# Complete Coprime Representatives, Phi of One and Coprime Multiplicativity

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261101-r06-euler-totient-function-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=1. 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.

View answer and explanation(1, [1], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (1, [1], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0001/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 2. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=2. 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.

View answer and explanation(2, [1], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (2, [1], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0002/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 3. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=3. 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.

View answer and explanation(3, [1, 2], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (3, [1, 2], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0003/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 4. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=4. 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.

View answer and explanation(4, [1, 3], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (4, [1, 3], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0004/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 5. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=5. 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.

View answer and explanation(5, [1, 2, 3, 4], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (5, [1, 2, 3, 4], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0005/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 6. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=6. 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.

View answer and explanation(6, [1, 5], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (6, [1, 5], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0006/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 7. Use Every Representative from 1 through n, Including the n=1 Convention

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.

View answer and explanation(7, [1, 2, 3, 4, 5, 6], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (7, [1, 2, 3, 4, 5, 6], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0007/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 8. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=8. 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.

View answer and explanation(8, [1, 3, 5, 7], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (8, [1, 3, 5, 7], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0008/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 9. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=9. 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.

View answer and explanation(9, [1, 2, 4, 5, 7, 8], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (9, [1, 2, 4, 5, 7, 8], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0009/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 10. Use Every Representative from 1 through n, Including the n=1 Convention

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.

View answer and explanation(10, [1, 3, 7, 9], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (10, [1, 3, 7, 9], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0010/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 11. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=11. 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.

View answer and explanation(11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0011/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 12. Use Every Representative from 1 through n, Including the n=1 Convention

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.

View answer and explanation(12, [1, 5, 7, 11], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (12, [1, 5, 7, 11], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0012/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 13. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=13. 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.

View answer and explanation(13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0013/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 14. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=14. 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.

View answer and explanation(14, [1, 3, 5, 9, 11, 13], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (14, [1, 3, 5, 9, 11, 13], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0014/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 15. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=15. 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.

View answer and explanation(15, [1, 2, 4, 7, 8, 11, 13, 14], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (15, [1, 2, 4, 7, 8, 11, 13, 14], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0015/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 16. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=16. 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.

View answer and explanation(16, [1, 3, 5, 7, 9, 11, 13, 15], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (16, [1, 3, 5, 7, 9, 11, 13, 15], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0016/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 17. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=17. 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.

View answer and explanation(17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0017/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 18. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=18. 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.

View answer and explanation(18, [1, 5, 7, 11, 13, 17], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (18, [1, 5, 7, 11, 13, 17], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0018/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 19. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=19. 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.

View answer and explanation(19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0019/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 20. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=20. 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.

View answer and explanation(20, [1, 3, 7, 9, 11, 13, 17, 19], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (20, [1, 3, 7, 9, 11, 13, 17, 19], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0020/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 21. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=21. 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.

View answer and explanation(21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0021/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 22. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=22. 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.

View answer and explanation(22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0022/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 23. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=23. 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.

View answer and explanation(23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0023/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 24. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=24. 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.

View answer and explanation(24, [1, 5, 7, 11, 13, 17, 19, 23], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (24, [1, 5, 7, 11, 13, 17, 19, 23], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0024/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 25. Use Every Representative from 1 through n, Including the n=1 Convention

Positive integer n=25. 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.

View answer and explanation(25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], representatives_1_through_n)

Enumerate the full range and compute exact gcd values. For n=1 the single representative is 1 and gcd(1,1)=1, preserving phi(1)=1. Do not mix this list with a second residue convention or count twice. The exact result, retaining the requested order, is (25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], representatives_1_through_n).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0025/)

Euler totient function · qf-mod32-totient-coprime-residues · candidate





## 26. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=1, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(1, [1], 1)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (1, [1], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0026/)

Euler totient function · qf-mod32-totient-value · candidate





## 27. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=2, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(2, [1], 1)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (2, [1], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0027/)

Euler totient function · qf-mod32-totient-value · candidate





## 28. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=3, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(3, [1, 2], 2)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (3, [1, 2], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0028/)

Euler totient function · qf-mod32-totient-value · candidate





## 29. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=4, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(4, [1, 3], 2)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (4, [1, 3], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0029/)

Euler totient function · qf-mod32-totient-value · candidate





## 30. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=5, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(5, [1, 2, 3, 4], 4)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (5, [1, 2, 3, 4], 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0030/)

Euler totient function · qf-mod32-totient-value · candidate





## 31. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=6, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(6, [1, 5], 2)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (6, [1, 5], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0031/)

Euler totient function · qf-mod32-totient-value · candidate





## 32. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=7, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(7, [1, 2, 3, 4, 5, 6], 6)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (7, [1, 2, 3, 4, 5, 6], 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0032/)

Euler totient function · qf-mod32-totient-value · candidate





## 33. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=8, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(8, [1, 3, 5, 7], 4)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (8, [1, 3, 5, 7], 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0033/)

Euler totient function · qf-mod32-totient-value · candidate





## 34. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=9, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(9, [1, 2, 4, 5, 7, 8], 6)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (9, [1, 2, 4, 5, 7, 8], 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0034/)

Euler totient function · qf-mod32-totient-value · candidate





## 35. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=10, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(10, [1, 3, 7, 9], 4)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (10, [1, 3, 7, 9], 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0035/)

Euler totient function · qf-mod32-totient-value · candidate





## 36. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=11, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 10)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0036/)

Euler totient function · qf-mod32-totient-value · candidate





## 37. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=12, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(12, [1, 5, 7, 11], 4)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (12, [1, 5, 7, 11], 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0037/)

Euler totient function · qf-mod32-totient-value · candidate





## 38. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=13, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 12)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0038/)

Euler totient function · qf-mod32-totient-value · candidate





## 39. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=14, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(14, [1, 3, 5, 9, 11, 13], 6)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (14, [1, 3, 5, 9, 11, 13], 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0039/)

Euler totient function · qf-mod32-totient-value · candidate





## 40. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=15, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(15, [1, 2, 4, 7, 8, 11, 13, 14], 8)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (15, [1, 2, 4, 7, 8, 11, 13, 14], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0040/)

Euler totient function · qf-mod32-totient-value · candidate





## 41. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=16, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(16, [1, 3, 5, 7, 9, 11, 13, 15], 8)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (16, [1, 3, 5, 7, 9, 11, 13, 15], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0041/)

Euler totient function · qf-mod32-totient-value · candidate





## 42. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=17, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 16)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (17, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 16).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0042/)

Euler totient function · qf-mod32-totient-value · candidate





## 43. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=18, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(18, [1, 5, 7, 11, 13, 17], 6)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (18, [1, 5, 7, 11, 13, 17], 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0043/)

Euler totient function · qf-mod32-totient-value · candidate





## 44. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=19, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 18)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 18).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0044/)

Euler totient function · qf-mod32-totient-value · candidate





## 45. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=20, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(20, [1, 3, 7, 9, 11, 13, 17, 19], 8)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (20, [1, 3, 7, 9, 11, 13, 17, 19], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0045/)

Euler totient function · qf-mod32-totient-value · candidate





## 46. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=21, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], 12)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (21, [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], 12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0046/)

Euler totient function · qf-mod32-totient-value · candidate





## 47. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=22, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 10)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (22, [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0047/)

Euler totient function · qf-mod32-totient-value · candidate





## 48. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=23, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 22)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (23, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 22).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0048/)

Euler totient function · qf-mod32-totient-value · candidate





## 49. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=24, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(24, [1, 5, 7, 11, 13, 17, 19, 23], 8)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (24, [1, 5, 7, 11, 13, 17, 19, 23], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0049/)

Euler totient function · qf-mod32-totient-value · candidate





## 50. Euler Totient Is the Exact Count of Coprime Representatives

Positive integer n=25, using representatives 1 through n. Give (n,coprime_residues,phi_n).

View answer and explanation(25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], 20)

List all representatives with gcd(k,n)=1 and count them. The exact count is for this n, not a proof of a general totient formula from sampled values. The exact result, retaining the requested order, is (25, [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], 20).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0050/)

Euler totient function · qf-mod32-totient-value · candidate





## 51. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=1, 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.

View answer and explanation(1, 1, phi_1_equals_1)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (1, 1, phi_1_equals_1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0051/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 52. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=2, 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.

View answer and explanation(2, 1, prime_phi_equals_n_minus_1)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (2, 1, prime_phi_equals_n_minus_1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0052/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 53. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=3, 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.

View answer and explanation(3, 2, prime_phi_equals_n_minus_1)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (3, 2, prime_phi_equals_n_minus_1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0053/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 54. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=4, 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.

View answer and explanation(4, 2, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (4, 2, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0054/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 55. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=5, 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.

View answer and explanation(5, 4, prime_phi_equals_n_minus_1)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (5, 4, prime_phi_equals_n_minus_1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0055/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 56. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=6, 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.

View answer and explanation(6, 2, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (6, 2, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0056/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 57. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=7, 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.

View answer and explanation(7, 6, prime_phi_equals_n_minus_1)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (7, 6, prime_phi_equals_n_minus_1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0057/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 58. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=8, 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.

View answer and explanation(8, 4, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (8, 4, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0058/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 59. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=9, 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.

View answer and explanation(9, 6, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (9, 6, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0059/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 60. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=10, 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.

View answer and explanation(10, 4, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (10, 4, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0060/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 61. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=11, 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.

View answer and explanation(11, 10, prime_phi_equals_n_minus_1)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (11, 10, prime_phi_equals_n_minus_1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0061/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 62. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=12, 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.

View answer and explanation(12, 4, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (12, 4, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0062/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 63. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=13, 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.

View answer and explanation(13, 12, prime_phi_equals_n_minus_1)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (13, 12, prime_phi_equals_n_minus_1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0063/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 64. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=14, 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.

View answer and explanation(14, 6, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (14, 6, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0064/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 65. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=15, 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.

View answer and explanation(15, 8, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (15, 8, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0065/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 66. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=16, 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.

View answer and explanation(16, 8, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (16, 8, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0066/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 67. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=17, 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.

View answer and explanation(17, 16, prime_phi_equals_n_minus_1)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (17, 16, prime_phi_equals_n_minus_1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0067/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 68. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=18, 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.

View answer and explanation(18, 6, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (18, 6, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0068/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 69. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=19, 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.

View answer and explanation(19, 18, prime_phi_equals_n_minus_1)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (19, 18, prime_phi_equals_n_minus_1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0069/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 70. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=20, 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.

View answer and explanation(20, 8, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (20, 8, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0070/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 71. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=21, 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.

View answer and explanation(21, 12, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (21, 12, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0071/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 72. Keep phi(1), the Prime Case and the Composite Boundary Distinct

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.

View answer and explanation(22, 10, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (22, 10, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0072/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 73. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=23, 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.

View answer and explanation(23, 22, prime_phi_equals_n_minus_1)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (23, 22, prime_phi_equals_n_minus_1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0073/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 74. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=24, 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.

View answer and explanation(24, 8, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (24, 8, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0074/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 75. Keep phi(1), the Prime Case and the Composite Boundary Distinct

Positive integer n=25, 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.

View answer and explanation(25, 20, composite_no_prime_shortcut)

Handle n=1 separately, then determine whether n is prime. The prime shortcut phi(p)=p−1 is not a rule for arbitrary composites. The exact result, retaining the requested order, is (25, 20, composite_no_prime_shortcut).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0075/)

Euler totient function · qf-mod32-totient-boundary-prime · candidate





## 76. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=3 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).

View answer and explanation(2, 3, 1, 2, 2, 2, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 3, 1, 2, 2, 2, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0076/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 77. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=5 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).

View answer and explanation(2, 5, 1, 4, 4, 4, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 5, 1, 4, 4, 4, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0077/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 78. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=7 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).

View answer and explanation(2, 7, 1, 6, 6, 6, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 7, 1, 6, 6, 6, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0078/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 79. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=9 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).

View answer and explanation(2, 9, 1, 6, 6, 6, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 9, 1, 6, 6, 6, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0079/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 80. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=11 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).

View answer and explanation(2, 11, 1, 10, 10, 10, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 11, 1, 10, 10, 10, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0080/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 81. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=13 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).

View answer and explanation(2, 13, 1, 12, 12, 12, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 13, 1, 12, 12, 12, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0081/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 82. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=15 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).

View answer and explanation(2, 15, 1, 8, 8, 8, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 15, 1, 8, 8, 8, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0082/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 83. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=17 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).

View answer and explanation(2, 17, 1, 16, 16, 16, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 17, 1, 16, 16, 16, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0083/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 84. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=19 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).

View answer and explanation(2, 19, 1, 18, 18, 18, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 19, 1, 18, 18, 18, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0084/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 85. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=21 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).

View answer and explanation(2, 21, 1, 12, 12, 12, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 21, 1, 12, 12, 12, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0085/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 86. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=23 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).

View answer and explanation(2, 23, 1, 22, 22, 22, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 23, 1, 22, 22, 22, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0086/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 87. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=25 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).

View answer and explanation(2, 25, 1, 20, 20, 20, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 25, 1, 20, 20, 20, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0087/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 88. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=27 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).

View answer and explanation(2, 27, 1, 18, 18, 18, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 27, 1, 18, 18, 18, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0088/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 89. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=29 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).

View answer and explanation(2, 29, 1, 28, 28, 28, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 29, 1, 28, 28, 28, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0089/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 90. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=31 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).

View answer and explanation(2, 31, 1, 30, 30, 30, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 31, 1, 30, 30, 30, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0090/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 91. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=33 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).

View answer and explanation(2, 33, 1, 20, 20, 20, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 33, 1, 20, 20, 20, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0091/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 92. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=35 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).

View answer and explanation(2, 35, 1, 24, 24, 24, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 35, 1, 24, 24, 24, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0092/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 93. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=37 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).

View answer and explanation(2, 37, 1, 36, 36, 36, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 37, 1, 36, 36, 36, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0093/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 94. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=39 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).

View answer and explanation(2, 39, 1, 24, 24, 24, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 39, 1, 24, 24, 24, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0094/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 95. Totient Multiplicativity Requires the Coprimality Premise

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).

View answer and explanation(2, 41, 1, 40, 40, 40, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 41, 1, 40, 40, 40, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0095/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 96. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=43 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).

View answer and explanation(2, 43, 1, 42, 42, 42, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 43, 1, 42, 42, 42, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0096/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 97. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=45 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).

View answer and explanation(2, 45, 1, 24, 24, 24, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 45, 1, 24, 24, 24, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0097/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 98. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=47 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).

View answer and explanation(2, 47, 1, 46, 46, 46, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 47, 1, 46, 46, 46, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0098/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 99. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=2, n=49 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).

View answer and explanation(2, 49, 1, 42, 42, 42, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 49, 1, 42, 42, 42, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0099/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate





## 100. Totient Multiplicativity Requires the Coprimality Premise

Positive integers m=3, n=4 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).

View answer and explanation(3, 4, 2, 2, 4, 4, Yes, instance_only_with_coprimality)

Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (3, 4, 2, 2, 4, 4, Yes, instance_only_with_coprimality).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r06-euler-totient-function-0100/)

Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r06-euler-totient-function-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261101-r06-euler-totient-function-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r06-euler-totient-function-foundations/validation-report.json)