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R0016-08776 · q-20261013-advance-20261018-r02-natural-divisibility-0051
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=0, b=0, bound=30. 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=0, b=0, bound=30. 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:a=0;b=0
After
0 divides b=Yes
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S001
Before
window0 second divisor:a=0;b=0
After
0 divides b=Yes
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S002
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, S003
Before
window1 first divisor:a=0;b=1
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S004
Before
window1 second divisor:a=0;b=1
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S005
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, S006
Before
window2 first divisor:a=0;b=2
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S007
Before
window2 second divisor:a=0;b=2
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S008
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, S009
Before
window3 first divisor:a=0;b=3
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S010
Before
window3 second divisor:a=0;b=3
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S011
Before
candidate=3;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, S012
Before
window4 first divisor:a=0;b=4
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S013
Before
window4 second divisor:a=0;b=4
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S014
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, S015
Before
window5 first divisor:a=0;b=5
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S016
Before
window5 second divisor:a=0;b=5
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S017
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, S018
Before
window6 first divisor:a=0;b=6
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S019
Before
window6 second divisor:a=0;b=6
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S020
Before
candidate=6;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, S021
Before
window7 first divisor:a=0;b=7
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S022
Before
window7 second divisor:a=0;b=7
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S023
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, S024
Before
window8 first divisor:a=0;b=8
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S025
Before
window8 second divisor:a=0;b=8
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S026
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, S027
Before
window9 first divisor:a=0;b=9
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S028
Before
window9 second divisor:a=0;b=9
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S029
Before
candidate=9;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
window10 first divisor:a=0;b=10
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S031
Before
window10 second divisor:a=0;b=10
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S032
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, S033
Before
window11 first divisor:a=0;b=11
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S034
Before
window11 second divisor:a=0;b=11
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S035
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, S036
Before
window12 first divisor:a=0;b=12
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S037
Before
window12 second divisor:a=0;b=12
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S038
Before
candidate=12;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, S039
Before
window13 first divisor:a=0;b=13
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S040
Before
window13 second divisor:a=0;b=13
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S041
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, S042
Before
window14 first divisor:a=0;b=14
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S043
Before
window14 second divisor:a=0;b=14
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S044
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, S045
Before
window15 first divisor:a=0;b=15
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S046
Before
window15 second divisor:a=0;b=15
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S047
Before
candidate=15;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, S048
Before
window16 first divisor:a=0;b=16
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S049
Before
window16 second divisor:a=0;b=16
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S050
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, S051
Before
window17 first divisor:a=0;b=17
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S052
Before
window17 second divisor:a=0;b=17
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S053
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, S054
Before
window18 first divisor:a=0;b=18
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S055
Before
window18 second divisor:a=0;b=18
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S056
Before
candidate=18;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, S057
Before
window19 first divisor:a=0;b=19
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S058
Before
window19 second divisor:a=0;b=19
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S059
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, S060
Before
window20 first divisor:a=0;b=20
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S061
Before
window20 second divisor:a=0;b=20
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S062
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, S063
Before
window21 first divisor:a=0;b=21
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S064
Before
window21 second divisor:a=0;b=21
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S065
Before
candidate=21;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, S066
Before
window22 first divisor:a=0;b=22
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S067
Before
window22 second divisor:a=0;b=22
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S068
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, S069
Before
window23 first divisor:a=0;b=23
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S070
Before
window23 second divisor:a=0;b=23
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S071
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, S072
Before
window24 first divisor:a=0;b=24
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S073
Before
window24 second divisor:a=0;b=24
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S074
Before
candidate=24;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
window25 first divisor:a=0;b=25
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S076
Before
window25 second divisor:a=0;b=25
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S077
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, S078
Before
window26 first divisor:a=0;b=26
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S079
Before
window26 second divisor:a=0;b=26
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S080
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, S081
Before
window27 first divisor:a=0;b=27
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S082
Before
window27 second divisor:a=0;b=27
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S083
Before
candidate=27;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, S084
Before
window28 first divisor:a=0;b=28
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S085
Before
window28 second divisor:a=0;b=28
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S086
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, S087
Before
window29 first divisor:a=0;b=29
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S088
Before
window29 second divisor:a=0;b=29
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S089
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, S090
Before
window30 first divisor:a=0;b=30
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S091
Before
window30 second divisor:a=0;b=30
After
0 divides b=No
Conditions: none
Natural divisibility means some natural k satisfies b=a*k.For a=0 the right side is always0,so only b=0 works;branch before any remainder modulo zero.
Rule: zero_divisor_natural_equality_guard · Uses: G001, S092
Before
candidate=30;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, S093
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.(0, 0, 30, [0])
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: f3cb11c4a456442028aed1ef819fa02c02adeab953e3b281f8062079bec2958ecandidate record SHA-256: c93c5e548afca918c1afd094f28af28696cff62949b68dcbe2e9b8b12cf49bcdsemantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1Lean: NOT_RUN · human review: NOT_PERFORMED