Question 970: Order Divides Phi without Automatically Equaling Phi

Original question and givens

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).

• G001 — 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).

Definition and scope

The units split into equal-length orbits, so the order divides phi(n); neither equality nor a primitive root follows.

• mko-multiplicative-order-mod-n

Step-by-step derivation

n=22; a=3

order is defined here only for n≥2 and units modulo n

a=3

normalized residue=3; gcd(3,22)=1

gcd(residue,n)=1; units=[1, 3, 5, 7, 9, 13, 15, 17, 19, 21]

phi(22)=10

power product at k=1 := mul(1, 3)

power product at k=1 = 3

product=3

a^1 mod 22=3

power product at k=2 := mul(3, 3)

power product at k=2 = 9

product=9

a^2 mod 22=9

power product at k=3 := mul(9, 3)

power product at k=3 = 27

product=27

a^3 mod 22=5

power product at k=4 := mul(5, 3)

power product at k=4 = 15

product=15

a^4 mod 22=15

power product at k=5 := mul(15, 3)

power product at k=5 = 45

product=45

a^5 mod 22=1

all positive k<=5: [[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]]

first positive return to1 occurs at k=5

multiply unit 1 successively by 3

orbit=[1, 3, 9, 5, 15]; length=5

multiply unit 7 successively by 3

orbit=[7, 21, 19, 13, 17]; length=5

complete disjoint unit orbits=[[1, 3, 9, 5, 15], [7, 21, 19, 13, 17]]

phi=10=2·5; remainder=0

Reverse checks

• S017 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S016 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S015 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S014 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S013 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S012 · premise_and_result_audit — Recompute power product at k=5 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S010 · premise_and_result_audit — Recompute power product at k=4 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S008 · premise_and_result_audit — Recompute power product at k=3 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S006 · premise_and_result_audit — Recompute power product at k=2 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S004 · premise_and_result_audit — Recompute power product at k=1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S002 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S001 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.

Result

(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)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.