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Question 16632: 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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R0032-16632 · q-20261014-advance-20261101-r06-euler-totient-function-0007

Original question and givens

Positive integer n=7. 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=7. 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, 7]

After

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

S002 · implies

Before

selected modulus:original representative k=1 in1..7;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, 7]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 7, 0, 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, S002

S004 · implies

Before

selected modulus:original representative k=2 in1..7;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

selected modulus representative3:original nonnegative Euclidean inputs=[3, 7]

After

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

S006 · implies

Before

selected modulus:original representative k=3 in1..7;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, 7]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 7, 0, 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, S006

S008 · implies

Before

selected modulus:original representative k=4 in1..7;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, S007

S009 · implies

Before

selected modulus representative5:original nonnegative Euclidean inputs=[5, 7]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 7, 0, 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, S008

S010 · implies

Before

selected modulus:original representative k=5 in1..7;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, 7]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 7, 0, 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, S010

S012 · implies

Before

selected modulus:original representative k=6 in1..7;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

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

After

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

Conditions: none

Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;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..7;verified gcd(k,n)=7

After

coprime indicator=0;include representative=No

Conditions: none

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

Step identifiers and dependencies

Rule: totient_every_original_representative_unit_gcd_test · Uses: G001, S013

S015 · calculation

Before

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

After

selected modulus exact coprime indicator count = 6

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, S014

S016 · implies

Before

selected modulus:complete representative range1..7;all selected coprime representatives=[1, 2, 3, 4, 5, 6]

After

phi(7)=6;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, S015

Reverse checks

  • 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 — Recompute selected modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals.
  • 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

(7, [1, 2, 3, 4, 5, 6], 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: e517b8bc7886423dad9fafee477ff4baf0044e8968265c383f4d764e84b6c67c
  • candidate record SHA-256: 7d50198c3a661c733882f03ff1afc900acad498849607996940dc492b921bc5b
  • semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e
  • question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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