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Question 16663: Totient Multiplicativity Requires the Coprimality Premise

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0032-16663 · q-20261014-advance-20261101-r06-euler-totient-function-0088

Original question and givens

Positive integers m=2, n=27 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=27 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.

Step-by-step derivation

S001 · implies

Before

totient multiplicativity coprimality premise:original nonnegative Euclidean inputs=[2, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 27, 0, 2], [27, 2, 13, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001

S002 · calculation

Before

totient multiplicativity original modulus product := mul(2, 27)

After

totient multiplicativity original modulus product = 54

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S003

S005 · implies

Before

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S004

S006 · implies

Before

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: totient_complete_finite_count_not_sample_formula · Uses: G001, S007

S009 · implies

Before

right modulus representative1:original nonnegative Euclidean inputs=[1, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 27, 0, 1], [27, 1, 27, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S008

S010 · implies

Before

right modulus:original representative k=1 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S009

S011 · implies

Before

right modulus representative2:original nonnegative Euclidean inputs=[2, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 27, 0, 2], [27, 2, 13, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S010

S012 · implies

Before

right modulus:original representative k=2 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S011

S013 · implies

Before

right modulus representative3:original nonnegative Euclidean inputs=[3, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 27, 0, 3], [27, 3, 9, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S012

S014 · implies

Before

right modulus:original representative k=3 in1..27;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S013

S015 · implies

Before

right modulus representative4:original nonnegative Euclidean inputs=[4, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 27, 0, 4], [27, 4, 6, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S014

S016 · implies

Before

right modulus:original representative k=4 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S015

S017 · implies

Before

right modulus representative5:original nonnegative Euclidean inputs=[5, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 27, 0, 5], [27, 5, 5, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S016

S018 · implies

Before

right modulus:original representative k=5 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S017

S019 · implies

Before

right modulus representative6:original nonnegative Euclidean inputs=[6, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 27, 0, 6], [27, 6, 4, 3], [6, 3, 2, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S018

S020 · implies

Before

right modulus:original representative k=6 in1..27;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S019

S021 · implies

Before

right modulus representative7:original nonnegative Euclidean inputs=[7, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 27, 0, 7], [27, 7, 3, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S020

S022 · implies

Before

right modulus:original representative k=7 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S021

S023 · implies

Before

right modulus representative8:original nonnegative Euclidean inputs=[8, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 27, 0, 8], [27, 8, 3, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S022

S024 · implies

Before

right modulus:original representative k=8 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S023

S025 · implies

Before

right modulus representative9:original nonnegative Euclidean inputs=[9, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 27, 0, 9], [27, 9, 3, 0]];terminal positive gcd=9

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S024

S026 · implies

Before

right modulus:original representative k=9 in1..27;verified gcd(k,n)=9

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S025

S027 · implies

Before

right modulus representative10:original nonnegative Euclidean inputs=[10, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 27, 0, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S026

S028 · implies

Before

right modulus:original representative k=10 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S027

S029 · implies

Before

right modulus representative11:original nonnegative Euclidean inputs=[11, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 27, 0, 11], [27, 11, 2, 5], [11, 5, 2, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S028

S030 · implies

Before

right modulus:original representative k=11 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S029

S031 · implies

Before

right modulus representative12:original nonnegative Euclidean inputs=[12, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 27, 0, 12], [27, 12, 2, 3], [12, 3, 4, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S030

S032 · implies

Before

right modulus:original representative k=12 in1..27;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S031

S033 · implies

Before

right modulus representative13:original nonnegative Euclidean inputs=[13, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 27, 0, 13], [27, 13, 2, 1], [13, 1, 13, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S032

S034 · implies

Before

right modulus:original representative k=13 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S033

S035 · implies

Before

right modulus representative14:original nonnegative Euclidean inputs=[14, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 27, 0, 14], [27, 14, 1, 13], [14, 13, 1, 1], [13, 1, 13, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S034

S036 · implies

Before

right modulus:original representative k=14 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S035

S037 · implies

Before

right modulus representative15:original nonnegative Euclidean inputs=[15, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 27, 0, 15], [27, 15, 1, 12], [15, 12, 1, 3], [12, 3, 4, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S036

S038 · implies

Before

right modulus:original representative k=15 in1..27;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S037

S039 · implies

Before

right modulus representative16:original nonnegative Euclidean inputs=[16, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 27, 0, 16], [27, 16, 1, 11], [16, 11, 1, 5], [11, 5, 2, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S038

S040 · implies

Before

right modulus:original representative k=16 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S039

S041 · implies

Before

right modulus representative17:original nonnegative Euclidean inputs=[17, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 27, 0, 17], [27, 17, 1, 10], [17, 10, 1, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S040

S042 · implies

Before

right modulus:original representative k=17 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S041

S043 · implies

Before

right modulus representative18:original nonnegative Euclidean inputs=[18, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 27, 0, 18], [27, 18, 1, 9], [18, 9, 2, 0]];terminal positive gcd=9

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S042

S044 · implies

Before

right modulus:original representative k=18 in1..27;verified gcd(k,n)=9

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S043

S045 · implies

Before

right modulus representative19:original nonnegative Euclidean inputs=[19, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 27, 0, 19], [27, 19, 1, 8], [19, 8, 2, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S044

S046 · implies

Before

right modulus:original representative k=19 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S045

S047 · implies

Before

right modulus representative20:original nonnegative Euclidean inputs=[20, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 27, 0, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S046

S048 · implies

Before

right modulus:original representative k=20 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S047

S049 · implies

Before

right modulus representative21:original nonnegative Euclidean inputs=[21, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 27, 0, 21], [27, 21, 1, 6], [21, 6, 3, 3], [6, 3, 2, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S048

S050 · implies

Before

right modulus:original representative k=21 in1..27;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S049

S051 · implies

Before

right modulus representative22:original nonnegative Euclidean inputs=[22, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 27, 0, 22], [27, 22, 1, 5], [22, 5, 4, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S050

S052 · implies

Before

right modulus:original representative k=22 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S051

S053 · implies

Before

right modulus representative23:original nonnegative Euclidean inputs=[23, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 27, 0, 23], [27, 23, 1, 4], [23, 4, 5, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S052

S054 · implies

Before

right modulus:original representative k=23 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S053

S055 · implies

Before

right modulus representative24:original nonnegative Euclidean inputs=[24, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 27, 0, 24], [27, 24, 1, 3], [24, 3, 8, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S054

S056 · implies

Before

right modulus:original representative k=24 in1..27;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S055

S057 · implies

Before

right modulus representative25:original nonnegative Euclidean inputs=[25, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 27, 0, 25], [27, 25, 1, 2], [25, 2, 12, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S056

S058 · implies

Before

right modulus:original representative k=25 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S057

S059 · implies

Before

right modulus representative26:original nonnegative Euclidean inputs=[26, 27]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 27, 0, 26], [27, 26, 1, 1], [26, 1, 26, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S058

S060 · implies

Before

right modulus:original representative k=26 in1..27;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S059

S061 · implies

Before

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

After

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

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S060

S062 · implies

Before

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

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S061

S063 · calculation

Before

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

After

right modulus exact coprime indicator count = 18

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · implies

Before

right modulus:complete representative range1..27;all selected coprime representatives=[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26]

After

phi(27)=18;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 dependencies

Rule: totient_complete_finite_count_not_sample_formula · Uses: G001, S063

S065 · implies

Before

product modulus representative1:original nonnegative Euclidean inputs=[1, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 54, 0, 1], [54, 1, 54, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S064

S066 · implies

Before

product modulus:original representative k=1 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S065

S067 · implies

Before

product modulus representative2:original nonnegative Euclidean inputs=[2, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 54, 0, 2], [54, 2, 27, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S066

S068 · implies

Before

product modulus:original representative k=2 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S067

S069 · implies

Before

product modulus representative3:original nonnegative Euclidean inputs=[3, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 54, 0, 3], [54, 3, 18, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S068

S070 · implies

Before

product modulus:original representative k=3 in1..54;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S069

S071 · implies

Before

product modulus representative4:original nonnegative Euclidean inputs=[4, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 54, 0, 4], [54, 4, 13, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S070

S072 · implies

Before

product modulus:original representative k=4 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S071

S073 · implies

Before

product modulus representative5:original nonnegative Euclidean inputs=[5, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 54, 0, 5], [54, 5, 10, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S072

S074 · implies

Before

product modulus:original representative k=5 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S073

S075 · implies

Before

product modulus representative6:original nonnegative Euclidean inputs=[6, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 54, 0, 6], [54, 6, 9, 0]];terminal positive gcd=6

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S074

S076 · implies

Before

product modulus:original representative k=6 in1..54;verified gcd(k,n)=6

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S075

S077 · implies

Before

product modulus representative7:original nonnegative Euclidean inputs=[7, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 54, 0, 7], [54, 7, 7, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S076

S078 · implies

Before

product modulus:original representative k=7 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S077

S079 · implies

Before

product modulus representative8:original nonnegative Euclidean inputs=[8, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 54, 0, 8], [54, 8, 6, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S078

S080 · implies

Before

product modulus:original representative k=8 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S079

S081 · implies

Before

product modulus representative9:original nonnegative Euclidean inputs=[9, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 54, 0, 9], [54, 9, 6, 0]];terminal positive gcd=9

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S080

S082 · implies

Before

product modulus:original representative k=9 in1..54;verified gcd(k,n)=9

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S081

S083 · implies

Before

product modulus representative10:original nonnegative Euclidean inputs=[10, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 54, 0, 10], [54, 10, 5, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S082

S084 · implies

Before

product modulus:original representative k=10 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S083

S085 · implies

Before

product modulus representative11:original nonnegative Euclidean inputs=[11, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 54, 0, 11], [54, 11, 4, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S084

S086 · implies

Before

product modulus:original representative k=11 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S085

S087 · implies

Before

product modulus representative12:original nonnegative Euclidean inputs=[12, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 54, 0, 12], [54, 12, 4, 6], [12, 6, 2, 0]];terminal positive gcd=6

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S086

S088 · implies

Before

product modulus:original representative k=12 in1..54;verified gcd(k,n)=6

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S087

S089 · implies

Before

product modulus representative13:original nonnegative Euclidean inputs=[13, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 54, 0, 13], [54, 13, 4, 2], [13, 2, 6, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S088

S090 · implies

Before

product modulus:original representative k=13 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S089

S091 · implies

Before

product modulus representative14:original nonnegative Euclidean inputs=[14, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 54, 0, 14], [54, 14, 3, 12], [14, 12, 1, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S090

S092 · implies

Before

product modulus:original representative k=14 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S091

S093 · implies

Before

product modulus representative15:original nonnegative Euclidean inputs=[15, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 54, 0, 15], [54, 15, 3, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S092

S094 · implies

Before

product modulus:original representative k=15 in1..54;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S093

S095 · implies

Before

product modulus representative16:original nonnegative Euclidean inputs=[16, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 54, 0, 16], [54, 16, 3, 6], [16, 6, 2, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S094

S096 · implies

Before

product modulus:original representative k=16 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S095

S097 · implies

Before

product modulus representative17:original nonnegative Euclidean inputs=[17, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 54, 0, 17], [54, 17, 3, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S096

S098 · implies

Before

product modulus:original representative k=17 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S097

S099 · implies

Before

product modulus representative18:original nonnegative Euclidean inputs=[18, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 54, 0, 18], [54, 18, 3, 0]];terminal positive gcd=18

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S098

S100 · implies

Before

product modulus:original representative k=18 in1..54;verified gcd(k,n)=18

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S099

S101 · implies

Before

product modulus representative19:original nonnegative Euclidean inputs=[19, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 54, 0, 19], [54, 19, 2, 16], [19, 16, 1, 3], [16, 3, 5, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S100

S102 · implies

Before

product modulus:original representative k=19 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S101

S103 · implies

Before

product modulus representative20:original nonnegative Euclidean inputs=[20, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 54, 0, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S102

S104 · implies

Before

product modulus:original representative k=20 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S103

S105 · implies

Before

product modulus representative21:original nonnegative Euclidean inputs=[21, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 54, 0, 21], [54, 21, 2, 12], [21, 12, 1, 9], [12, 9, 1, 3], [9, 3, 3, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S104

S106 · implies

Before

product modulus:original representative k=21 in1..54;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S105

S107 · implies

Before

product modulus representative22:original nonnegative Euclidean inputs=[22, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 54, 0, 22], [54, 22, 2, 10], [22, 10, 2, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S106

S108 · implies

Before

product modulus:original representative k=22 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S107

S109 · implies

Before

product modulus representative23:original nonnegative Euclidean inputs=[23, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 54, 0, 23], [54, 23, 2, 8], [23, 8, 2, 7], [8, 7, 1, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S108

S110 · implies

Before

product modulus:original representative k=23 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S109

S111 · implies

Before

product modulus representative24:original nonnegative Euclidean inputs=[24, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 54, 0, 24], [54, 24, 2, 6], [24, 6, 4, 0]];terminal positive gcd=6

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S110

S112 · implies

Before

product modulus:original representative k=24 in1..54;verified gcd(k,n)=6

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S111

S113 · implies

Before

product modulus representative25:original nonnegative Euclidean inputs=[25, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 54, 0, 25], [54, 25, 2, 4], [25, 4, 6, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S112

S114 · implies

Before

product modulus:original representative k=25 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S113

S115 · implies

Before

product modulus representative26:original nonnegative Euclidean inputs=[26, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 54, 0, 26], [54, 26, 2, 2], [26, 2, 13, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S114

S116 · implies

Before

product modulus:original representative k=26 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S115

S117 · implies

Before

product modulus representative27:original nonnegative Euclidean inputs=[27, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 54, 0, 27], [54, 27, 2, 0]];terminal positive gcd=27

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S116

S118 · implies

Before

product modulus:original representative k=27 in1..54;verified gcd(k,n)=27

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S117

S119 · implies

Before

product modulus representative28:original nonnegative Euclidean inputs=[28, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 54, 0, 28], [54, 28, 1, 26], [28, 26, 1, 2], [26, 2, 13, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S118

S120 · implies

Before

product modulus:original representative k=28 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S119

S121 · implies

Before

product modulus representative29:original nonnegative Euclidean inputs=[29, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 54, 0, 29], [54, 29, 1, 25], [29, 25, 1, 4], [25, 4, 6, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S120

S122 · implies

Before

product modulus:original representative k=29 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S121

S123 · implies

Before

product modulus representative30:original nonnegative Euclidean inputs=[30, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 54, 0, 30], [54, 30, 1, 24], [30, 24, 1, 6], [24, 6, 4, 0]];terminal positive gcd=6

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S122

S124 · implies

Before

product modulus:original representative k=30 in1..54;verified gcd(k,n)=6

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S123

S125 · implies

Before

product modulus representative31:original nonnegative Euclidean inputs=[31, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[31, 54, 0, 31], [54, 31, 1, 23], [31, 23, 1, 8], [23, 8, 2, 7], [8, 7, 1, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S124

S126 · implies

Before

product modulus:original representative k=31 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S125

S127 · implies

Before

product modulus representative32:original nonnegative Euclidean inputs=[32, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[32, 54, 0, 32], [54, 32, 1, 22], [32, 22, 1, 10], [22, 10, 2, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S126

S128 · implies

Before

product modulus:original representative k=32 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S127

S129 · implies

Before

product modulus representative33:original nonnegative Euclidean inputs=[33, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[33, 54, 0, 33], [54, 33, 1, 21], [33, 21, 1, 12], [21, 12, 1, 9], [12, 9, 1, 3], [9, 3, 3, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S128

S130 · implies

Before

product modulus:original representative k=33 in1..54;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S129

S131 · implies

Before

product modulus representative34:original nonnegative Euclidean inputs=[34, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[34, 54, 0, 34], [54, 34, 1, 20], [34, 20, 1, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S130

S132 · implies

Before

product modulus:original representative k=34 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S131

S133 · implies

Before

product modulus representative35:original nonnegative Euclidean inputs=[35, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[35, 54, 0, 35], [54, 35, 1, 19], [35, 19, 1, 16], [19, 16, 1, 3], [16, 3, 5, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S132

S134 · implies

Before

product modulus:original representative k=35 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S133

S135 · implies

Before

product modulus representative36:original nonnegative Euclidean inputs=[36, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[36, 54, 0, 36], [54, 36, 1, 18], [36, 18, 2, 0]];terminal positive gcd=18

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S134

S136 · implies

Before

product modulus:original representative k=36 in1..54;verified gcd(k,n)=18

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S135

S137 · implies

Before

product modulus representative37:original nonnegative Euclidean inputs=[37, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[37, 54, 0, 37], [54, 37, 1, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S136

S138 · implies

Before

product modulus:original representative k=37 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S137

S139 · implies

Before

product modulus representative38:original nonnegative Euclidean inputs=[38, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[38, 54, 0, 38], [54, 38, 1, 16], [38, 16, 2, 6], [16, 6, 2, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S138

S140 · implies

Before

product modulus:original representative k=38 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S139

S141 · implies

Before

product modulus representative39:original nonnegative Euclidean inputs=[39, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[39, 54, 0, 39], [54, 39, 1, 15], [39, 15, 2, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S140

S142 · implies

Before

product modulus:original representative k=39 in1..54;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S141

S143 · implies

Before

product modulus representative40:original nonnegative Euclidean inputs=[40, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[40, 54, 0, 40], [54, 40, 1, 14], [40, 14, 2, 12], [14, 12, 1, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S142

S144 · implies

Before

product modulus:original representative k=40 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S143

S145 · implies

Before

product modulus representative41:original nonnegative Euclidean inputs=[41, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[41, 54, 0, 41], [54, 41, 1, 13], [41, 13, 3, 2], [13, 2, 6, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S144

S146 · implies

Before

product modulus:original representative k=41 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S145

S147 · implies

Before

product modulus representative42:original nonnegative Euclidean inputs=[42, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[42, 54, 0, 42], [54, 42, 1, 12], [42, 12, 3, 6], [12, 6, 2, 0]];terminal positive gcd=6

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S146

S148 · implies

Before

product modulus:original representative k=42 in1..54;verified gcd(k,n)=6

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S147

S149 · implies

Before

product modulus representative43:original nonnegative Euclidean inputs=[43, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[43, 54, 0, 43], [54, 43, 1, 11], [43, 11, 3, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S148

S150 · implies

Before

product modulus:original representative k=43 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S149

S151 · implies

Before

product modulus representative44:original nonnegative Euclidean inputs=[44, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[44, 54, 0, 44], [54, 44, 1, 10], [44, 10, 4, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S150

S152 · implies

Before

product modulus:original representative k=44 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S151

S153 · implies

Before

product modulus representative45:original nonnegative Euclidean inputs=[45, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[45, 54, 0, 45], [54, 45, 1, 9], [45, 9, 5, 0]];terminal positive gcd=9

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S152

S154 · implies

Before

product modulus:original representative k=45 in1..54;verified gcd(k,n)=9

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S153

S155 · implies

Before

product modulus representative46:original nonnegative Euclidean inputs=[46, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[46, 54, 0, 46], [54, 46, 1, 8], [46, 8, 5, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S154

S156 · implies

Before

product modulus:original representative k=46 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S155

S157 · implies

Before

product modulus representative47:original nonnegative Euclidean inputs=[47, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[47, 54, 0, 47], [54, 47, 1, 7], [47, 7, 6, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S156

S158 · implies

Before

product modulus:original representative k=47 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S157

S159 · implies

Before

product modulus representative48:original nonnegative Euclidean inputs=[48, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[48, 54, 0, 48], [54, 48, 1, 6], [48, 6, 8, 0]];terminal positive gcd=6

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S158

S160 · implies

Before

product modulus:original representative k=48 in1..54;verified gcd(k,n)=6

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S159

S161 · implies

Before

product modulus representative49:original nonnegative Euclidean inputs=[49, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[49, 54, 0, 49], [54, 49, 1, 5], [49, 5, 9, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S160

S162 · implies

Before

product modulus:original representative k=49 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S161

S163 · implies

Before

product modulus representative50:original nonnegative Euclidean inputs=[50, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[50, 54, 0, 50], [54, 50, 1, 4], [50, 4, 12, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S162

S164 · implies

Before

product modulus:original representative k=50 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S163

S165 · implies

Before

product modulus representative51:original nonnegative Euclidean inputs=[51, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[51, 54, 0, 51], [54, 51, 1, 3], [51, 3, 17, 0]];terminal positive gcd=3

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S164

S166 · implies

Before

product modulus:original representative k=51 in1..54;verified gcd(k,n)=3

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S165

S167 · implies

Before

product modulus representative52:original nonnegative Euclidean inputs=[52, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[52, 54, 0, 52], [54, 52, 1, 2], [52, 2, 26, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S166

S168 · implies

Before

product modulus:original representative k=52 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S167

S169 · implies

Before

product modulus representative53:original nonnegative Euclidean inputs=[53, 54]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[53, 54, 0, 53], [54, 53, 1, 1], [53, 1, 53, 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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S168

S170 · implies

Before

product modulus:original representative k=53 in1..54;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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S169

S171 · implies

Before

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

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[54, 54, 1, 0]];terminal positive gcd=54

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 dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S170

S172 · implies

Before

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

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 dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S171

S173 · calculation

Before

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

After

product modulus exact coprime indicator count = 18

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S172

S174 · implies

Before

product modulus:complete representative range1..54;all selected coprime representatives=[1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53]

After

phi(54)=18;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 dependencies

Rule: totient_complete_finite_count_not_sample_formula · Uses: G001, S173

S175 · calculation

Before

totient supplied instance product of counts := mul(1, 18)

After

totient supplied instance product of counts = 18

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S174

S176 · implies

Before

original coprime inputs=[2, 27];gcd=1;product modulus=54;exact totients=[1, 18, 18]

After

phi(m)*phi(n)=18;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 dependencies

Rule: totient_coprime_instance_not_unrestricted_multiplicativity_proof · Uses: G001, S175

Reverse checks

  • 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 — Recompute totient supplied instance product of counts with exact rational arithmetic, not rounded decimals.
  • 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 — Recompute product modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
  • S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S169 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S160 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S157 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S149 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S063 · premise_and_result_audit — Recompute right modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
  • 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, 27, 1, 18, 18, 18, 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: b5f37f8e2cbfdeb939753d331367c24ac7692ccd22a520c094831822bf295247
  • candidate record SHA-256: 67827bea1014ae537a9589b39c1638f660637e09fd6f6a236d9219367aacd770
  • semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e
  • question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON