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Question 16670: 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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R0032-16670 · q-20261014-advance-20261101-r06-euler-totient-function-0095

Original question and givens

Positive integers m=2, n=41 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=41 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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 41, 0, 2], [41, 2, 20, 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, 41)

After

totient multiplicativity original modulus product = 82

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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 41, 0, 1], [41, 1, 41, 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..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 41, 0, 2], [41, 2, 20, 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..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 41, 0, 3], [41, 3, 13, 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, S012

S014 · implies

Before

right modulus:original representative k=3 in1..41;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, S013

S015 · implies

Before

right modulus representative4:original nonnegative Euclidean inputs=[4, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 41, 0, 4], [41, 4, 10, 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, S014

S016 · implies

Before

right modulus:original representative k=4 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 41, 0, 5], [41, 5, 8, 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, S016

S018 · implies

Before

right modulus:original representative k=5 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 41, 0, 6], [41, 6, 6, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S018

S020 · implies

Before

right modulus:original representative k=6 in1..41;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, S019

S021 · implies

Before

right modulus representative7:original nonnegative Euclidean inputs=[7, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 41, 0, 7], [41, 7, 5, 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..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 41, 0, 8], [41, 8, 5, 1], [8, 1, 8, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S022

S024 · implies

Before

right modulus:original representative k=8 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 41, 0, 9], [41, 9, 4, 5], [9, 5, 1, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S024

S026 · implies

Before

right modulus:original representative k=9 in1..41;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, S025

S027 · implies

Before

right modulus representative10:original nonnegative Euclidean inputs=[10, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 41, 0, 10], [41, 10, 4, 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, S026

S028 · implies

Before

right modulus:original representative k=10 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 41, 0, 11], [41, 11, 3, 8], [11, 8, 1, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S028

S030 · implies

Before

right modulus:original representative k=11 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 41, 0, 12], [41, 12, 3, 5], [12, 5, 2, 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, S030

S032 · implies

Before

right modulus:original representative k=12 in1..41;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, S031

S033 · implies

Before

right modulus representative13:original nonnegative Euclidean inputs=[13, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 41, 0, 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, S032

S034 · implies

Before

right modulus:original representative k=13 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 41, 0, 14], [41, 14, 2, 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..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 41, 0, 15], [41, 15, 2, 11], [15, 11, 1, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S036

S038 · implies

Before

right modulus:original representative k=15 in1..41;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, S037

S039 · implies

Before

right modulus representative16:original nonnegative Euclidean inputs=[16, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 41, 0, 16], [41, 16, 2, 9], [16, 9, 1, 7], [9, 7, 1, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S038

S040 · implies

Before

right modulus:original representative k=16 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 41, 0, 17], [41, 17, 2, 7], [17, 7, 2, 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..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 41, 0, 18], [41, 18, 2, 5], [18, 5, 3, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S042

S044 · implies

Before

right modulus:original representative k=18 in1..41;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, S043

S045 · implies

Before

right modulus representative19:original nonnegative Euclidean inputs=[19, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 41, 0, 19], [41, 19, 2, 3], [19, 3, 6, 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, S044

S046 · implies

Before

right modulus:original representative k=19 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 41, 0, 20], [41, 20, 2, 1], [20, 1, 20, 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..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 41, 0, 21], [41, 21, 1, 20], [21, 20, 1, 1], [20, 1, 20, 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, S048

S050 · implies

Before

right modulus:original representative k=21 in1..41;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, S049

S051 · implies

Before

right modulus representative22:original nonnegative Euclidean inputs=[22, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 41, 0, 22], [41, 22, 1, 19], [22, 19, 1, 3], [19, 3, 6, 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, S050

S052 · implies

Before

right modulus:original representative k=22 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 41, 0, 23], [41, 23, 1, 18], [23, 18, 1, 5], [18, 5, 3, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S052

S054 · implies

Before

right modulus:original representative k=23 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 41, 0, 24], [41, 24, 1, 17], [24, 17, 1, 7], [17, 7, 2, 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, S054

S056 · implies

Before

right modulus:original representative k=24 in1..41;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, S055

S057 · implies

Before

right modulus representative25:original nonnegative Euclidean inputs=[25, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 41, 0, 25], [41, 25, 1, 16], [25, 16, 1, 9], [16, 9, 1, 7], [9, 7, 1, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S056

S058 · implies

Before

right modulus:original representative k=25 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 41, 0, 26], [41, 26, 1, 15], [26, 15, 1, 11], [15, 11, 1, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S058

S060 · implies

Before

right modulus:original representative k=26 in1..41;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, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 41, 0, 27], [41, 27, 1, 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, S060

S062 · implies

Before

right modulus:original representative k=27 in1..41;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, S061

S063 · implies

Before

right modulus representative28:original nonnegative Euclidean inputs=[28, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 41, 0, 28], [41, 28, 1, 13], [28, 13, 2, 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, S062

S064 · implies

Before

right modulus:original representative k=28 in1..41;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, S063

S065 · implies

Before

right modulus representative29:original nonnegative Euclidean inputs=[29, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 41, 0, 29], [41, 29, 1, 12], [29, 12, 2, 5], [12, 5, 2, 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, S064

S066 · implies

Before

right modulus:original representative k=29 in1..41;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

right modulus representative30:original nonnegative Euclidean inputs=[30, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 41, 0, 30], [41, 30, 1, 11], [30, 11, 2, 8], [11, 8, 1, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S066

S068 · implies

Before

right modulus:original representative k=30 in1..41;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, S067

S069 · implies

Before

right modulus representative31:original nonnegative Euclidean inputs=[31, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[31, 41, 0, 31], [41, 31, 1, 10], [31, 10, 3, 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, S068

S070 · implies

Before

right modulus:original representative k=31 in1..41;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, S069

S071 · implies

Before

right modulus representative32:original nonnegative Euclidean inputs=[32, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[32, 41, 0, 32], [41, 32, 1, 9], [32, 9, 3, 5], [9, 5, 1, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S070

S072 · implies

Before

right modulus:original representative k=32 in1..41;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, S071

S073 · implies

Before

right modulus representative33:original nonnegative Euclidean inputs=[33, 41]

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S072

S074 · implies

Before

right modulus:original representative k=33 in1..41;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

right modulus representative34:original nonnegative Euclidean inputs=[34, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[34, 41, 0, 34], [41, 34, 1, 7], [34, 7, 4, 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, S074

S076 · implies

Before

right modulus:original representative k=34 in1..41;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, S075

S077 · implies

Before

right modulus representative35:original nonnegative Euclidean inputs=[35, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[35, 41, 0, 35], [41, 35, 1, 6], [35, 6, 5, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S076

S078 · implies

Before

right modulus:original representative k=35 in1..41;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

right modulus representative36:original nonnegative Euclidean inputs=[36, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[36, 41, 0, 36], [41, 36, 1, 5], [36, 5, 7, 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, S078

S080 · implies

Before

right modulus:original representative k=36 in1..41;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, S079

S081 · implies

Before

right modulus representative37:original nonnegative Euclidean inputs=[37, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[37, 41, 0, 37], [41, 37, 1, 4], [37, 4, 9, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S080

S082 · implies

Before

right modulus:original representative k=37 in1..41;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, S081

S083 · implies

Before

right modulus representative38:original nonnegative Euclidean inputs=[38, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[38, 41, 0, 38], [41, 38, 1, 3], [38, 3, 12, 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, S082

S084 · implies

Before

right modulus:original representative k=38 in1..41;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, S083

S085 · implies

Before

right modulus representative39:original nonnegative Euclidean inputs=[39, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[39, 41, 0, 39], [41, 39, 1, 2], [39, 2, 19, 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, S084

S086 · implies

Before

right modulus:original representative k=39 in1..41;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

right modulus representative40:original nonnegative Euclidean inputs=[40, 41]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[40, 41, 0, 40], [41, 40, 1, 1], [40, 1, 40, 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, S086

S088 · implies

Before

right modulus:original representative k=40 in1..41;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, S087

S089 · implies

Before

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

After

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

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

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

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, S089

S091 · calculation

Before

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

After

right modulus exact coprime indicator count = 40

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, S090

S092 · implies

Before

right modulus:complete representative range1..41;all selected coprime representatives=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]

After

phi(41)=40;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, S091

S093 · implies

Before

product modulus representative1:original nonnegative Euclidean inputs=[1, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 82, 0, 1], [82, 1, 82, 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, S092

S094 · implies

Before

product modulus:original representative k=1 in1..82;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, S093

S095 · implies

Before

product modulus representative2:original nonnegative Euclidean inputs=[2, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 82, 0, 2], [82, 2, 41, 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=2 in1..82;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 representative3:original nonnegative Euclidean inputs=[3, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 82, 0, 3], [82, 3, 27, 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, S096

S098 · implies

Before

product modulus:original representative k=3 in1..82;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 representative4:original nonnegative Euclidean inputs=[4, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 82, 0, 4], [82, 4, 20, 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, S098

S100 · implies

Before

product modulus:original representative k=4 in1..82;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, S099

S101 · implies

Before

product modulus representative5:original nonnegative Euclidean inputs=[5, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 82, 0, 5], [82, 5, 16, 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, S100

S102 · implies

Before

product modulus:original representative k=5 in1..82;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 representative6:original nonnegative Euclidean inputs=[6, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 82, 0, 6], [82, 6, 13, 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, S102

S104 · implies

Before

product modulus:original representative k=6 in1..82;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 representative7:original nonnegative Euclidean inputs=[7, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 82, 0, 7], [82, 7, 11, 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, S104

S106 · implies

Before

product modulus:original representative k=7 in1..82;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, S105

S107 · implies

Before

product modulus representative8:original nonnegative Euclidean inputs=[8, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 82, 0, 8], [82, 8, 10, 2], [8, 2, 4, 0]];terminal positive gcd=2

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S106

S108 · implies

Before

product modulus:original representative k=8 in1..82;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 representative9:original nonnegative Euclidean inputs=[9, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 82, 0, 9], [82, 9, 9, 1], [9, 1, 9, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S108

S110 · implies

Before

product modulus:original representative k=9 in1..82;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 representative10:original nonnegative Euclidean inputs=[10, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 82, 0, 10], [82, 10, 8, 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, S110

S112 · implies

Before

product modulus:original representative k=10 in1..82;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, S111

S113 · implies

Before

product modulus representative11:original nonnegative Euclidean inputs=[11, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 82, 0, 11], [82, 11, 7, 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, S112

S114 · implies

Before

product modulus:original representative k=11 in1..82;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 representative12:original nonnegative Euclidean inputs=[12, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 82, 0, 12], [82, 12, 6, 10], [12, 10, 1, 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, S114

S116 · implies

Before

product modulus:original representative k=12 in1..82;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 representative13:original nonnegative Euclidean inputs=[13, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 82, 0, 13], [82, 13, 6, 4], [13, 4, 3, 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, S116

S118 · implies

Before

product modulus:original representative k=13 in1..82;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, S117

S119 · implies

Before

product modulus representative14:original nonnegative Euclidean inputs=[14, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 82, 0, 14], [82, 14, 5, 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, S118

S120 · implies

Before

product modulus:original representative k=14 in1..82;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 representative15:original nonnegative Euclidean inputs=[15, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 82, 0, 15], [82, 15, 5, 7], [15, 7, 2, 1], [7, 1, 7, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S120

S122 · implies

Before

product modulus:original representative k=15 in1..82;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 representative16:original nonnegative Euclidean inputs=[16, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 82, 0, 16], [82, 16, 5, 2], [16, 2, 8, 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, S122

S124 · implies

Before

product modulus:original representative k=16 in1..82;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, S123

S125 · implies

Before

product modulus representative17:original nonnegative Euclidean inputs=[17, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 82, 0, 17], [82, 17, 4, 14], [17, 14, 1, 3], [14, 3, 4, 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, S124

S126 · implies

Before

product modulus:original representative k=17 in1..82;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 representative18:original nonnegative Euclidean inputs=[18, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 82, 0, 18], [82, 18, 4, 10], [18, 10, 1, 8], [10, 8, 1, 2], [8, 2, 4, 0]];terminal positive gcd=2

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S126

S128 · implies

Before

product modulus:original representative k=18 in1..82;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 representative19:original nonnegative Euclidean inputs=[19, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 82, 0, 19], [82, 19, 4, 6], [19, 6, 3, 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, S128

S130 · implies

Before

product modulus:original representative k=19 in1..82;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, S129

S131 · implies

Before

product modulus representative20:original nonnegative Euclidean inputs=[20, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 82, 0, 20], [82, 20, 4, 2], [20, 2, 10, 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=20 in1..82;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 representative21:original nonnegative Euclidean inputs=[21, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 82, 0, 21], [82, 21, 3, 19], [21, 19, 1, 2], [19, 2, 9, 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, S132

S134 · implies

Before

product modulus:original representative k=21 in1..82;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 representative22:original nonnegative Euclidean inputs=[22, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 82, 0, 22], [82, 22, 3, 16], [22, 16, 1, 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, S134

S136 · implies

Before

product modulus:original representative k=22 in1..82;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, S135

S137 · implies

Before

product modulus representative23:original nonnegative Euclidean inputs=[23, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[23, 82, 0, 23], [82, 23, 3, 13], [23, 13, 1, 10], [13, 10, 1, 3], [10, 3, 3, 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, S136

S138 · implies

Before

product modulus:original representative k=23 in1..82;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 representative24:original nonnegative Euclidean inputs=[24, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[24, 82, 0, 24], [82, 24, 3, 10], [24, 10, 2, 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, S138

S140 · implies

Before

product modulus:original representative k=24 in1..82;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 representative25:original nonnegative Euclidean inputs=[25, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[25, 82, 0, 25], [82, 25, 3, 7], [25, 7, 3, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S140

S142 · implies

Before

product modulus:original representative k=25 in1..82;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, S141

S143 · implies

Before

product modulus representative26:original nonnegative Euclidean inputs=[26, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[26, 82, 0, 26], [82, 26, 3, 4], [26, 4, 6, 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, S142

S144 · implies

Before

product modulus:original representative k=26 in1..82;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 representative27:original nonnegative Euclidean inputs=[27, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[27, 82, 0, 27], [82, 27, 3, 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, S144

S146 · implies

Before

product modulus:original representative k=27 in1..82;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 representative28:original nonnegative Euclidean inputs=[28, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[28, 82, 0, 28], [82, 28, 2, 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, S146

S148 · implies

Before

product modulus:original representative k=28 in1..82;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, S147

S149 · implies

Before

product modulus representative29:original nonnegative Euclidean inputs=[29, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[29, 82, 0, 29], [82, 29, 2, 24], [29, 24, 1, 5], [24, 5, 4, 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, S148

S150 · implies

Before

product modulus:original representative k=29 in1..82;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 representative30:original nonnegative Euclidean inputs=[30, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[30, 82, 0, 30], [82, 30, 2, 22], [30, 22, 1, 8], [22, 8, 2, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S150

S152 · implies

Before

product modulus:original representative k=30 in1..82;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 representative31:original nonnegative Euclidean inputs=[31, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[31, 82, 0, 31], [82, 31, 2, 20], [31, 20, 1, 11], [20, 11, 1, 9], [11, 9, 1, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S152

S154 · implies

Before

product modulus:original representative k=31 in1..82;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, S153

S155 · implies

Before

product modulus representative32:original nonnegative Euclidean inputs=[32, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[32, 82, 0, 32], [82, 32, 2, 18], [32, 18, 1, 14], [18, 14, 1, 4], [14, 4, 3, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S154

S156 · implies

Before

product modulus:original representative k=32 in1..82;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 representative33:original nonnegative Euclidean inputs=[33, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[33, 82, 0, 33], [82, 33, 2, 16], [33, 16, 2, 1], [16, 1, 16, 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=33 in1..82;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 representative34:original nonnegative Euclidean inputs=[34, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[34, 82, 0, 34], [82, 34, 2, 14], [34, 14, 2, 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, S158

S160 · implies

Before

product modulus:original representative k=34 in1..82;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, S159

S161 · implies

Before

product modulus representative35:original nonnegative Euclidean inputs=[35, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[35, 82, 0, 35], [82, 35, 2, 12], [35, 12, 2, 11], [12, 11, 1, 1], [11, 1, 11, 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=35 in1..82;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 representative36:original nonnegative Euclidean inputs=[36, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[36, 82, 0, 36], [82, 36, 2, 10], [36, 10, 3, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S162

S164 · implies

Before

product modulus:original representative k=36 in1..82;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 representative37:original nonnegative Euclidean inputs=[37, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[37, 82, 0, 37], [82, 37, 2, 8], [37, 8, 4, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S164

S166 · implies

Before

product modulus:original representative k=37 in1..82;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, S165

S167 · implies

Before

product modulus representative38:original nonnegative Euclidean inputs=[38, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[38, 82, 0, 38], [82, 38, 2, 6], [38, 6, 6, 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, S166

S168 · implies

Before

product modulus:original representative k=38 in1..82;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 representative39:original nonnegative Euclidean inputs=[39, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[39, 82, 0, 39], [82, 39, 2, 4], [39, 4, 9, 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, S168

S170 · implies

Before

product modulus:original representative k=39 in1..82;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 representative40:original nonnegative Euclidean inputs=[40, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[40, 82, 0, 40], [82, 40, 2, 2], [40, 2, 20, 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, S170

S172 · implies

Before

product modulus:original representative k=40 in1..82;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, S171

S173 · implies

Before

product modulus representative41:original nonnegative Euclidean inputs=[41, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[41, 82, 0, 41], [82, 41, 2, 0]];terminal positive gcd=41

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, S172

S174 · implies

Before

product modulus:original representative k=41 in1..82;verified gcd(k,n)=41

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, S173

S175 · implies

Before

product modulus representative42:original nonnegative Euclidean inputs=[42, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[42, 82, 0, 42], [82, 42, 1, 40], [42, 40, 1, 2], [40, 2, 20, 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, S174

S176 · implies

Before

product modulus:original representative k=42 in1..82;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, S175

S177 · implies

Before

product modulus representative43:original nonnegative Euclidean inputs=[43, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[43, 82, 0, 43], [82, 43, 1, 39], [43, 39, 1, 4], [39, 4, 9, 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, S176

S178 · implies

Before

product modulus:original representative k=43 in1..82;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, S177

S179 · implies

Before

product modulus representative44:original nonnegative Euclidean inputs=[44, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[44, 82, 0, 44], [82, 44, 1, 38], [44, 38, 1, 6], [38, 6, 6, 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, S178

S180 · implies

Before

product modulus:original representative k=44 in1..82;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, S179

S181 · implies

Before

product modulus representative45:original nonnegative Euclidean inputs=[45, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[45, 82, 0, 45], [82, 45, 1, 37], [45, 37, 1, 8], [37, 8, 4, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S180

S182 · implies

Before

product modulus:original representative k=45 in1..82;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, S181

S183 · implies

Before

product modulus representative46:original nonnegative Euclidean inputs=[46, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[46, 82, 0, 46], [82, 46, 1, 36], [46, 36, 1, 10], [36, 10, 3, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S182

S184 · implies

Before

product modulus:original representative k=46 in1..82;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, S183

S185 · implies

Before

product modulus representative47:original nonnegative Euclidean inputs=[47, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[47, 82, 0, 47], [82, 47, 1, 35], [47, 35, 1, 12], [35, 12, 2, 11], [12, 11, 1, 1], [11, 1, 11, 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, S184

S186 · implies

Before

product modulus:original representative k=47 in1..82;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, S185

S187 · implies

Before

product modulus representative48:original nonnegative Euclidean inputs=[48, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[48, 82, 0, 48], [82, 48, 1, 34], [48, 34, 1, 14], [34, 14, 2, 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, S186

S188 · implies

Before

product modulus:original representative k=48 in1..82;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, S187

S189 · implies

Before

product modulus representative49:original nonnegative Euclidean inputs=[49, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[49, 82, 0, 49], [82, 49, 1, 33], [49, 33, 1, 16], [33, 16, 2, 1], [16, 1, 16, 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, S188

S190 · implies

Before

product modulus:original representative k=49 in1..82;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, S189

S191 · implies

Before

product modulus representative50:original nonnegative Euclidean inputs=[50, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[50, 82, 0, 50], [82, 50, 1, 32], [50, 32, 1, 18], [32, 18, 1, 14], [18, 14, 1, 4], [14, 4, 3, 2], [4, 2, 2, 0]];terminal positive gcd=2

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S190

S192 · implies

Before

product modulus:original representative k=50 in1..82;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, S191

S193 · implies

Before

product modulus representative51:original nonnegative Euclidean inputs=[51, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[51, 82, 0, 51], [82, 51, 1, 31], [51, 31, 1, 20], [31, 20, 1, 11], [20, 11, 1, 9], [11, 9, 1, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S192

S194 · implies

Before

product modulus:original representative k=51 in1..82;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, S193

S195 · implies

Before

product modulus representative52:original nonnegative Euclidean inputs=[52, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[52, 82, 0, 52], [82, 52, 1, 30], [52, 30, 1, 22], [30, 22, 1, 8], [22, 8, 2, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S194

S196 · implies

Before

product modulus:original representative k=52 in1..82;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, S195

S197 · implies

Before

product modulus representative53:original nonnegative Euclidean inputs=[53, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[53, 82, 0, 53], [82, 53, 1, 29], [53, 29, 1, 24], [29, 24, 1, 5], [24, 5, 4, 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, S196

S198 · implies

Before

product modulus:original representative k=53 in1..82;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, S197

S199 · implies

Before

product modulus representative54:original nonnegative Euclidean inputs=[54, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[54, 82, 0, 54], [82, 54, 1, 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, S198

S200 · implies

Before

product modulus:original representative k=54 in1..82;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, S199

S201 · implies

Before

product modulus representative55:original nonnegative Euclidean inputs=[55, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[55, 82, 0, 55], [82, 55, 1, 27], [55, 27, 2, 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, S200

S202 · implies

Before

product modulus:original representative k=55 in1..82;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, S201

S203 · implies

Before

product modulus representative56:original nonnegative Euclidean inputs=[56, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[56, 82, 0, 56], [82, 56, 1, 26], [56, 26, 2, 4], [26, 4, 6, 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, S202

S204 · implies

Before

product modulus:original representative k=56 in1..82;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, S203

S205 · implies

Before

product modulus representative57:original nonnegative Euclidean inputs=[57, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[57, 82, 0, 57], [82, 57, 1, 25], [57, 25, 2, 7], [25, 7, 3, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S204

S206 · implies

Before

product modulus:original representative k=57 in1..82;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, S205

S207 · implies

Before

product modulus representative58:original nonnegative Euclidean inputs=[58, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[58, 82, 0, 58], [82, 58, 1, 24], [58, 24, 2, 10], [24, 10, 2, 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, S206

S208 · implies

Before

product modulus:original representative k=58 in1..82;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, S207

S209 · implies

Before

product modulus representative59:original nonnegative Euclidean inputs=[59, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[59, 82, 0, 59], [82, 59, 1, 23], [59, 23, 2, 13], [23, 13, 1, 10], [13, 10, 1, 3], [10, 3, 3, 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, S208

S210 · implies

Before

product modulus:original representative k=59 in1..82;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, S209

S211 · implies

Before

product modulus representative60:original nonnegative Euclidean inputs=[60, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[60, 82, 0, 60], [82, 60, 1, 22], [60, 22, 2, 16], [22, 16, 1, 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, S210

S212 · implies

Before

product modulus:original representative k=60 in1..82;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, S211

S213 · implies

Before

product modulus representative61:original nonnegative Euclidean inputs=[61, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[61, 82, 0, 61], [82, 61, 1, 21], [61, 21, 2, 19], [21, 19, 1, 2], [19, 2, 9, 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, S212

S214 · implies

Before

product modulus:original representative k=61 in1..82;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, S213

S215 · implies

Before

product modulus representative62:original nonnegative Euclidean inputs=[62, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[62, 82, 0, 62], [82, 62, 1, 20], [62, 20, 3, 2], [20, 2, 10, 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, S214

S216 · implies

Before

product modulus:original representative k=62 in1..82;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, S215

S217 · implies

Before

product modulus representative63:original nonnegative Euclidean inputs=[63, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[63, 82, 0, 63], [82, 63, 1, 19], [63, 19, 3, 6], [19, 6, 3, 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, S216

S218 · implies

Before

product modulus:original representative k=63 in1..82;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, S217

S219 · implies

Before

product modulus representative64:original nonnegative Euclidean inputs=[64, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[64, 82, 0, 64], [82, 64, 1, 18], [64, 18, 3, 10], [18, 10, 1, 8], [10, 8, 1, 2], [8, 2, 4, 0]];terminal positive gcd=2

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S218

S220 · implies

Before

product modulus:original representative k=64 in1..82;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, S219

S221 · implies

Before

product modulus representative65:original nonnegative Euclidean inputs=[65, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[65, 82, 0, 65], [82, 65, 1, 17], [65, 17, 3, 14], [17, 14, 1, 3], [14, 3, 4, 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, S220

S222 · implies

Before

product modulus:original representative k=65 in1..82;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, S221

S223 · implies

Before

product modulus representative66:original nonnegative Euclidean inputs=[66, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[66, 82, 0, 66], [82, 66, 1, 16], [66, 16, 4, 2], [16, 2, 8, 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, S222

S224 · implies

Before

product modulus:original representative k=66 in1..82;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, S223

S225 · implies

Before

product modulus representative67:original nonnegative Euclidean inputs=[67, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[67, 82, 0, 67], [82, 67, 1, 15], [67, 15, 4, 7], [15, 7, 2, 1], [7, 1, 7, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S224

S226 · implies

Before

product modulus:original representative k=67 in1..82;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, S225

S227 · implies

Before

product modulus representative68:original nonnegative Euclidean inputs=[68, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[68, 82, 0, 68], [82, 68, 1, 14], [68, 14, 4, 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, S226

S228 · implies

Before

product modulus:original representative k=68 in1..82;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, S227

S229 · implies

Before

product modulus representative69:original nonnegative Euclidean inputs=[69, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[69, 82, 0, 69], [82, 69, 1, 13], [69, 13, 5, 4], [13, 4, 3, 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, S228

S230 · implies

Before

product modulus:original representative k=69 in1..82;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, S229

S231 · implies

Before

product modulus representative70:original nonnegative Euclidean inputs=[70, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[70, 82, 0, 70], [82, 70, 1, 12], [70, 12, 5, 10], [12, 10, 1, 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, S230

S232 · implies

Before

product modulus:original representative k=70 in1..82;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, S231

S233 · implies

Before

product modulus representative71:original nonnegative Euclidean inputs=[71, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[71, 82, 0, 71], [82, 71, 1, 11], [71, 11, 6, 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, S232

S234 · implies

Before

product modulus:original representative k=71 in1..82;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, S233

S235 · implies

Before

product modulus representative72:original nonnegative Euclidean inputs=[72, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[72, 82, 0, 72], [82, 72, 1, 10], [72, 10, 7, 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, S234

S236 · implies

Before

product modulus:original representative k=72 in1..82;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, S235

S237 · implies

Before

product modulus representative73:original nonnegative Euclidean inputs=[73, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[73, 82, 0, 73], [82, 73, 1, 9], [73, 9, 8, 1], [9, 1, 9, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S236

S238 · implies

Before

product modulus:original representative k=73 in1..82;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, S237

S239 · implies

Before

product modulus representative74:original nonnegative Euclidean inputs=[74, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[74, 82, 0, 74], [82, 74, 1, 8], [74, 8, 9, 2], [8, 2, 4, 0]];terminal positive gcd=2

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S238

S240 · implies

Before

product modulus:original representative k=74 in1..82;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, S239

S241 · implies

Before

product modulus representative75:original nonnegative Euclidean inputs=[75, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[75, 82, 0, 75], [82, 75, 1, 7], [75, 7, 10, 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, S240

S242 · implies

Before

product modulus:original representative k=75 in1..82;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, S241

S243 · implies

Before

product modulus representative76:original nonnegative Euclidean inputs=[76, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[76, 82, 0, 76], [82, 76, 1, 6], [76, 6, 12, 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, S242

S244 · implies

Before

product modulus:original representative k=76 in1..82;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, S243

S245 · implies

Before

product modulus representative77:original nonnegative Euclidean inputs=[77, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[77, 82, 0, 77], [82, 77, 1, 5], [77, 5, 15, 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, S244

S246 · implies

Before

product modulus:original representative k=77 in1..82;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, S245

S247 · implies

Before

product modulus representative78:original nonnegative Euclidean inputs=[78, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[78, 82, 0, 78], [82, 78, 1, 4], [78, 4, 19, 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, S246

S248 · implies

Before

product modulus:original representative k=78 in1..82;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, S247

S249 · implies

Before

product modulus representative79:original nonnegative Euclidean inputs=[79, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[79, 82, 0, 79], [82, 79, 1, 3], [79, 3, 26, 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, S248

S250 · implies

Before

product modulus:original representative k=79 in1..82;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, S249

S251 · implies

Before

product modulus representative80:original nonnegative Euclidean inputs=[80, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[80, 82, 0, 80], [82, 80, 1, 2], [80, 2, 40, 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, S250

S252 · implies

Before

product modulus:original representative k=80 in1..82;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, S251

S253 · implies

Before

product modulus representative81:original nonnegative Euclidean inputs=[81, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[81, 82, 0, 81], [82, 81, 1, 1], [81, 1, 81, 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, S252

S254 · implies

Before

product modulus:original representative k=81 in1..82;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, S253

S255 · implies

Before

product modulus representative82:original nonnegative Euclidean inputs=[82, 82]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[82, 82, 1, 0]];terminal positive gcd=82

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, S254

S256 · implies

Before

product modulus:original representative k=82 in1..82;verified gcd(k,n)=82

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, S255

S257 · calculation

Before

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

After

product modulus exact coprime indicator count = 40

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, S256

S258 · implies

Before

product modulus:complete representative range1..82;all selected coprime representatives=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81]

After

phi(82)=40;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, S257

S259 · calculation

Before

totient supplied instance product of counts := mul(1, 40)

After

totient supplied instance product of counts = 40

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, S258

S260 · implies

Before

original coprime inputs=[2, 41];gcd=1;product modulus=82;exact totients=[1, 40, 40]

After

phi(m)*phi(n)=40;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, S259

Reverse checks

  • S260 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S259 · premise_and_result_audit — Recompute totient supplied instance product of counts with exact rational arithmetic, not rounded decimals.
  • S258 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S257 · premise_and_result_audit — Recompute product modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
  • S256 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S255 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S254 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S253 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S252 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S251 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S250 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S249 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S248 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S247 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S246 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S245 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S244 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S243 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S242 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S241 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S240 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S239 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S238 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S237 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S236 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S235 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S234 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S233 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S232 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S231 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S230 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S229 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S228 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S227 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S226 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S225 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S224 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S223 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S222 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S221 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S220 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S219 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S218 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S217 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S216 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S215 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S214 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S213 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S212 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S211 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S210 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S209 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S208 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S206 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S205 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S204 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S203 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S202 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S201 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S200 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S199 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S198 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S197 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S196 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S195 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S194 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S193 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S190 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S189 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S188 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S187 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S185 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S183 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S182 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S181 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S180 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S179 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S178 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S177 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S175 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S173 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S169 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S167 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S161 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S160 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S157 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S155 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S149 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S091 · premise_and_result_audit — Recompute right modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
  • S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — 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, 41, 1, 40, 40, 40, 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: 4eb21ea63aa8ebf62b5e767c89c4e5d88794947fc1b7be8f6d3e84e2c287d28a
  • candidate record SHA-256: a66c692fb98f3545aab0f8fcf14227d64ba83e84f56c66d2e872c17851b1069b
  • semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e
  • question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON