# Question 16637: Use Every Representative from 1 through n, Including the n=1 Convention

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0032-16637 · q-20261014-advance-20261101-r06-euler-totient-function-0012



## Original question and givens

Positive integer n=12. 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=12. 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.


- [mko-euler-totient-function](https://ocme.evemisslab.com/en/mko/mko-euler-totient-function/)





## Step-by-step derivation

S001 · impliesBefore

selected modulus representative1:original nonnegative Euclidean inputs=[1, 12]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 12, 0, 1], [12, 1, 12, 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 dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001





S002 · impliesBefore

selected modulus:original representative k=1 in1..12;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 dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S001





S003 · impliesBefore

selected modulus representative2:original nonnegative Euclidean inputs=[2, 12]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 12, 0, 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 dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S002





S004 · impliesBefore

selected modulus:original representative k=2 in1..12;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 dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S003





S005 · impliesBefore

selected modulus representative3:original nonnegative Euclidean inputs=[3, 12]

After

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

Conditions: none

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

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S004





S006 · impliesBefore

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

After

coprime indicator=0;include representative=No

Conditions: none

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

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S005





S007 · impliesBefore

selected modulus representative4:original nonnegative Euclidean inputs=[4, 12]

After

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

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 dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S006





S008 · impliesBefore

selected modulus:original representative k=4 in1..12;verified gcd(k,n)=4

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 dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S007





S009 · impliesBefore

selected modulus representative5:original nonnegative Euclidean inputs=[5, 12]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 12, 0, 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 dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S008





S010 · impliesBefore

selected modulus:original representative k=5 in1..12;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 dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S009





S011 · impliesBefore

selected modulus representative6:original nonnegative Euclidean inputs=[6, 12]

After

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

Conditions: none

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

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S010





S012 · impliesBefore

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

After

coprime indicator=0;include representative=No

Conditions: none

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

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S011





S013 · impliesBefore

selected modulus representative7:original nonnegative Euclidean inputs=[7, 12]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 12, 0, 7], [12, 7, 1, 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 dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S012





S014 · impliesBefore

selected modulus:original representative k=7 in1..12;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 dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S013





S015 · impliesBefore

selected modulus representative8:original nonnegative Euclidean inputs=[8, 12]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 12, 0, 8], [12, 8, 1, 4], [8, 4, 2, 0]];terminal positive gcd=4

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 dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S014





S016 · impliesBefore

selected modulus:original representative k=8 in1..12;verified gcd(k,n)=4

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 dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S015





S017 · impliesBefore

selected modulus representative9:original nonnegative Euclidean inputs=[9, 12]

After

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

Conditions: none

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

Step identifiers and dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S016





S018 · impliesBefore

selected modulus:original representative k=9 in1..12;verified gcd(k,n)=3

After

coprime indicator=0;include representative=No

Conditions: none

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

Step identifiers and dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S017





S019 · impliesBefore

selected modulus representative10:original nonnegative Euclidean inputs=[10, 12]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 12, 0, 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 dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S018





S020 · impliesBefore

selected modulus:original representative k=10 in1..12;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 dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S019





S021 · impliesBefore

selected modulus representative11:original nonnegative Euclidean inputs=[11, 12]

After

division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 12, 0, 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 dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S020





S022 · impliesBefore

selected modulus:original representative k=11 in1..12;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 dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S021





S023 · impliesBefore

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

After

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

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 dependenciesRule: totient_Euclidean_division_chain_preserves_common_divisors · Uses: G001, S022





S024 · impliesBefore

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

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 dependenciesRule: totient_every_original_representative_unit_gcd_test · Uses: G001, S023





S025 · calculationBefore

selected modulus exact coprime indicator count := sum(1, 0, 0, 0, 1, 0, 1, 0, 0, 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · impliesBefore

selected modulus:complete representative range1..12;all selected coprime representatives=[1, 5, 7, 11]

After

phi(12)=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 dependenciesRule: totient_complete_finite_count_not_sample_formula · Uses: G001, S025









## Reverse checks


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

(12, [1, 5, 7, 11], 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: a372cdc249be8443a769d02a5284e40f684737dca12159c7e0d106ecea29bfbe
- candidate record SHA-256: 0ee7fbf843b5972428d21db5d254dac54c8f4f176354fea7423275413abf4f14
- semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e
- question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8
- Lean: NOT_RUN · human review: NOT_PERFORMED

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