# Question 9153: Every Prime in an Inclusive Bounded Interval

[Return to catalogue page 46](https://ocme.evemisslab.com/en/questions/page/46/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261018-r04-prime-number-foundations/#q-20261013-advance-20261018-r04-prime-number-0078)

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.

R0017-09153 · q-20261013-advance-20261018-r04-prime-number-0078



## Original question and givens

Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=10, hi=30, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).


- G001 — Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=10, hi=30, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).





## 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-prime-number](https://ocme.evemisslab.com/en/mko/mko-prime-number/)





## Step-by-step derivation

S001 · impliesBefore

natural-number prime protocol;0 and1 are not prime

After

a failed candidate witness alone is not a primality proof

Conditions: none

Naturals include0.Primality retains both n≥2 and a complete factor bound.A failed single factor candidate only invalidates that witness;it does not certify primality.

Step identifiers and dependenciesRule: prime_complete_evidence_scope · Uses: G001





S002 · impliesBefore

phase=interval value10;n=10

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S003 · impliesBefore

interval value10 trial divisor2:natural dividend=10;positive divisor=2

After

10=5·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S004 · impliesBefore

phase=interval value10;tried ascending divisors=[2];sqrt floor=3

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S005 · impliesBefore

phase=interval value11;n=11

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S006 · impliesBefore

interval value11 trial divisor2:natural dividend=11;positive divisor=2

After

11=5·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S007 · impliesBefore

interval value11 trial divisor3:natural dividend=11;positive divisor=3

After

11=3·3+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S008 · impliesBefore

phase=interval value11;tried ascending divisors=[2, 3];sqrt floor=3

After

least factor at least2=11;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S009 · impliesBefore

phase=interval value12;n=12

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S010 · impliesBefore

interval value12 trial divisor2:natural dividend=12;positive divisor=2

After

12=6·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S011 · impliesBefore

phase=interval value12;tried ascending divisors=[2];sqrt floor=3

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S012 · impliesBefore

phase=interval value13;n=13

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S013 · impliesBefore

interval value13 trial divisor2:natural dividend=13;positive divisor=2

After

13=6·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S014 · impliesBefore

interval value13 trial divisor3:natural dividend=13;positive divisor=3

After

13=4·3+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S015 · impliesBefore

phase=interval value13;tried ascending divisors=[2, 3];sqrt floor=3

After

least factor at least2=13;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S016 · impliesBefore

phase=interval value14;n=14

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S017 · impliesBefore

interval value14 trial divisor2:natural dividend=14;positive divisor=2

After

14=7·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S018 · impliesBefore

phase=interval value14;tried ascending divisors=[2];sqrt floor=3

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S019 · impliesBefore

phase=interval value15;n=15

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S020 · impliesBefore

interval value15 trial divisor2:natural dividend=15;positive divisor=2

After

15=7·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S021 · impliesBefore

interval value15 trial divisor3:natural dividend=15;positive divisor=3

After

15=5·3+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S022 · impliesBefore

phase=interval value15;tried ascending divisors=[2, 3];sqrt floor=3

After

least factor at least2=3;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S023 · impliesBefore

phase=interval value16;n=16

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S024 · impliesBefore

interval value16 trial divisor2:natural dividend=16;positive divisor=2

After

16=8·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S025 · impliesBefore

phase=interval value16;tried ascending divisors=[2];sqrt floor=4

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S026 · impliesBefore

phase=interval value17;n=17

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S027 · impliesBefore

interval value17 trial divisor2:natural dividend=17;positive divisor=2

After

17=8·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S028 · impliesBefore

interval value17 trial divisor3:natural dividend=17;positive divisor=3

After

17=5·3+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S029 · impliesBefore

interval value17 trial divisor4:natural dividend=17;positive divisor=4

After

17=4·4+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S030 · impliesBefore

phase=interval value17;tried ascending divisors=[2, 3, 4];sqrt floor=4

After

least factor at least2=17;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S031 · impliesBefore

phase=interval value18;n=18

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S032 · impliesBefore

interval value18 trial divisor2:natural dividend=18;positive divisor=2

After

18=9·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S033 · impliesBefore

phase=interval value18;tried ascending divisors=[2];sqrt floor=4

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S034 · impliesBefore

phase=interval value19;n=19

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S035 · impliesBefore

interval value19 trial divisor2:natural dividend=19;positive divisor=2

After

19=9·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S036 · impliesBefore

interval value19 trial divisor3:natural dividend=19;positive divisor=3

After

19=6·3+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S037 · impliesBefore

interval value19 trial divisor4:natural dividend=19;positive divisor=4

After

19=4·4+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S038 · impliesBefore

phase=interval value19;tried ascending divisors=[2, 3, 4];sqrt floor=4

After

least factor at least2=19;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S039 · impliesBefore

phase=interval value20;n=20

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S040 · impliesBefore

interval value20 trial divisor2:natural dividend=20;positive divisor=2

After

20=10·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S041 · impliesBefore

phase=interval value20;tried ascending divisors=[2];sqrt floor=4

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S042 · impliesBefore

phase=interval value21;n=21

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S043 · impliesBefore

interval value21 trial divisor2:natural dividend=21;positive divisor=2

After

21=10·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S044 · impliesBefore

interval value21 trial divisor3:natural dividend=21;positive divisor=3

After

21=7·3+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S045 · impliesBefore

phase=interval value21;tried ascending divisors=[2, 3];sqrt floor=4

After

least factor at least2=3;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S046 · impliesBefore

phase=interval value22;n=22

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S047 · impliesBefore

interval value22 trial divisor2:natural dividend=22;positive divisor=2

After

22=11·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S048 · impliesBefore

phase=interval value22;tried ascending divisors=[2];sqrt floor=4

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S049 · impliesBefore

phase=interval value23;n=23

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S050 · impliesBefore

interval value23 trial divisor2:natural dividend=23;positive divisor=2

After

23=11·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S051 · impliesBefore

interval value23 trial divisor3:natural dividend=23;positive divisor=3

After

23=7·3+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S052 · impliesBefore

interval value23 trial divisor4:natural dividend=23;positive divisor=4

After

23=5·4+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S053 · impliesBefore

phase=interval value23;tried ascending divisors=[2, 3, 4];sqrt floor=4

After

least factor at least2=23;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S054 · impliesBefore

phase=interval value24;n=24

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S055 · impliesBefore

interval value24 trial divisor2:natural dividend=24;positive divisor=2

After

24=12·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S056 · impliesBefore

phase=interval value24;tried ascending divisors=[2];sqrt floor=4

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S057 · impliesBefore

phase=interval value25;n=25

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S058 · impliesBefore

interval value25 trial divisor2:natural dividend=25;positive divisor=2

After

25=12·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S059 · impliesBefore

interval value25 trial divisor3:natural dividend=25;positive divisor=3

After

25=8·3+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S060 · impliesBefore

interval value25 trial divisor4:natural dividend=25;positive divisor=4

After

25=6·4+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S061 · impliesBefore

interval value25 trial divisor5:natural dividend=25;positive divisor=5

After

25=5·5+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S062 · impliesBefore

phase=interval value25;tried ascending divisors=[2, 3, 4, 5];sqrt floor=5

After

least factor at least2=5;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S063 · impliesBefore

phase=interval value26;n=26

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S064 · impliesBefore

interval value26 trial divisor2:natural dividend=26;positive divisor=2

After

26=13·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S065 · impliesBefore

phase=interval value26;tried ascending divisors=[2];sqrt floor=5

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S066 · impliesBefore

phase=interval value27;n=27

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S067 · impliesBefore

interval value27 trial divisor2:natural dividend=27;positive divisor=2

After

27=13·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S068 · impliesBefore

interval value27 trial divisor3:natural dividend=27;positive divisor=3

After

27=9·3+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S069 · impliesBefore

phase=interval value27;tried ascending divisors=[2, 3];sqrt floor=5

After

least factor at least2=3;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S070 · impliesBefore

phase=interval value28;n=28

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S071 · impliesBefore

interval value28 trial divisor2:natural dividend=28;positive divisor=2

After

28=14·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S072 · impliesBefore

phase=interval value28;tried ascending divisors=[2];sqrt floor=5

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S073 · impliesBefore

phase=interval value29;n=29

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S074 · impliesBefore

interval value29 trial divisor2:natural dividend=29;positive divisor=2

After

29=14·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S075 · impliesBefore

interval value29 trial divisor3:natural dividend=29;positive divisor=3

After

29=9·3+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S076 · impliesBefore

interval value29 trial divisor4:natural dividend=29;positive divisor=4

After

29=7·4+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S077 · impliesBefore

interval value29 trial divisor5:natural dividend=29;positive divisor=5

After

29=5·5+4

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S078 · impliesBefore

phase=interval value29;tried ascending divisors=[2, 3, 4, 5];sqrt floor=5

After

least factor at least2=29;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S079 · impliesBefore

phase=interval value30;n=30

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

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





S080 · impliesBefore

interval value30 trial divisor2:natural dividend=30;positive divisor=2

After

30=15·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

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





S081 · impliesBefore

phase=interval value30;tried ascending divisors=[2];sqrt floor=5

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

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





S082 · impliesBefore

inclusive interval=[10, 30]

After

all primes=[11, 13, 17, 19, 23, 29];count=6

Conditions: none

Check both interval endpoints and retain exactly those values passing the complete primality test.This finite enumeration makes no distribution claim outside the interval.

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









## Reverse checks


- 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

(10, 30, [11, 13, 17, 19, 23, 29], 6)

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: 0a0cc84cce809122590e99dbb91a4dd1f3c3739f5a5307905ea1223d7371f8aa
- candidate record SHA-256: 00d33b6453899856e6bd033a15ee8a8460e22669fe5f2d9071d793d2f4ef0d26
- semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
- question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261018-r04-prime-number-0078.json)