Complete Coprime Representatives, Phi of One and Coprime Multiplicativity
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Euler totient function · qf-mod32-totient-multiplicativity-instance · candidate
Canonical question JSONL · English JSONL · Validation report