# Qualified Units, First Positive Returns and Phi Divisibility

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

batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations · 2026-10-15 · 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. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=3 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (3, 2, 2, 1, 2, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0001/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 2. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=4 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (4, 3, 3, 1, 2, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0002/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 3. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=5 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (5, 2, 2, 1, 4, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0003/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 4. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=6 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (6, 5, 5, 1, 2, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0004/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 5. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=7 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (7, 2, 2, 1, 6, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0005/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 6. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=8 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

View answer and explanation(8, 3, 3, 1, 4, Yes, unit_mod_n_qualified)

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (8, 3, 3, 1, 4, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0006/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 7. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=9 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (9, 2, 2, 1, 6, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0007/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 8. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=10 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

View answer and explanation(10, 3, 3, 1, 4, Yes, unit_mod_n_qualified)

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (10, 3, 3, 1, 4, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0008/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 9. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=11 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (11, 2, 2, 1, 10, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0009/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 10. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=12 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

View answer and explanation(12, 5, 5, 1, 4, Yes, unit_mod_n_qualified)

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (12, 5, 5, 1, 4, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0010/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 11. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=13 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (13, 2, 2, 1, 12, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0011/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 12. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=14 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

View answer and explanation(14, 3, 3, 1, 6, Yes, unit_mod_n_qualified)

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (14, 3, 3, 1, 6, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0012/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 13. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=15 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (15, 2, 2, 1, 8, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0013/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 14. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=16 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

View answer and explanation(16, 3, 3, 1, 8, Yes, unit_mod_n_qualified)

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (16, 3, 3, 1, 8, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0014/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 15. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=17 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (17, 2, 2, 1, 16, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0015/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 16. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=18 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

View answer and explanation(18, 5, 5, 1, 6, Yes, unit_mod_n_qualified)

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (18, 5, 5, 1, 6, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0016/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 17. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=19 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (19, 2, 2, 1, 18, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0017/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 18. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=20 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

View answer and explanation(20, 3, 3, 1, 8, Yes, unit_mod_n_qualified)

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (20, 3, 3, 1, 8, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0018/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 19. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=21 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (21, 2, 2, 1, 12, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0019/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 20. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=22 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

View answer and explanation(22, 3, 3, 1, 10, Yes, unit_mod_n_qualified)

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (22, 3, 3, 1, 10, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0020/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 21. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=23 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (23, 2, 2, 1, 22, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0021/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 22. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=24 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

View answer and explanation(24, 5, 5, 1, 8, Yes, unit_mod_n_qualified)

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (24, 5, 5, 1, 8, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0022/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 23. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=25 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (25, 2, 2, 1, 20, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0023/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 24. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=26 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

View answer and explanation(26, 3, 3, 1, 12, Yes, unit_mod_n_qualified)

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (26, 3, 3, 1, 12, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0024/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 25. Normalize and Qualify a Unit before Defining Multiplicative Order

Use exact modulus n=27 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (n, a, normalized_residue, gcd, phi, unit_qualified, certificate).

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

Retain gcd and phi(n) from the complete unit residue carrier. Euler theorem supplies the return bound for a unit, not a claim that its order always equals phi(n) or that it is a primitive root. The exact result, retaining the requested order, is (27, 2, 2, 1, 18, Yes, unit_mod_n_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0025/)

Multiplicative order modulo n · qf-mod47-morder-qualification · candidate





## 26. The First Positive Power Returning to One Defines the Order

Use exact modulus n=3 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0026/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 27. The First Positive Power Returning to One Defines the Order

Use exact modulus n=4 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0027/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 28. The First Positive Power Returning to One Defines the Order

Use exact modulus n=5 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 3], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0028/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 29. The First Positive Power Returning to One Defines the Order

Use exact modulus n=6 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 5], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0029/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 30. The First Positive Power Returning to One Defines the Order

Use exact modulus n=7 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 1]], 3, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0030/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 31. The First Positive Power Returning to One Defines the Order

Use exact modulus n=8 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0031/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 32. The First Positive Power Returning to One Defines the Order

Use exact modulus n=9 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 7], [5, 5], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0032/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 33. The First Positive Power Returning to One Defines the Order

Use exact modulus n=10 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

View answer and explanation([[1, 3], [2, 9], [3, 7], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 7], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0033/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 34. The First Positive Power Returning to One Defines the Order

Use exact modulus n=11 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 5], [5, 10], [6, 9], [7, 7], [8, 3], [9, 6], [10, 1]], 10, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0034/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 35. The First Positive Power Returning to One Defines the Order

Use exact modulus n=12 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 5], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0035/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 36. The First Positive Power Returning to One Defines the Order

Use exact modulus n=13 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]], 12, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0036/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 37. The First Positive Power Returning to One Defines the Order

Use exact modulus n=14 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

View answer and explanation([[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return)

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0037/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 38. The First Positive Power Returning to One Defines the Order

Use exact modulus n=15 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

View answer and explanation([[1, 2], [2, 4], [3, 8], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0038/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 39. The First Positive Power Returning to One Defines the Order

Use exact modulus n=16 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

View answer and explanation([[1, 3], [2, 9], [3, 11], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 11], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0039/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 40. The First Positive Power Returning to One Defines the Order

Use exact modulus n=17 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

View answer and explanation([[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], 8, Yes, Yes, order_is_the_first_positive_return)

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], 8, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0040/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 41. The First Positive Power Returning to One Defines the Order

Use exact modulus n=18 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

View answer and explanation([[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return)

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0041/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 42. The First Positive Power Returning to One Defines the Order

Use exact modulus n=19 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 13], [6, 7], [7, 14], [8, 9], [9, 18], [10, 17], [11, 15], [12, 11], [13, 3], [14, 6], [15, 12], [16, 5], [17, 10], [18, 1]], 18, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0042/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 43. The First Positive Power Returning to One Defines the Order

Use exact modulus n=20 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

View answer and explanation([[1, 3], [2, 9], [3, 7], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return)

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 7], [4, 1]], 4, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0043/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 44. The First Positive Power Returning to One Defines the Order

Use exact modulus n=21 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

View answer and explanation([[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return)

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], 6, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0044/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 45. The First Positive Power Returning to One Defines the Order

Use exact modulus n=22 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

View answer and explanation([[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], 5, Yes, Yes, order_is_the_first_positive_return)

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], 5, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0045/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 46. The First Positive Power Returning to One Defines the Order

Use exact modulus n=23 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 9], [6, 18], [7, 13], [8, 3], [9, 6], [10, 12], [11, 1]], 11, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0046/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 47. The First Positive Power Returning to One Defines the Order

Use exact modulus n=24 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 5], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0047/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 48. The First Positive Power Returning to One Defines the Order

Use exact modulus n=25 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]], 20, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0048/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 49. The First Positive Power Returning to One Defines the Order

Use exact modulus n=26 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

View answer and explanation([[1, 3], [2, 9], [3, 1]], 3, Yes, Yes, order_is_the_first_positive_return)

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 3], [2, 9], [3, 1]], 3, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0049/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 50. The First Positive Power Returning to One Defines the Order

Use exact modulus n=27 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

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

Start at exponent1, retain every power through the first return and verify that no earlier exponent returned1. Exponent0 is not the order, and a later multiple of the order is not the least exponent. The exact result, retaining the requested order, is ([[1, 2], [2, 4], [3, 8], [4, 16], [5, 5], [6, 10], [7, 20], [8, 13], [9, 26], [10, 25], [11, 23], [12, 19], [13, 11], [14, 22], [15, 17], [16, 7], [17, 14], [18, 1]], 18, Yes, Yes, order_is_the_first_positive_return).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0050/)

Multiplicative order modulo n · qf-mod47-morder-power-first-return · candidate





## 51. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=3 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 2, Yes, [[1, 2], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0051/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 52. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=4 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 2, Yes, [[1, 3], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0052/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 53. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=5 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (4, 4, Yes, [[1, 2], [2, 4], [3, 3], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0053/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 54. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=6 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 2, Yes, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0054/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 55. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=7 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (3, 6, Yes, [[1, 2], [2, 4], [3, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0055/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 56. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=8 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 4, Yes, [[1, 3], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0056/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 57. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=9 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (6, 6, Yes, [[1, 2], [2, 4], [3, 8], [4, 7], [5, 5], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0057/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 58. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=10 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(4, 4, Yes, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (4, 4, Yes, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0058/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 59. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=11 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (10, 10, Yes, [[1, 2], [2, 4], [3, 8], [4, 5], [5, 10], [6, 9], [7, 7], [8, 3], [9, 6], [10, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0059/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 60. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=12 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 4, Yes, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0060/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 61. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=13 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (12, 12, Yes, [[1, 2], [2, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0061/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 62. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=14 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(6, 6, Yes, [[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (6, 6, Yes, [[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0062/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 63. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=15 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(4, 8, Yes, [[1, 2], [2, 4], [3, 8], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (4, 8, Yes, [[1, 2], [2, 4], [3, 8], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0063/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 64. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=16 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(4, 8, Yes, [[1, 3], [2, 9], [3, 11], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (4, 8, Yes, [[1, 3], [2, 9], [3, 11], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0064/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 65. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=17 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(8, 16, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (8, 16, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0065/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 66. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=18 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(6, 6, Yes, [[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (6, 6, Yes, [[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0066/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 67. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=19 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (18, 18, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 13], [6, 7], [7, 14], [8, 9], [9, 18], [10, 17], [11, 15], [12, 11], [13, 3], [14, 6], [15, 12], [16, 5], [17, 10], [18, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0067/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 68. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=20 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(4, 8, Yes, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (4, 8, Yes, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0068/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 69. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=21 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(6, 12, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (6, 12, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0069/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 70. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=22 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(5, 10, Yes, [[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (5, 10, Yes, [[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0070/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 71. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=23 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(11, 22, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 9], [6, 18], [7, 13], [8, 3], [9, 6], [10, 12], [11, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (11, 22, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 9], [6, 18], [7, 13], [8, 3], [9, 6], [10, 12], [11, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0071/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 72. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=24 and integer a=5. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(2, 8, Yes, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (2, 8, Yes, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0072/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 73. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=25 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (20, 20, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0073/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 74. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=26 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

View answer and explanation(3, 12, Yes, [[1, 3], [2, 9], [3, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (3, 12, Yes, [[1, 3], [2, 9], [3, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0074/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 75. Order Divides Phi without Automatically Equaling Phi

Use exact modulus n=27 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (order, phi, order_divides_phi, power_rows, certificate).

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

Retain the actual least return, the complete phi count and every power through that return. Check divisibility, not equality. No primitive-root or additive-order claim follows automatically. The exact result, retaining the requested order, is (18, 18, Yes, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 5], [6, 10], [7, 20], [8, 13], [9, 26], [10, 25], [11, 23], [12, 19], [13, 11], [14, 22], [15, 17], [16, 7], [17, 14], [18, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0075/)

Multiplicative order modulo n · qf-mod47-morder-phi-divisibility · candidate





## 76. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=8 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(noncoprime, 8, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 8, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0076/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 77. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=1 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(invalid_modulus, 1, 3, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 3, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0077/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 78. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=5 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

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

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 5, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0078/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 79. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=8 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(negative_normalization, 8, -1, 7, 1, Yes, [[1, 7], [2, 1]], 2, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 8, -1, 7, 1, Yes, [[1, 7], [2, 1]], 2, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0079/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 80. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=16 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(noncoprime, 16, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 16, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0080/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 81. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=1 and integer a=7. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(invalid_modulus, 1, 7, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 7, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0081/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 82. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=9 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(order_one, 9, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 9, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0082/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 83. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=12 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(negative_normalization, 12, -1, 11, 1, Yes, [[1, 11], [2, 1]], 2, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 12, -1, 11, 1, Yes, [[1, 11], [2, 1]], 2, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0083/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 84. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=24 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(noncoprime, 24, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 24, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0084/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 85. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=1 and integer a=11. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(invalid_modulus, 1, 11, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 11, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0085/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 86. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=13 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(order_one, 13, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 13, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0086/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 87. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=16 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(negative_normalization, 16, -1, 15, 1, Yes, [[1, 15], [2, 1]], 2, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 16, -1, 15, 1, Yes, [[1, 15], [2, 1]], 2, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0087/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 88. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=32 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(noncoprime, 32, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 32, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0088/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 89. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=1 and integer a=15. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(invalid_modulus, 1, 15, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 15, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0089/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 90. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=17 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(order_one, 17, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 17, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0090/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 91. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=20 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(negative_normalization, 20, -1, 19, 1, Yes, [[1, 19], [2, 1]], 2, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 20, -1, 19, 1, Yes, [[1, 19], [2, 1]], 2, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0091/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 92. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=40 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(noncoprime, 40, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 40, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0092/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 93. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=1 and integer a=19. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(invalid_modulus, 1, 19, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 19, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0093/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 94. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=21 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(order_one, 21, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 21, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0094/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 95. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=24 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(negative_normalization, 24, -1, 23, 1, Yes, [[1, 23], [2, 1]], 2, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 24, -1, 23, 1, Yes, [[1, 23], [2, 1]], 2, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0095/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 96. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=48 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(noncoprime, 48, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 48, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0096/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 97. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=1 and integer a=23. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=invalid_modulus. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(invalid_modulus, 1, 23, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (invalid_modulus, 1, 23, null, null, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0097/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 98. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=25 and integer a=1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(order_one, 25, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (order_one, 25, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0098/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 99. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=28 and integer a=-1. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=negative_normalization. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(negative_normalization, 28, -1, 27, 1, Yes, [[1, 27], [2, 1]], 2, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (negative_normalization, 28, -1, 27, 1, Yes, [[1, 27], [2, 1]], 2, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0099/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate





## 100. Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Use exact modulus n=56 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables.

View answer and explanation(noncoprime, 56, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)

For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 56, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0100/)

Multiplicative order modulo n · qf-mod47-morder-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations/validation-report.json)