# Question 8794: All Common Multiples in an Inclusive Natural Window

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0016-08794 · q-20261013-advance-20261018-r02-natural-divisibility-0069



## Original question and givens

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=4, b=0, bound=48. 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=4, b=0, bound=48. Give (a, b, bound, all common multiples in [0,bound] in increasing order).





## Definition and scope

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.


- [mko-natural-divisibility](https://ocme.evemisslab.com/en/mko/mko-natural-divisibility/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: natural_divisibility_zero_scope · Uses: G001





S002 · impliesBefore

window0 first divisor:divide natural 0 by positive 4

After

0=0·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S001





S003 · impliesBefore

window0 first divisor:a=4;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S002





S004 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S003





S005 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S004





S006 · impliesBefore

window1 first divisor:divide natural 1 by positive 4

After

1=0·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S005





S007 · impliesBefore

window1 first divisor:a=4;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S006





S008 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S007





S009 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S008





S010 · impliesBefore

window2 first divisor:divide natural 2 by positive 4

After

2=0·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S009





S011 · impliesBefore

window2 first divisor:a=4;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S010





S012 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S011





S013 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S012





S014 · impliesBefore

window3 first divisor:divide natural 3 by positive 4

After

3=0·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S013





S015 · impliesBefore

window3 first divisor:a=4;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S014





S016 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S015





S017 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S016





S018 · impliesBefore

window4 first divisor:divide natural 4 by positive 4

After

4=1·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S017





S019 · impliesBefore

window4 first divisor:a=4;b=4;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S018





S020 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S019





S021 · impliesBefore

candidate=4;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S020





S022 · impliesBefore

window5 first divisor:divide natural 5 by positive 4

After

5=1·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S021





S023 · impliesBefore

window5 first divisor:a=4;b=5;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S022





S024 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S023





S025 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S024





S026 · impliesBefore

window6 first divisor:divide natural 6 by positive 4

After

6=1·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S025





S027 · impliesBefore

window6 first divisor:a=4;b=6;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S026





S028 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S027





S029 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S028





S030 · impliesBefore

window7 first divisor:divide natural 7 by positive 4

After

7=1·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S029





S031 · impliesBefore

window7 first divisor:a=4;b=7;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S030





S032 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S031





S033 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S032





S034 · impliesBefore

window8 first divisor:divide natural 8 by positive 4

After

8=2·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S033





S035 · impliesBefore

window8 first divisor:a=4;b=8;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S034





S036 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S035





S037 · impliesBefore

candidate=8;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S036





S038 · impliesBefore

window9 first divisor:divide natural 9 by positive 4

After

9=2·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S037





S039 · impliesBefore

window9 first divisor:a=4;b=9;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S038





S040 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S039





S041 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S040





S042 · impliesBefore

window10 first divisor:divide natural 10 by positive 4

After

10=2·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S041





S043 · impliesBefore

window10 first divisor:a=4;b=10;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S042





S044 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S043





S045 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S044





S046 · impliesBefore

window11 first divisor:divide natural 11 by positive 4

After

11=2·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S045





S047 · impliesBefore

window11 first divisor:a=4;b=11;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S046





S048 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S047





S049 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S048





S050 · impliesBefore

window12 first divisor:divide natural 12 by positive 4

After

12=3·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S049





S051 · impliesBefore

window12 first divisor:a=4;b=12;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S050





S052 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S051





S053 · impliesBefore

candidate=12;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S052





S054 · impliesBefore

window13 first divisor:divide natural 13 by positive 4

After

13=3·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S053





S055 · impliesBefore

window13 first divisor:a=4;b=13;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S054





S056 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S055





S057 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S056





S058 · impliesBefore

window14 first divisor:divide natural 14 by positive 4

After

14=3·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S057





S059 · impliesBefore

window14 first divisor:a=4;b=14;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S058





S060 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S059





S061 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S060





S062 · impliesBefore

window15 first divisor:divide natural 15 by positive 4

After

15=3·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S061





S063 · impliesBefore

window15 first divisor:a=4;b=15;quotient=3;remainder=3

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S062





S064 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S063





S065 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S064





S066 · impliesBefore

window16 first divisor:divide natural 16 by positive 4

After

16=4·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S065





S067 · impliesBefore

window16 first divisor:a=4;b=16;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S066





S068 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S067





S069 · impliesBefore

candidate=16;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S068





S070 · impliesBefore

window17 first divisor:divide natural 17 by positive 4

After

17=4·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S069





S071 · impliesBefore

window17 first divisor:a=4;b=17;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S070





S072 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S071





S073 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S072





S074 · impliesBefore

window18 first divisor:divide natural 18 by positive 4

After

18=4·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S073





S075 · impliesBefore

window18 first divisor:a=4;b=18;quotient=4;remainder=2

After

divides=No;witness/status=does not exist

Conditions: none

For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S074





S076 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S075





S077 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S076





S078 · impliesBefore

window19 first divisor:divide natural 19 by positive 4

After

19=4·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S077





S079 · impliesBefore

window19 first divisor:a=4;b=19;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S078





S080 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S079





S081 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S080





S082 · impliesBefore

window20 first divisor:divide natural 20 by positive 4

After

20=5·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S081





S083 · impliesBefore

window20 first divisor:a=4;b=20;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S082





S084 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S083





S085 · impliesBefore

candidate=20;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S084





S086 · impliesBefore

window21 first divisor:divide natural 21 by positive 4

After

21=5·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S085





S087 · impliesBefore

window21 first divisor:a=4;b=21;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S086





S088 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S087





S089 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S088





S090 · impliesBefore

window22 first divisor:divide natural 22 by positive 4

After

22=5·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S089





S091 · impliesBefore

window22 first divisor:a=4;b=22;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S090





S092 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S091





S093 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S092





S094 · impliesBefore

window23 first divisor:divide natural 23 by positive 4

After

23=5·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S093





S095 · impliesBefore

window23 first divisor:a=4;b=23;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S094





S096 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S095





S097 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S096





S098 · impliesBefore

window24 first divisor:divide natural 24 by positive 4

After

24=6·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S097





S099 · impliesBefore

window24 first divisor:a=4;b=24;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S098





S100 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S099





S101 · impliesBefore

candidate=24;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S100





S102 · impliesBefore

window25 first divisor:divide natural 25 by positive 4

After

25=6·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S101





S103 · impliesBefore

window25 first divisor:a=4;b=25;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S102





S104 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S103





S105 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S104





S106 · impliesBefore

window26 first divisor:divide natural 26 by positive 4

After

26=6·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S105





S107 · impliesBefore

window26 first divisor:a=4;b=26;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S106





S108 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S107





S109 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S108





S110 · impliesBefore

window27 first divisor:divide natural 27 by positive 4

After

27=6·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S109





S111 · impliesBefore

window27 first divisor:a=4;b=27;quotient=6;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S110





S112 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S111





S113 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S112





S114 · impliesBefore

window28 first divisor:divide natural 28 by positive 4

After

28=7·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S113





S115 · impliesBefore

window28 first divisor:a=4;b=28;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S114





S116 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S115





S117 · impliesBefore

candidate=28;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S116





S118 · impliesBefore

window29 first divisor:divide natural 29 by positive 4

After

29=7·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S117





S119 · impliesBefore

window29 first divisor:a=4;b=29;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S118





S120 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S119





S121 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S120





S122 · impliesBefore

window30 first divisor:divide natural 30 by positive 4

After

30=7·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S121





S123 · impliesBefore

window30 first divisor:a=4;b=30;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S122





S124 · impliesBefore

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S123





S125 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S124





S126 · impliesBefore

window31 first divisor:divide natural 31 by positive 4

After

31=7·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S125





S127 · impliesBefore

window31 first divisor:a=4;b=31;quotient=7;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S126





S128 · impliesBefore

window31 second divisor:a=0;b=31

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S127





S129 · impliesBefore

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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S128





S130 · impliesBefore

window32 first divisor:divide natural 32 by positive 4

After

32=8·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S129





S131 · impliesBefore

window32 first divisor:a=4;b=32;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S130





S132 · impliesBefore

window32 second divisor:a=0;b=32

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S131





S133 · impliesBefore

candidate=32;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S132





S134 · impliesBefore

window33 first divisor:divide natural 33 by positive 4

After

33=8·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S133





S135 · impliesBefore

window33 first divisor:a=4;b=33;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S134





S136 · impliesBefore

window33 second divisor:a=0;b=33

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S135





S137 · impliesBefore

candidate=33;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S136





S138 · impliesBefore

window34 first divisor:divide natural 34 by positive 4

After

34=8·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S137





S139 · impliesBefore

window34 first divisor:a=4;b=34;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S138





S140 · impliesBefore

window34 second divisor:a=0;b=34

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S139





S141 · impliesBefore

candidate=34;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S140





S142 · impliesBefore

window35 first divisor:divide natural 35 by positive 4

After

35=8·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S141





S143 · impliesBefore

window35 first divisor:a=4;b=35;quotient=8;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S142





S144 · impliesBefore

window35 second divisor:a=0;b=35

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S143





S145 · impliesBefore

candidate=35;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S144





S146 · impliesBefore

window36 first divisor:divide natural 36 by positive 4

After

36=9·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S145





S147 · impliesBefore

window36 first divisor:a=4;b=36;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S146





S148 · impliesBefore

window36 second divisor:a=0;b=36

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S147





S149 · impliesBefore

candidate=36;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S148





S150 · impliesBefore

window37 first divisor:divide natural 37 by positive 4

After

37=9·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S149





S151 · impliesBefore

window37 first divisor:a=4;b=37;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S150





S152 · impliesBefore

window37 second divisor:a=0;b=37

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S151





S153 · impliesBefore

candidate=37;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S152





S154 · impliesBefore

window38 first divisor:divide natural 38 by positive 4

After

38=9·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S153





S155 · impliesBefore

window38 first divisor:a=4;b=38;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S154





S156 · impliesBefore

window38 second divisor:a=0;b=38

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S155





S157 · impliesBefore

candidate=38;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S156





S158 · impliesBefore

window39 first divisor:divide natural 39 by positive 4

After

39=9·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S157





S159 · impliesBefore

window39 first divisor:a=4;b=39;quotient=9;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S158





S160 · impliesBefore

window39 second divisor:a=0;b=39

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S159





S161 · impliesBefore

candidate=39;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S160





S162 · impliesBefore

window40 first divisor:divide natural 40 by positive 4

After

40=10·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S161





S163 · impliesBefore

window40 first divisor:a=4;b=40;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S162





S164 · impliesBefore

window40 second divisor:a=0;b=40

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S163





S165 · impliesBefore

candidate=40;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S164





S166 · impliesBefore

window41 first divisor:divide natural 41 by positive 4

After

41=10·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S165





S167 · impliesBefore

window41 first divisor:a=4;b=41;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S166





S168 · impliesBefore

window41 second divisor:a=0;b=41

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S167





S169 · impliesBefore

candidate=41;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S168





S170 · impliesBefore

window42 first divisor:divide natural 42 by positive 4

After

42=10·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S169





S171 · impliesBefore

window42 first divisor:a=4;b=42;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S170





S172 · impliesBefore

window42 second divisor:a=0;b=42

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S171





S173 · impliesBefore

candidate=42;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S172





S174 · impliesBefore

window43 first divisor:divide natural 43 by positive 4

After

43=10·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S173





S175 · impliesBefore

window43 first divisor:a=4;b=43;quotient=10;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S174





S176 · impliesBefore

window43 second divisor:a=0;b=43

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S175





S177 · impliesBefore

candidate=43;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S176





S178 · impliesBefore

window44 first divisor:divide natural 44 by positive 4

After

44=11·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S177





S179 · impliesBefore

window44 first divisor:a=4;b=44;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S178





S180 · impliesBefore

window44 second divisor:a=0;b=44

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S179





S181 · impliesBefore

candidate=44;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S180





S182 · impliesBefore

window45 first divisor:divide natural 45 by positive 4

After

45=11·4+1

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S181





S183 · impliesBefore

window45 first divisor:a=4;b=45;quotient=11;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S182





S184 · impliesBefore

window45 second divisor:a=0;b=45

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S183





S185 · impliesBefore

candidate=45;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S184





S186 · impliesBefore

window46 first divisor:divide natural 46 by positive 4

After

46=11·4+2

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S185





S187 · impliesBefore

window46 first divisor:a=4;b=46;quotient=11;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S186





S188 · impliesBefore

window46 second divisor:a=0;b=46

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S187





S189 · impliesBefore

candidate=46;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S188





S190 · impliesBefore

window47 first divisor:divide natural 47 by positive 4

After

47=11·4+3

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S189





S191 · impliesBefore

window47 first divisor:a=4;b=47;quotient=11;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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S190





S192 · impliesBefore

window47 second divisor:a=0;b=47

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S191





S193 · impliesBefore

candidate=47;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S192





S194 · impliesBefore

window48 first divisor:divide natural 48 by positive 4

After

48=12·4+0

Conditions: none

A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero.

Step identifiers and dependenciesRule: natural_euclidean_division_witness · Uses: G001, S193





S195 · impliesBefore

window48 first divisor:a=4;b=48;quotient=12;remainder=0

After

divides=Yes;witness/status=12

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.

Step identifiers and dependenciesRule: positive_divisor_remainder_criterion · Uses: G001, S194





S196 · impliesBefore

window48 second divisor:a=0;b=48

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.

Step identifiers and dependenciesRule: zero_divisor_natural_equality_guard · Uses: G001, S195





S197 · impliesBefore

candidate=48;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.

Step identifiers and dependenciesRule: both_divisibility_window_guards · Uses: G001, S196









## Reverse checks


- S197 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S196 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S195 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S194 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S193 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S190 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S189 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S188 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S187 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S185 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S183 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S182 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S181 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S180 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S179 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S178 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S177 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S175 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S173 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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.





## Result

(4, 0, 48, [0])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 0ce488f69fa34201fa4a4d2adb9f2101c897754ff4fcb6ce88b5a4fbe3157358
- candidate record SHA-256: e92968f1588c042fb424c41281fff64ea5353ce89456a30975a751105adb2ee7
- semantic review receipt SHA-256: 00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2
- question-first witness SHA-256: 198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261018-r02-natural-divisibility-0069.json)