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Question 16635: Use Every Representative from 1 through n, Including the n=1 Convention

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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← Question derivation index · Original question batch

R0032-16635 · q-20261014-advance-20261101-r06-euler-totient-function-0010

Original question and givens

Positive integer n=10. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.

  • G001 — Positive integer n=10. Use the complete representative range 1≤k≤n. Give (n,coprime_residues,representatives_1_through_n), listing every k with gcd(k,n)=1 increasingly.

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, 10]

After

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

S002 · implies

Before

selected modulus:original representative k=1 in1..10;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, 10]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 10, 0, 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, S002

S004 · implies

Before

selected modulus:original representative k=2 in1..10;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, 10]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 10, 0, 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, S004

S006 · implies

Before

selected modulus:original representative k=3 in1..10;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, 10]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 10, 0, 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, S006

S008 · implies

Before

selected modulus:original representative k=4 in1..10;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, 10]

After

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

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;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..10;verified gcd(k,n)=5

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, S009

S011 · implies

Before

selected modulus representative6:original nonnegative Euclidean inputs=[6, 10]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 10, 0, 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, S010

S012 · implies

Before

selected modulus:original representative k=6 in1..10;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, 10]

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S012

S014 · implies

Before

selected modulus:original representative k=7 in1..10;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, 10]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 10, 0, 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, S014

S016 · implies

Before

selected modulus:original representative k=8 in1..10;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, 10]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 10, 0, 9], [10, 9, 1, 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, S016

S018 · implies

Before

selected modulus:original representative k=9 in1..10;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, 10]

After

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

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;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..10;verified gcd(k,n)=10

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 · calculation

Before

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

After

selected modulus exact coprime indicator count = 4

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, S020

S022 · implies

Before

selected modulus:complete representative range1..10;all selected coprime representatives=[1, 3, 7, 9]

After

phi(10)=4;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, S021

Reverse checks

  • 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 — Recompute selected modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(10, [1, 3, 7, 9], representatives_1_through_n)

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: a69bed639d425ece69167a03ccd76658ad36a5252395623d23cd6f62544b2f9d
  • candidate record SHA-256: 246c8f981ff37fb1b892878592809b93be1fd5496e0845c9e13ba3f5294a4497
  • semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e
  • question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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