before
window21 second divisor:a=6;b=21;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, S109
Before
candidate=21;a divides=Yes;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S110
Before
window22 first divisor:divide natural 22 by positive 3
After
22=7·3+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, S111
Before
window22 first divisor:a=3;b=22;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, S112
Before
window22 second divisor:divide natural 22 by positive 6
After
22=3·6+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, S113
Before
window22 second divisor:a=6;b=22;quotient=3;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, S114
Before
candidate=22;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S115
Before
window23 first divisor:divide natural 23 by positive 3
After
23=7·3+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, S116
Before
window23 first divisor:a=3;b=23;quotient=7;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, S117
Before
window23 second divisor:divide natural 23 by positive 6
After
23=3·6+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, S118
Before
window23 second divisor:a=6;b=23;quotient=3;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, S119
Before
candidate=23;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S120
Before
window24 first divisor:divide natural 24 by positive 3
After
24=8·3+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, S121
Before
window24 first divisor:a=3;b=24;quotient=8;remainder=0
After
divides=Yes;witness/status=8
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, S122
Before
window24 second divisor:divide natural 24 by positive 6
After
24=4·6+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, S123
Before
window24 second divisor:a=6;b=24;quotient=4;remainder=0
After
divides=Yes;witness/status=4
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, S124
Before
candidate=24;a divides=Yes;b divides=Yes
After
common multiple=Yes
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S125
Before
window25 first divisor:divide natural 25 by positive 3
After
25=8·3+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, S126
Before
window25 first divisor:a=3;b=25;quotient=8;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, S127
Before
window25 second divisor:divide natural 25 by positive 6
After
25=4·6+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, S128
Before
window25 second divisor:a=6;b=25;quotient=4;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, S129
Before
candidate=25;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S130
Before
window26 first divisor:divide natural 26 by positive 3
After
26=8·3+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, S131
Before
window26 first divisor:a=3;b=26;quotient=8;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, S132
Before
window26 second divisor:divide natural 26 by positive 6
After
26=4·6+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, S133
Before
window26 second divisor:a=6;b=26;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, S134
Before
candidate=26;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S135
Before
window27 first divisor:divide natural 27 by positive 3
After
27=9·3+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, S136
Before
window27 first divisor:a=3;b=27;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, S137
Before
window27 second divisor:divide natural 27 by positive 6
After
27=4·6+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, S138
Before
window27 second divisor:a=6;b=27;quotient=4;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, S139
Before
candidate=27;a divides=Yes;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S140
Before
window28 first divisor:divide natural 28 by positive 3
After
28=9·3+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, S141
Before
window28 first divisor:a=3;b=28;quotient=9;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, S142
Before
window28 second divisor:divide natural 28 by positive 6
After
28=4·6+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, S143
Before
window28 second divisor:a=6;b=28;quotient=4;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, S144
Before
candidate=28;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S145
Before
window29 first divisor:divide natural 29 by positive 3
After
29=9·3+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, S146
Before
window29 first divisor:a=3;b=29;quotient=9;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, S147
Before
window29 second divisor:divide natural 29 by positive 6
After
29=4·6+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, S148
Before
window29 second divisor:a=6;b=29;quotient=4;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, S149
Before
candidate=29;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S150
Before
window30 first divisor:divide natural 30 by positive 3
After
30=10·3+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, S151
Before
window30 first divisor:a=3;b=30;quotient=10;remainder=0
After
divides=Yes;witness/status=10
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, S152
Before
window30 second divisor:divide natural 30 by positive 6
After
30=5·6+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, S153
Before
window30 second divisor:a=6;b=30;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, S154
Before
candidate=30;a divides=Yes;b divides=Yes
After
common multiple=Yes
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S155
Before
window31 first divisor:divide natural 31 by positive 3
After
31=10·3+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, S156
Before
window31 first divisor:a=3;b=31;quotient=10;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, S157
Before
window31 second divisor:divide natural 31 by positive 6
After
31=5·6+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, S158
Before
window31 second divisor:a=6;b=31;quotient=5;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, S159
Before
candidate=31;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S160
Before
window32 first divisor:divide natural 32 by positive 3
After
32=10·3+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, S161
Before
window32 first divisor:a=3;b=32;quotient=10;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, S162
Before
window32 second divisor:divide natural 32 by positive 6
After
32=5·6+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, S163
Before
window32 second divisor:a=6;b=32;quotient=5;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, S164
Before
candidate=32;a divides=No;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S165
Before
window33 first divisor:divide natural 33 by positive 3
After
33=11·3+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, S166
Before
window33 first divisor:a=3;b=33;quotient=11;remainder=0
After
divides=Yes;witness/status=11
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, S167
Before
window33 second divisor:divide natural 33 by positive 6
After
33=5·6+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, S168
Before
window33 second divisor:a=6;b=33;quotient=5;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, S169
Before
candidate=33;a divides=Yes;b divides=No
After
common multiple=No
Conditions: none
The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition.
Rule: both_divisibility_window_guards · Uses: G001, S170
S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S169 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S160 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S157 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S149 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(3, 6, 33, [0, 6, 12, 18, 24, 30])
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: fbc2c998b5e6ed99d84bf1b48152bfa685b2e357f68e9fb000da72f79fca4740candidate record SHA-256: 90049fcf0356bd75c77f2259ebcd097f2958b6e43ed849693d014705be9547fbsemantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1Lean: NOT_RUN · human review: NOT_PERFORMED