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Question 16622: Keep phi(1), the Prime Case and the Composite Boundary Distinct

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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R0032-16622 · q-20261014-advance-20261101-r06-euler-totient-function-0072

Original question and givens

Positive integer n=22, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.

  • G001 — Positive integer n=22, with phi computed from the complete 1-through-n representative convention. Give (n,phi_n,classification). Classification is phi_1_equals_1, prime_phi_equals_n_minus_1 or composite_no_prime_shortcut.

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

selected modulus representative1:original nonnegative Euclidean inputs=[1, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 22, 0, 1], [22, 1, 22, 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 · implies

Before

selected modulus:original representative k=1 in1..22;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, S001

S003 · implies

Before

selected modulus representative2:original nonnegative Euclidean inputs=[2, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 22, 0, 2], [22, 2, 11, 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, S002

S004 · implies

Before

selected modulus:original representative k=2 in1..22;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, S003

S005 · implies

Before

selected modulus representative3:original nonnegative Euclidean inputs=[3, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 22, 0, 3], [22, 3, 7, 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, S004

S006 · implies

Before

selected modulus:original representative k=3 in1..22;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, S005

S007 · implies

Before

selected modulus representative4:original nonnegative Euclidean inputs=[4, 22]

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S006

S008 · implies

Before

selected modulus:original representative k=4 in1..22;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, S007

S009 · implies

Before

selected modulus representative5:original nonnegative Euclidean inputs=[5, 22]

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S008

S010 · implies

Before

selected modulus:original representative k=5 in1..22;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

selected modulus representative6:original nonnegative Euclidean inputs=[6, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 22, 0, 6], [22, 6, 3, 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, S010

S012 · implies

Before

selected modulus:original representative k=6 in1..22;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, S011

S013 · implies

Before

selected modulus representative7:original nonnegative Euclidean inputs=[7, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 22, 0, 7], [22, 7, 3, 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, S012

S014 · implies

Before

selected modulus:original representative k=7 in1..22;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

selected modulus representative8:original nonnegative Euclidean inputs=[8, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 22, 0, 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, S014

S016 · implies

Before

selected modulus:original representative k=8 in1..22;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, S015

S017 · implies

Before

selected modulus representative9:original nonnegative Euclidean inputs=[9, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 22, 0, 9], [22, 9, 2, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S016

S018 · implies

Before

selected modulus:original representative k=9 in1..22;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

selected modulus representative10:original nonnegative Euclidean inputs=[10, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 22, 0, 10], [22, 10, 2, 2], [10, 2, 5, 0]];terminal positive gcd=2

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S018

S020 · implies

Before

selected modulus:original representative k=10 in1..22;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, S019

S021 · implies

Before

selected modulus representative11:original nonnegative Euclidean inputs=[11, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 22, 0, 11], [22, 11, 2, 0]];terminal positive gcd=11

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;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

selected modulus:original representative k=11 in1..22;verified gcd(k,n)=11

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, S021

S023 · implies

Before

selected modulus representative12:original nonnegative Euclidean inputs=[12, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 22, 0, 12], [22, 12, 1, 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, S022

S024 · implies

Before

selected modulus:original representative k=12 in1..22;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, S023

S025 · implies

Before

selected modulus representative13:original nonnegative Euclidean inputs=[13, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 22, 0, 13], [22, 13, 1, 9], [13, 9, 1, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal positive gcd=1

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S024

S026 · implies

Before

selected modulus:original representative k=13 in1..22;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

selected modulus representative14:original nonnegative Euclidean inputs=[14, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 22, 0, 14], [22, 14, 1, 8], [14, 8, 1, 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, S026

S028 · implies

Before

selected modulus:original representative k=14 in1..22;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, S027

S029 · implies

Before

selected modulus representative15:original nonnegative Euclidean inputs=[15, 22]

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S028

S030 · implies

Before

selected modulus:original representative k=15 in1..22;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

selected modulus representative16:original nonnegative Euclidean inputs=[16, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 22, 0, 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, S030

S032 · implies

Before

selected modulus:original representative k=16 in1..22;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, S031

S033 · implies

Before

selected modulus representative17:original nonnegative Euclidean inputs=[17, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 22, 0, 17], [22, 17, 1, 5], [17, 5, 3, 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, S032

S034 · implies

Before

selected modulus:original representative k=17 in1..22;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

selected modulus representative18:original nonnegative Euclidean inputs=[18, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 22, 0, 18], [22, 18, 1, 4], [18, 4, 4, 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, S034

S036 · implies

Before

selected modulus:original representative k=18 in1..22;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, S035

S037 · implies

Before

selected modulus representative19:original nonnegative Euclidean inputs=[19, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 22, 0, 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, S036

S038 · implies

Before

selected modulus:original representative k=19 in1..22;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

selected modulus representative20:original nonnegative Euclidean inputs=[20, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 22, 0, 20], [22, 20, 1, 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, S038

S040 · implies

Before

selected modulus:original representative k=20 in1..22;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, S039

S041 · implies

Before

selected modulus representative21:original nonnegative Euclidean inputs=[21, 22]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 22, 0, 21], [22, 21, 1, 1], [21, 1, 21, 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

selected modulus:original representative k=21 in1..22;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

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

After

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

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;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

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

After

coprime indicator=0;include representative=No

Conditions: none

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

Step identifiers and dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S043

S045 · calculation

Before

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

After

selected modulus exact coprime indicator count = 10

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, S044

S046 · implies

Before

selected modulus:complete representative range1..22;all selected coprime representatives=[1, 3, 5, 7, 9, 13, 15, 17, 19, 21]

After

phi(22)=10;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, S045

S047 · implies

Before

original n=22;exact phi=10;first proper trial divisor up to floor(sqrt(n))=2

After

n=1 handled before primality;classification=composite_no_prime_shortcut

Conditions: none

One is not prime but phi(1)=1.For n≥2,no divisor up to its square root means primality;then every representative1 through n−1 is coprime and phi=n−1.Composite inputs retain the complete count rather than a prime-only shortcut.

Step identifiers and dependencies

Rule: totient_one_prime_composite_distinct_cases · Uses: G001, S046

Reverse checks

  • 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 — Recompute selected modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(22, 10, composite_no_prime_shortcut)

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: 369ee713225f5557ac25a0db8ab7388aa1c9577bb15b7725b694ef3723bfaedb
  • candidate record SHA-256: 1e46548af1c2a07b0516b91e73449d349f651bef21cb89e6807ba5ed0ad9de77
  • semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e
  • question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON