# Unit Residues, Exact Totient Powers and Signed Base Normalization

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-20261107-r06-euler-theorem-number-theory-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.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=5≥2 and original integer base a=2. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

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

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (5, 2, 4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0001/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 2. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=6≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(6, 5, 2, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (6, 5, 2, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0002/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 3. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=7≥2 and original integer base a=4. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(7, 4, 6, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (7, 4, 6, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0003/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 4. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=8≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(8, 5, 4, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (8, 5, 4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0004/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 5. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=9≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(9, 7, 6, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (9, 7, 6, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0005/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 6. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=10≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(10, 7, 4, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (10, 7, 4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0006/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 7. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=11≥2 and original integer base a=8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(11, 8, 10, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (11, 8, 10, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0007/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 8. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=12≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(12, 11, 4, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (12, 11, 4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0008/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 9. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=13≥2 and original integer base a=10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(13, 10, 12, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (13, 10, 12, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0009/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 10. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=14≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(14, 11, 6, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (14, 11, 6, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0010/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 11. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=15≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(15, 13, 8, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (15, 13, 8, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0011/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 12. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=16≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(16, 13, 8, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (16, 13, 8, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0012/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 13. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=17≥2 and original integer base a=14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(17, 14, 16, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (17, 14, 16, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0013/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 14. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=18≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(18, 17, 6, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (18, 17, 6, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0014/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 15. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=19≥2 and original integer base a=16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(19, 16, 18, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (19, 16, 18, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0015/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 16. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=20≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(20, 17, 8, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (20, 17, 8, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0016/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 17. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=21≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(21, 19, 12, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (21, 19, 12, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0017/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 18. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=22≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(22, 19, 10, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (22, 19, 10, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0018/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 19. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=23≥2 and original integer base a=20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(23, 20, 22, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (23, 20, 22, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0019/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 20. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=24≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(24, 23, 8, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (24, 23, 8, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0020/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 21. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=25≥2 and original integer base a=22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(25, 22, 20, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (25, 22, 20, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0021/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 22. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=26≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(26, 23, 12, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (26, 23, 12, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0022/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 23. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=27≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(27, 25, 18, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (27, 25, 18, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0023/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 24. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=28≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(28, 25, 12, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (28, 25, 12, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0024/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 25. Euler Congruence Requires a Unit Residue, Not a Prime Modulus

Use integer modulus n=29≥2 and original integer base a=26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).

View answer and explanation(29, 26, 28, 1, Yes)

Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (29, 26, 28, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0025/)

Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate





## 26. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=5≥2 and original integer base a=2. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4], [1, 2, 3, 4], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0026/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 27. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=6≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

View answer and explanation([1, 5], [1, 5], Yes, multiplication_by_a_permutes_reduced_residues)

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5], [1, 5], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0027/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 28. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=7≥2 and original integer base a=4. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0028/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 29. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=8≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

View answer and explanation([1, 3, 5, 7], [1, 3, 5, 7], Yes, multiplication_by_a_permutes_reduced_residues)

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7], [1, 3, 5, 7], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0029/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 30. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=9≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

View answer and explanation([1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 8], Yes, multiplication_by_a_permutes_reduced_residues)

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 8], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0030/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 31. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=10≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

View answer and explanation([1, 3, 7, 9], [1, 3, 7, 9], Yes, multiplication_by_a_permutes_reduced_residues)

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 7, 9], [1, 3, 7, 9], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0031/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 32. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=11≥2 and original integer base a=8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0032/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 33. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=12≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

View answer and explanation([1, 5, 7, 11], [1, 5, 7, 11], Yes, multiplication_by_a_permutes_reduced_residues)

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5, 7, 11], [1, 5, 7, 11], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0033/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 34. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=13≥2 and original integer base a=10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0034/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 35. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=14≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

View answer and explanation([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], Yes, multiplication_by_a_permutes_reduced_residues)

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0035/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 36. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=15≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 7, 8, 11, 13, 14], [1, 2, 4, 7, 8, 11, 13, 14], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0036/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 37. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=16≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0037/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 38. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=17≥2 and original integer base a=14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0038/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 39. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=18≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

View answer and explanation([1, 5, 7, 11, 13, 17], [1, 5, 7, 11, 13, 17], Yes, multiplication_by_a_permutes_reduced_residues)

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5, 7, 11, 13, 17], [1, 5, 7, 11, 13, 17], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0039/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 40. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=19≥2 and original integer base a=16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0040/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 41. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=20≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 7, 9, 11, 13, 17, 19], [1, 3, 7, 9, 11, 13, 17, 19], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0041/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 42. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=21≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0042/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 43. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=22≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0043/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 44. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=23≥2 and original integer base a=20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0044/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 45. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=24≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5, 7, 11, 13, 17, 19, 23], [1, 5, 7, 11, 13, 17, 19, 23], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0045/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 46. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=25≥2 and original integer base a=22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

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

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0046/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 47. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=26≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], Yes, multiplication_by_a_permutes_reduced_residues)

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0047/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 48. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=27≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

View answer and explanation([1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], Yes, multiplication_by_a_permutes_reduced_residues)

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0048/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 49. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=28≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

View answer and explanation([1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], [1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], Yes, multiplication_by_a_permutes_reduced_residues)

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], [1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0049/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 50. Multiplication by a Unit Permutes the Complete Reduced Residues

Use integer modulus n=29≥2 and original integer base a=26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

View answer and explanation([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], Yes, multiplication_by_a_permutes_reduced_residues)

List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], Yes, multiplication_by_a_permutes_reduced_residues).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0050/)

Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate





## 51. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=5≥2 and original integer base a=-2. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-2, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-2, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0051/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 52. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=6≥2 and original integer base a=-5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-5, 1, 2, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-5, 1, 2, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0052/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 53. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=7≥2 and original integer base a=-4. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-4, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-4, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0053/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 54. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=8≥2 and original integer base a=-5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-5, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-5, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0054/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 55. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=9≥2 and original integer base a=-7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

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

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-7, 2, 6, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0055/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 56. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=10≥2 and original integer base a=-7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-7, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-7, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0056/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 57. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=11≥2 and original integer base a=-8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-8, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-8, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0057/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 58. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=12≥2 and original integer base a=-11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-11, 1, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-11, 1, 4, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0058/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 59. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=13≥2 and original integer base a=-10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-10, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-10, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0059/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 60. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=14≥2 and original integer base a=-11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-11, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-11, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0060/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 61. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=15≥2 and original integer base a=-13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-13, 2, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-13, 2, 8, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0061/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 62. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=16≥2 and original integer base a=-13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-13, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-13, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0062/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 63. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=17≥2 and original integer base a=-14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-14, 3, 16, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-14, 3, 16, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0063/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 64. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=18≥2 and original integer base a=-17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-17, 1, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-17, 1, 6, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0064/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 65. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=19≥2 and original integer base a=-16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-16, 3, 18, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-16, 3, 18, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0065/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 66. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=20≥2 and original integer base a=-17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-17, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-17, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0066/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 67. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=21≥2 and original integer base a=-19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-19, 2, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-19, 2, 12, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0067/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 68. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=22≥2 and original integer base a=-19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-19, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-19, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0068/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 69. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=23≥2 and original integer base a=-20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-20, 3, 22, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-20, 3, 22, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0069/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 70. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=24≥2 and original integer base a=-23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-23, 1, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-23, 1, 8, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0070/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 71. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=25≥2 and original integer base a=-22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-22, 3, 20, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-22, 3, 20, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0071/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 72. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=26≥2 and original integer base a=-23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-23, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-23, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0072/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 73. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=27≥2 and original integer base a=-25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-25, 2, 18, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-25, 2, 18, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0073/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 74. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=28≥2 and original integer base a=-25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-25, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-25, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0074/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 75. Normalize the Signed Base before Exact Modular Exponentiation

Use integer modulus n=29≥2 and original integer base a=-26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).

View answer and explanation(-26, 3, 28, 1, Yes, negative_base_normalized_before_exact_modular_power)

Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-26, 3, 28, 1, Yes, negative_base_normalized_before_exact_modular_power).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0075/)

Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate





## 76. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=6≥2 and original integer base a=6. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(6, 6, 6, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (6, 6, 6, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0076/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 77. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=7≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(7, 7, 7, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (7, 7, 7, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0077/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 78. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=8≥2 and original integer base a=8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(8, 8, 8, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (8, 8, 8, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0078/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 79. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=9≥2 and original integer base a=9. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(9, 9, 9, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (9, 9, 9, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0079/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 80. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=10≥2 and original integer base a=10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(10, 10, 10, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (10, 10, 10, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0080/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 81. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=11≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(11, 11, 11, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (11, 11, 11, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0081/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 82. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=12≥2 and original integer base a=12. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(12, 12, 12, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (12, 12, 12, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0082/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 83. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=13≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(13, 13, 13, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (13, 13, 13, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0083/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 84. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=14≥2 and original integer base a=14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(14, 14, 14, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (14, 14, 14, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0084/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 85. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=15≥2 and original integer base a=15. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(15, 15, 15, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (15, 15, 15, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0085/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 86. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=16≥2 and original integer base a=16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(16, 16, 16, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (16, 16, 16, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0086/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 87. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=17≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(17, 17, 17, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (17, 17, 17, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0087/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 88. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=18≥2 and original integer base a=18. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(18, 18, 18, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (18, 18, 18, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0088/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 89. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=19≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(19, 19, 19, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (19, 19, 19, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0089/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 90. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=20≥2 and original integer base a=20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(20, 20, 20, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (20, 20, 20, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0090/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 91. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=21≥2 and original integer base a=21. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(21, 21, 21, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (21, 21, 21, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0091/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 92. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=22≥2 and original integer base a=22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(22, 22, 22, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (22, 22, 22, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0092/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 93. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=23≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(23, 23, 23, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (23, 23, 23, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0093/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 94. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=24≥2 and original integer base a=24. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(24, 24, 24, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (24, 24, 24, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0094/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 95. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=25≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(25, 25, 25, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (25, 25, 25, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0095/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 96. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=26≥2 and original integer base a=26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(26, 26, 26, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (26, 26, 26, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0096/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 97. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=27≥2 and original integer base a=27. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(27, 27, 27, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (27, 27, 27, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0097/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 98. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=28≥2 and original integer base a=28. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(28, 28, 28, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (28, 28, 28, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0098/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 99. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=29≥2 and original integer base a=29. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(29, 29, 29, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (29, 29, 29, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0099/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate





## 100. A Nonunit Fails Euler Applicability, Not a Primality Test

Use integer modulus n=30≥2 and original integer base a=30. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).

View answer and explanation(30, 30, 30, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)

Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (30, 30, 30, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r06-euler-theorem-number-theory-0100/)

Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations/validation-report.json)