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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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).
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Multiplicative order modulo n · qf-mod47-morder-boundary · candidate
Canonical question JSONL · English JSONL · Validation report