# Question 16668: Totient Multiplicativity Requires the Coprimality Premise

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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.

R0032-16668 · q-20261014-advance-20261101-r06-euler-totient-function-0093



## Original question and givens

Positive integers m=2, n=37 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).


- G001 — Positive integers m=2, n=37 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).





## 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-euler-totient-function](https://ocme.evemisslab.com/en/mko/mko-euler-totient-function/)





## Step-by-step derivation

S001 · impliesBefore

totient multiplicativity coprimality premise:original nonnegative Euclidean inputs=[2, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 37, 0, 2], [37, 2, 18, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001





S002 · calculationBefore

totient multiplicativity original modulus product := mul(2, 37)

After

totient multiplicativity original modulus product = 74

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.

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





S003 · impliesBefore

left modulus representative1:original nonnegative Euclidean inputs=[1, 2]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 2, 0, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S004 · impliesBefore

left modulus:original representative k=1 in1..2;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S005 · impliesBefore

left modulus representative2:original nonnegative Euclidean inputs=[2, 2]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 2, 1, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S006 · impliesBefore

left modulus:original representative k=2 in1..2;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S007 · calculationBefore

left modulus exact coprime indicator count := sum(1, 0)

After

left modulus exact coprime indicator count = 1

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.

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





S008 · impliesBefore

left modulus:complete representative range1..2;all selected coprime representatives=[1]

After

phi(2)=1;representatives_1_through_n

Conditions: none

Sum every indicator for the exact count.This is complete finite counting for the supplied n,not a general formula extrapolated from samples.

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





S009 · impliesBefore

right modulus representative1:original nonnegative Euclidean inputs=[1, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 37, 0, 1], [37, 1, 37, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S010 · impliesBefore

right modulus:original representative k=1 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S011 · impliesBefore

right modulus representative2:original nonnegative Euclidean inputs=[2, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 37, 0, 2], [37, 2, 18, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S012 · impliesBefore

right modulus:original representative k=2 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S013 · impliesBefore

right modulus representative3:original nonnegative Euclidean inputs=[3, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 37, 0, 3], [37, 3, 12, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S014 · impliesBefore

right modulus:original representative k=3 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S015 · impliesBefore

right modulus representative4:original nonnegative Euclidean inputs=[4, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 37, 0, 4], [37, 4, 9, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S016 · impliesBefore

right modulus:original representative k=4 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S017 · impliesBefore

right modulus representative5:original nonnegative Euclidean inputs=[5, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 37, 0, 5], [37, 5, 7, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S018 · impliesBefore

right modulus:original representative k=5 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S019 · impliesBefore

right modulus representative6:original nonnegative Euclidean inputs=[6, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 37, 0, 6], [37, 6, 6, 1], [6, 1, 6, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S020 · impliesBefore

right modulus:original representative k=6 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S021 · impliesBefore

right modulus representative7:original nonnegative Euclidean inputs=[7, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 37, 0, 7], [37, 7, 5, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S022 · impliesBefore

right modulus:original representative k=7 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S023 · impliesBefore

right modulus representative8:original nonnegative Euclidean inputs=[8, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 37, 0, 8], [37, 8, 4, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S024 · impliesBefore

right modulus:original representative k=8 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S025 · impliesBefore

right modulus representative9:original nonnegative Euclidean inputs=[9, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 37, 0, 9], [37, 9, 4, 1], [9, 1, 9, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S026 · impliesBefore

right modulus:original representative k=9 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S027 · impliesBefore

right modulus representative10:original nonnegative Euclidean inputs=[10, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 37, 0, 10], [37, 10, 3, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S028 · impliesBefore

right modulus:original representative k=10 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S029 · impliesBefore

right modulus representative11:original nonnegative Euclidean inputs=[11, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 37, 0, 11], [37, 11, 3, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S030 · impliesBefore

right modulus:original representative k=11 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S031 · impliesBefore

right modulus representative12:original nonnegative Euclidean inputs=[12, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 37, 0, 12], [37, 12, 3, 1], [12, 1, 12, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S032 · impliesBefore

right modulus:original representative k=12 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S033 · impliesBefore

right modulus representative13:original nonnegative Euclidean inputs=[13, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 37, 0, 13], [37, 13, 2, 11], [13, 11, 1, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S034 · impliesBefore

right modulus:original representative k=13 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S035 · impliesBefore

right modulus representative14:original nonnegative Euclidean inputs=[14, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 37, 0, 14], [37, 14, 2, 9], [14, 9, 1, 5], [9, 5, 1, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S036 · impliesBefore

right modulus:original representative k=14 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S037 · impliesBefore

right modulus representative15:original nonnegative Euclidean inputs=[15, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 37, 0, 15], [37, 15, 2, 7], [15, 7, 2, 1], [7, 1, 7, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S038 · impliesBefore

right modulus:original representative k=15 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S039 · impliesBefore

right modulus representative16:original nonnegative Euclidean inputs=[16, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 37, 0, 16], [37, 16, 2, 5], [16, 5, 3, 1], [5, 1, 5, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S040 · impliesBefore

right modulus:original representative k=16 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S041 · impliesBefore

right modulus representative17:original nonnegative Euclidean inputs=[17, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 37, 0, 17], [37, 17, 2, 3], [17, 3, 5, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S042 · impliesBefore

right modulus:original representative k=17 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S043 · impliesBefore

right modulus representative18:original nonnegative Euclidean inputs=[18, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 37, 0, 18], [37, 18, 2, 1], [18, 1, 18, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S044 · impliesBefore

right modulus:original representative k=18 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S045 · impliesBefore

right modulus representative19:original nonnegative Euclidean inputs=[19, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 37, 0, 19], [37, 19, 1, 18], [19, 18, 1, 1], [18, 1, 18, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S046 · impliesBefore

right modulus:original representative k=19 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S047 · impliesBefore

right modulus representative20:original nonnegative Euclidean inputs=[20, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 37, 0, 20], [37, 20, 1, 17], [20, 17, 1, 3], [17, 3, 5, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S048 · impliesBefore

right modulus:original representative k=20 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S049 · impliesBefore

right modulus representative21:original nonnegative Euclidean inputs=[21, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 37, 0, 21], [37, 21, 1, 16], [21, 16, 1, 5], [16, 5, 3, 1], [5, 1, 5, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S050 · impliesBefore

right modulus:original representative k=21 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S051 · impliesBefore

right modulus representative22:original nonnegative Euclidean inputs=[22, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 37, 0, 22], [37, 22, 1, 15], [22, 15, 1, 7], [15, 7, 2, 1], [7, 1, 7, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S052 · impliesBefore

right modulus:original representative k=22 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S053 · impliesBefore

right modulus representative23:original nonnegative Euclidean inputs=[23, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 37, 0, 23], [37, 23, 1, 14], [23, 14, 1, 9], [14, 9, 1, 5], [9, 5, 1, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S054 · impliesBefore

right modulus:original representative k=23 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S055 · impliesBefore

right modulus representative24:original nonnegative Euclidean inputs=[24, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 37, 0, 24], [37, 24, 1, 13], [24, 13, 1, 11], [13, 11, 1, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S056 · impliesBefore

right modulus:original representative k=24 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S057 · impliesBefore

right modulus representative25:original nonnegative Euclidean inputs=[25, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 37, 0, 25], [37, 25, 1, 12], [25, 12, 2, 1], [12, 1, 12, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S058 · impliesBefore

right modulus:original representative k=25 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S059 · impliesBefore

right modulus representative26:original nonnegative Euclidean inputs=[26, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 37, 0, 26], [37, 26, 1, 11], [26, 11, 2, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S060 · impliesBefore

right modulus:original representative k=26 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S061 · impliesBefore

right modulus representative27:original nonnegative Euclidean inputs=[27, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 37, 0, 27], [37, 27, 1, 10], [27, 10, 2, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S062 · impliesBefore

right modulus:original representative k=27 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S063 · impliesBefore

right modulus representative28:original nonnegative Euclidean inputs=[28, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 37, 0, 28], [37, 28, 1, 9], [28, 9, 3, 1], [9, 1, 9, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S064 · impliesBefore

right modulus:original representative k=28 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S065 · impliesBefore

right modulus representative29:original nonnegative Euclidean inputs=[29, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 37, 0, 29], [37, 29, 1, 8], [29, 8, 3, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S066 · impliesBefore

right modulus:original representative k=29 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S067 · impliesBefore

right modulus representative30:original nonnegative Euclidean inputs=[30, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 37, 0, 30], [37, 30, 1, 7], [30, 7, 4, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S068 · impliesBefore

right modulus:original representative k=30 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S069 · impliesBefore

right modulus representative31:original nonnegative Euclidean inputs=[31, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[31, 37, 0, 31], [37, 31, 1, 6], [31, 6, 5, 1], [6, 1, 6, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S070 · impliesBefore

right modulus:original representative k=31 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S071 · impliesBefore

right modulus representative32:original nonnegative Euclidean inputs=[32, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[32, 37, 0, 32], [37, 32, 1, 5], [32, 5, 6, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S072 · impliesBefore

right modulus:original representative k=32 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S073 · impliesBefore

right modulus representative33:original nonnegative Euclidean inputs=[33, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[33, 37, 0, 33], [37, 33, 1, 4], [33, 4, 8, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S074 · impliesBefore

right modulus:original representative k=33 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S075 · impliesBefore

right modulus representative34:original nonnegative Euclidean inputs=[34, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[34, 37, 0, 34], [37, 34, 1, 3], [34, 3, 11, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S076 · impliesBefore

right modulus:original representative k=34 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S077 · impliesBefore

right modulus representative35:original nonnegative Euclidean inputs=[35, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[35, 37, 0, 35], [37, 35, 1, 2], [35, 2, 17, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S078 · impliesBefore

right modulus:original representative k=35 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S079 · impliesBefore

right modulus representative36:original nonnegative Euclidean inputs=[36, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[36, 37, 0, 36], [37, 36, 1, 1], [36, 1, 36, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S080 · impliesBefore

right modulus:original representative k=36 in1..37;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S081 · impliesBefore

right modulus representative37:original nonnegative Euclidean inputs=[37, 37]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[37, 37, 1, 0]];terminal positive gcd=37

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S082 · impliesBefore

right modulus:original representative k=37 in1..37;verified gcd(k,n)=37

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S083 · calculationBefore

right modulus exact coprime indicator count := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0)

After

right modulus exact coprime indicator count = 36

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.

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





S084 · impliesBefore

right modulus:complete representative range1..37;all selected coprime representatives=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]

After

phi(37)=36;representatives_1_through_n

Conditions: none

Sum every indicator for the exact count.This is complete finite counting for the supplied n,not a general formula extrapolated from samples.

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





S085 · impliesBefore

product modulus representative1:original nonnegative Euclidean inputs=[1, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 74, 0, 1], [74, 1, 74, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S086 · impliesBefore

product modulus:original representative k=1 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S087 · impliesBefore

product modulus representative2:original nonnegative Euclidean inputs=[2, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 74, 0, 2], [74, 2, 37, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S088 · impliesBefore

product modulus:original representative k=2 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S089 · impliesBefore

product modulus representative3:original nonnegative Euclidean inputs=[3, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 74, 0, 3], [74, 3, 24, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S090 · impliesBefore

product modulus:original representative k=3 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S091 · impliesBefore

product modulus representative4:original nonnegative Euclidean inputs=[4, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 74, 0, 4], [74, 4, 18, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S092 · impliesBefore

product modulus:original representative k=4 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S093 · impliesBefore

product modulus representative5:original nonnegative Euclidean inputs=[5, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 74, 0, 5], [74, 5, 14, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S094 · impliesBefore

product modulus:original representative k=5 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S095 · impliesBefore

product modulus representative6:original nonnegative Euclidean inputs=[6, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 74, 0, 6], [74, 6, 12, 2], [6, 2, 3, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S096 · impliesBefore

product modulus:original representative k=6 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S097 · impliesBefore

product modulus representative7:original nonnegative Euclidean inputs=[7, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 74, 0, 7], [74, 7, 10, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S098 · impliesBefore

product modulus:original representative k=7 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S099 · impliesBefore

product modulus representative8:original nonnegative Euclidean inputs=[8, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 74, 0, 8], [74, 8, 9, 2], [8, 2, 4, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S100 · impliesBefore

product modulus:original representative k=8 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S101 · impliesBefore

product modulus representative9:original nonnegative Euclidean inputs=[9, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 74, 0, 9], [74, 9, 8, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S102 · impliesBefore

product modulus:original representative k=9 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S103 · impliesBefore

product modulus representative10:original nonnegative Euclidean inputs=[10, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 74, 0, 10], [74, 10, 7, 4], [10, 4, 2, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S104 · impliesBefore

product modulus:original representative k=10 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S105 · impliesBefore

product modulus representative11:original nonnegative Euclidean inputs=[11, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 74, 0, 11], [74, 11, 6, 8], [11, 8, 1, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S106 · impliesBefore

product modulus:original representative k=11 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S107 · impliesBefore

product modulus representative12:original nonnegative Euclidean inputs=[12, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 74, 0, 12], [74, 12, 6, 2], [12, 2, 6, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S108 · impliesBefore

product modulus:original representative k=12 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S109 · impliesBefore

product modulus representative13:original nonnegative Euclidean inputs=[13, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 74, 0, 13], [74, 13, 5, 9], [13, 9, 1, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S110 · impliesBefore

product modulus:original representative k=13 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S111 · impliesBefore

product modulus representative14:original nonnegative Euclidean inputs=[14, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 74, 0, 14], [74, 14, 5, 4], [14, 4, 3, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S112 · impliesBefore

product modulus:original representative k=14 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S113 · impliesBefore

product modulus representative15:original nonnegative Euclidean inputs=[15, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 74, 0, 15], [74, 15, 4, 14], [15, 14, 1, 1], [14, 1, 14, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S114 · impliesBefore

product modulus:original representative k=15 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S115 · impliesBefore

product modulus representative16:original nonnegative Euclidean inputs=[16, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 74, 0, 16], [74, 16, 4, 10], [16, 10, 1, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S116 · impliesBefore

product modulus:original representative k=16 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S117 · impliesBefore

product modulus representative17:original nonnegative Euclidean inputs=[17, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 74, 0, 17], [74, 17, 4, 6], [17, 6, 2, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S118 · impliesBefore

product modulus:original representative k=17 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S119 · impliesBefore

product modulus representative18:original nonnegative Euclidean inputs=[18, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 74, 0, 18], [74, 18, 4, 2], [18, 2, 9, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S120 · impliesBefore

product modulus:original representative k=18 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S121 · impliesBefore

product modulus representative19:original nonnegative Euclidean inputs=[19, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 74, 0, 19], [74, 19, 3, 17], [19, 17, 1, 2], [17, 2, 8, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S122 · impliesBefore

product modulus:original representative k=19 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S123 · impliesBefore

product modulus representative20:original nonnegative Euclidean inputs=[20, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 74, 0, 20], [74, 20, 3, 14], [20, 14, 1, 6], [14, 6, 2, 2], [6, 2, 3, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S124 · impliesBefore

product modulus:original representative k=20 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S125 · impliesBefore

product modulus representative21:original nonnegative Euclidean inputs=[21, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 74, 0, 21], [74, 21, 3, 11], [21, 11, 1, 10], [11, 10, 1, 1], [10, 1, 10, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S126 · impliesBefore

product modulus:original representative k=21 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S127 · impliesBefore

product modulus representative22:original nonnegative Euclidean inputs=[22, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 74, 0, 22], [74, 22, 3, 8], [22, 8, 2, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S128 · impliesBefore

product modulus:original representative k=22 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S129 · impliesBefore

product modulus representative23:original nonnegative Euclidean inputs=[23, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 74, 0, 23], [74, 23, 3, 5], [23, 5, 4, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S130 · impliesBefore

product modulus:original representative k=23 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S131 · impliesBefore

product modulus representative24:original nonnegative Euclidean inputs=[24, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 74, 0, 24], [74, 24, 3, 2], [24, 2, 12, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S132 · impliesBefore

product modulus:original representative k=24 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S133 · impliesBefore

product modulus representative25:original nonnegative Euclidean inputs=[25, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 74, 0, 25], [74, 25, 2, 24], [25, 24, 1, 1], [24, 1, 24, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S134 · impliesBefore

product modulus:original representative k=25 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S135 · impliesBefore

product modulus representative26:original nonnegative Euclidean inputs=[26, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 74, 0, 26], [74, 26, 2, 22], [26, 22, 1, 4], [22, 4, 5, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S136 · impliesBefore

product modulus:original representative k=26 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S137 · impliesBefore

product modulus representative27:original nonnegative Euclidean inputs=[27, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 74, 0, 27], [74, 27, 2, 20], [27, 20, 1, 7], [20, 7, 2, 6], [7, 6, 1, 1], [6, 1, 6, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S138 · impliesBefore

product modulus:original representative k=27 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S139 · impliesBefore

product modulus representative28:original nonnegative Euclidean inputs=[28, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 74, 0, 28], [74, 28, 2, 18], [28, 18, 1, 10], [18, 10, 1, 8], [10, 8, 1, 2], [8, 2, 4, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S140 · impliesBefore

product modulus:original representative k=28 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S141 · impliesBefore

product modulus representative29:original nonnegative Euclidean inputs=[29, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 74, 0, 29], [74, 29, 2, 16], [29, 16, 1, 13], [16, 13, 1, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S142 · impliesBefore

product modulus:original representative k=29 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S143 · impliesBefore

product modulus representative30:original nonnegative Euclidean inputs=[30, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 74, 0, 30], [74, 30, 2, 14], [30, 14, 2, 2], [14, 2, 7, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S144 · impliesBefore

product modulus:original representative k=30 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S145 · impliesBefore

product modulus representative31:original nonnegative Euclidean inputs=[31, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[31, 74, 0, 31], [74, 31, 2, 12], [31, 12, 2, 7], [12, 7, 1, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S146 · impliesBefore

product modulus:original representative k=31 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S147 · impliesBefore

product modulus representative32:original nonnegative Euclidean inputs=[32, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[32, 74, 0, 32], [74, 32, 2, 10], [32, 10, 3, 2], [10, 2, 5, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S148 · impliesBefore

product modulus:original representative k=32 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S149 · impliesBefore

product modulus representative33:original nonnegative Euclidean inputs=[33, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[33, 74, 0, 33], [74, 33, 2, 8], [33, 8, 4, 1], [8, 1, 8, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S150 · impliesBefore

product modulus:original representative k=33 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S151 · impliesBefore

product modulus representative34:original nonnegative Euclidean inputs=[34, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[34, 74, 0, 34], [74, 34, 2, 6], [34, 6, 5, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S152 · impliesBefore

product modulus:original representative k=34 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S153 · impliesBefore

product modulus representative35:original nonnegative Euclidean inputs=[35, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[35, 74, 0, 35], [74, 35, 2, 4], [35, 4, 8, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S154 · impliesBefore

product modulus:original representative k=35 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S155 · impliesBefore

product modulus representative36:original nonnegative Euclidean inputs=[36, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[36, 74, 0, 36], [74, 36, 2, 2], [36, 2, 18, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S156 · impliesBefore

product modulus:original representative k=36 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S157 · impliesBefore

product modulus representative37:original nonnegative Euclidean inputs=[37, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[37, 74, 0, 37], [74, 37, 2, 0]];terminal positive gcd=37

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S158 · impliesBefore

product modulus:original representative k=37 in1..74;verified gcd(k,n)=37

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S159 · impliesBefore

product modulus representative38:original nonnegative Euclidean inputs=[38, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[38, 74, 0, 38], [74, 38, 1, 36], [38, 36, 1, 2], [36, 2, 18, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S160 · impliesBefore

product modulus:original representative k=38 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S161 · impliesBefore

product modulus representative39:original nonnegative Euclidean inputs=[39, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[39, 74, 0, 39], [74, 39, 1, 35], [39, 35, 1, 4], [35, 4, 8, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S162 · impliesBefore

product modulus:original representative k=39 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S163 · impliesBefore

product modulus representative40:original nonnegative Euclidean inputs=[40, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[40, 74, 0, 40], [74, 40, 1, 34], [40, 34, 1, 6], [34, 6, 5, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S164 · impliesBefore

product modulus:original representative k=40 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S165 · impliesBefore

product modulus representative41:original nonnegative Euclidean inputs=[41, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[41, 74, 0, 41], [74, 41, 1, 33], [41, 33, 1, 8], [33, 8, 4, 1], [8, 1, 8, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S166 · impliesBefore

product modulus:original representative k=41 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S167 · impliesBefore

product modulus representative42:original nonnegative Euclidean inputs=[42, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[42, 74, 0, 42], [74, 42, 1, 32], [42, 32, 1, 10], [32, 10, 3, 2], [10, 2, 5, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S168 · impliesBefore

product modulus:original representative k=42 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S169 · impliesBefore

product modulus representative43:original nonnegative Euclidean inputs=[43, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[43, 74, 0, 43], [74, 43, 1, 31], [43, 31, 1, 12], [31, 12, 2, 7], [12, 7, 1, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S170 · impliesBefore

product modulus:original representative k=43 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S171 · impliesBefore

product modulus representative44:original nonnegative Euclidean inputs=[44, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[44, 74, 0, 44], [74, 44, 1, 30], [44, 30, 1, 14], [30, 14, 2, 2], [14, 2, 7, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S172 · impliesBefore

product modulus:original representative k=44 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S173 · impliesBefore

product modulus representative45:original nonnegative Euclidean inputs=[45, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[45, 74, 0, 45], [74, 45, 1, 29], [45, 29, 1, 16], [29, 16, 1, 13], [16, 13, 1, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S174 · impliesBefore

product modulus:original representative k=45 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S175 · impliesBefore

product modulus representative46:original nonnegative Euclidean inputs=[46, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[46, 74, 0, 46], [74, 46, 1, 28], [46, 28, 1, 18], [28, 18, 1, 10], [18, 10, 1, 8], [10, 8, 1, 2], [8, 2, 4, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S176 · impliesBefore

product modulus:original representative k=46 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S177 · impliesBefore

product modulus representative47:original nonnegative Euclidean inputs=[47, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[47, 74, 0, 47], [74, 47, 1, 27], [47, 27, 1, 20], [27, 20, 1, 7], [20, 7, 2, 6], [7, 6, 1, 1], [6, 1, 6, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S178 · impliesBefore

product modulus:original representative k=47 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S179 · impliesBefore

product modulus representative48:original nonnegative Euclidean inputs=[48, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[48, 74, 0, 48], [74, 48, 1, 26], [48, 26, 1, 22], [26, 22, 1, 4], [22, 4, 5, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S180 · impliesBefore

product modulus:original representative k=48 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S181 · impliesBefore

product modulus representative49:original nonnegative Euclidean inputs=[49, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[49, 74, 0, 49], [74, 49, 1, 25], [49, 25, 1, 24], [25, 24, 1, 1], [24, 1, 24, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S182 · impliesBefore

product modulus:original representative k=49 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S183 · impliesBefore

product modulus representative50:original nonnegative Euclidean inputs=[50, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[50, 74, 0, 50], [74, 50, 1, 24], [50, 24, 2, 2], [24, 2, 12, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S184 · impliesBefore

product modulus:original representative k=50 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S185 · impliesBefore

product modulus representative51:original nonnegative Euclidean inputs=[51, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[51, 74, 0, 51], [74, 51, 1, 23], [51, 23, 2, 5], [23, 5, 4, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S186 · impliesBefore

product modulus:original representative k=51 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S187 · impliesBefore

product modulus representative52:original nonnegative Euclidean inputs=[52, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[52, 74, 0, 52], [74, 52, 1, 22], [52, 22, 2, 8], [22, 8, 2, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S188 · impliesBefore

product modulus:original representative k=52 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S189 · impliesBefore

product modulus representative53:original nonnegative Euclidean inputs=[53, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[53, 74, 0, 53], [74, 53, 1, 21], [53, 21, 2, 11], [21, 11, 1, 10], [11, 10, 1, 1], [10, 1, 10, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S190 · impliesBefore

product modulus:original representative k=53 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S191 · impliesBefore

product modulus representative54:original nonnegative Euclidean inputs=[54, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[54, 74, 0, 54], [74, 54, 1, 20], [54, 20, 2, 14], [20, 14, 1, 6], [14, 6, 2, 2], [6, 2, 3, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S192 · impliesBefore

product modulus:original representative k=54 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S193 · impliesBefore

product modulus representative55:original nonnegative Euclidean inputs=[55, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[55, 74, 0, 55], [74, 55, 1, 19], [55, 19, 2, 17], [19, 17, 1, 2], [17, 2, 8, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S194 · impliesBefore

product modulus:original representative k=55 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S195 · impliesBefore

product modulus representative56:original nonnegative Euclidean inputs=[56, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[56, 74, 0, 56], [74, 56, 1, 18], [56, 18, 3, 2], [18, 2, 9, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S196 · impliesBefore

product modulus:original representative k=56 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

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





S197 · impliesBefore

product modulus representative57:original nonnegative Euclidean inputs=[57, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[57, 74, 0, 57], [74, 57, 1, 17], [57, 17, 3, 6], [17, 6, 2, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

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





S198 · impliesBefore

product modulus:original representative k=57 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S197





S199 · impliesBefore

product modulus representative58:original nonnegative Euclidean inputs=[58, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[58, 74, 0, 58], [74, 58, 1, 16], [58, 16, 3, 10], [16, 10, 1, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S198





S200 · impliesBefore

product modulus:original representative k=58 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S199





S201 · impliesBefore

product modulus representative59:original nonnegative Euclidean inputs=[59, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[59, 74, 0, 59], [74, 59, 1, 15], [59, 15, 3, 14], [15, 14, 1, 1], [14, 1, 14, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S200





S202 · impliesBefore

product modulus:original representative k=59 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S201





S203 · impliesBefore

product modulus representative60:original nonnegative Euclidean inputs=[60, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[60, 74, 0, 60], [74, 60, 1, 14], [60, 14, 4, 4], [14, 4, 3, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S202





S204 · impliesBefore

product modulus:original representative k=60 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S203





S205 · impliesBefore

product modulus representative61:original nonnegative Euclidean inputs=[61, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[61, 74, 0, 61], [74, 61, 1, 13], [61, 13, 4, 9], [13, 9, 1, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S204





S206 · impliesBefore

product modulus:original representative k=61 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S205





S207 · impliesBefore

product modulus representative62:original nonnegative Euclidean inputs=[62, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[62, 74, 0, 62], [74, 62, 1, 12], [62, 12, 5, 2], [12, 2, 6, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S206





S208 · impliesBefore

product modulus:original representative k=62 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S207





S209 · impliesBefore

product modulus representative63:original nonnegative Euclidean inputs=[63, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[63, 74, 0, 63], [74, 63, 1, 11], [63, 11, 5, 8], [11, 8, 1, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S208





S210 · impliesBefore

product modulus:original representative k=63 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S209





S211 · impliesBefore

product modulus representative64:original nonnegative Euclidean inputs=[64, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[64, 74, 0, 64], [74, 64, 1, 10], [64, 10, 6, 4], [10, 4, 2, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S210





S212 · impliesBefore

product modulus:original representative k=64 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S211





S213 · impliesBefore

product modulus representative65:original nonnegative Euclidean inputs=[65, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[65, 74, 0, 65], [74, 65, 1, 9], [65, 9, 7, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S212





S214 · impliesBefore

product modulus:original representative k=65 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S213





S215 · impliesBefore

product modulus representative66:original nonnegative Euclidean inputs=[66, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[66, 74, 0, 66], [74, 66, 1, 8], [66, 8, 8, 2], [8, 2, 4, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S214





S216 · impliesBefore

product modulus:original representative k=66 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S215





S217 · impliesBefore

product modulus representative67:original nonnegative Euclidean inputs=[67, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[67, 74, 0, 67], [74, 67, 1, 7], [67, 7, 9, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S216





S218 · impliesBefore

product modulus:original representative k=67 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S217





S219 · impliesBefore

product modulus representative68:original nonnegative Euclidean inputs=[68, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[68, 74, 0, 68], [74, 68, 1, 6], [68, 6, 11, 2], [6, 2, 3, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S218





S220 · impliesBefore

product modulus:original representative k=68 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S219





S221 · impliesBefore

product modulus representative69:original nonnegative Euclidean inputs=[69, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[69, 74, 0, 69], [74, 69, 1, 5], [69, 5, 13, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S220





S222 · impliesBefore

product modulus:original representative k=69 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S221





S223 · impliesBefore

product modulus representative70:original nonnegative Euclidean inputs=[70, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[70, 74, 0, 70], [74, 70, 1, 4], [70, 4, 17, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S222





S224 · impliesBefore

product modulus:original representative k=70 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S223





S225 · impliesBefore

product modulus representative71:original nonnegative Euclidean inputs=[71, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[71, 74, 0, 71], [74, 71, 1, 3], [71, 3, 23, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S224





S226 · impliesBefore

product modulus:original representative k=71 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S225





S227 · impliesBefore

product modulus representative72:original nonnegative Euclidean inputs=[72, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[72, 74, 0, 72], [74, 72, 1, 2], [72, 2, 36, 0]];terminal positive gcd=2

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S226





S228 · impliesBefore

product modulus:original representative k=72 in1..74;verified gcd(k,n)=2

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S227





S229 · impliesBefore

product modulus representative73:original nonnegative Euclidean inputs=[73, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[73, 74, 0, 73], [74, 73, 1, 1], [73, 1, 73, 0]];terminal positive gcd=1

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S228





S230 · impliesBefore

product modulus:original representative k=73 in1..74;verified gcd(k,n)=1

After

coprime indicator=1;include representative=Yes

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S229





S231 · impliesBefore

product modulus representative74:original nonnegative Euclidean inputs=[74, 74]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[74, 74, 1, 0]];terminal positive gcd=74

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer.

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S230





S232 · impliesBefore

product modulus:original representative k=74 in1..74;verified gcd(k,n)=74

After

coprime indicator=0;include representative=No

Conditions: none

Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list.

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S231





S233 · calculationBefore

product modulus exact coprime indicator count := sum(1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0)

After

product modulus exact coprime indicator count = 36

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S232





S234 · impliesBefore

product modulus:complete representative range1..74;all selected coprime representatives=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73]

After

phi(74)=36;representatives_1_through_n

Conditions: none

Sum every indicator for the exact count.This is complete finite counting for the supplied n,not a general formula extrapolated from samples.

Step identifiers and dependenciesRule: totient_complete_finite_count_not_sample_formula · Uses: G001, S233





S235 · calculationBefore

totient supplied instance product of counts := mul(1, 36)

After

totient supplied instance product of counts = 36

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S234





S236 · impliesBefore

original coprime inputs=[2, 37];gcd=1;product modulus=74;exact totients=[1, 36, 36]

After

phi(m)*phi(n)=36;same supplied-instance equality=Yes;instance_only_with_coprimality

Conditions: none

First verify coprimality and all three allowed moduli,then count the three totients completely and compare.This checks the supplied instance only.Non-coprime inputs do not qualify,and numeric agreement is not a proof of the universal multiplicativity theorem.

Step identifiers and dependenciesRule: totient_coprime_instance_not_unrestricted_multiplicativity_proof · Uses: G001, S235









## Reverse checks


- S236 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S235 · premise_and_result_audit — Recompute totient supplied instance product of counts with exact rational arithmetic, not rounded decimals.
- S234 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S233 · premise_and_result_audit — Recompute product modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
- S232 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S231 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S230 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S229 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S228 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S227 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S226 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S225 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S224 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S223 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S222 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S221 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S220 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S219 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S218 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S217 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S216 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S215 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S214 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S213 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S212 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S211 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S210 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S209 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S208 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S206 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S205 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S204 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S203 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S202 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S201 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S200 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S199 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S198 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Recompute right modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
- 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 — Recompute left modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
- 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 — Recompute totient multiplicativity original modulus product with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(2, 37, 1, 36, 36, 36, Yes, instance_only_with_coprimality)

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: 8d0e9453b24cb1eab0e86953772619c60566afad333daaf2835335599b5c330a
- candidate record SHA-256: f7affc6ea05351460c7e5e371867ceb49f71ab5633c9cc2a35fd98da5008d0d4
- semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e
- question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261101-r06-euler-totient-function-0093.json)