Question 969: Order Divides Phi without Automatically Equaling Phi

Original question and givens

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).

• G001 — 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).

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=21; a=2

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

a=2

normalized residue=2; gcd(2,21)=1

gcd(residue,n)=1; units=[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]

phi(21)=12

power product at k=1 := mul(1, 2)

power product at k=1 = 2

product=2

a^1 mod 21=2

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

power product at k=2 = 4

product=4

a^2 mod 21=4

power product at k=3 := mul(4, 2)

power product at k=3 = 8

product=8

a^3 mod 21=8

power product at k=4 := mul(8, 2)

power product at k=4 = 16

product=16

a^4 mod 21=16

power product at k=5 := mul(16, 2)

power product at k=5 = 32

product=32

a^5 mod 21=11

power product at k=6 := mul(11, 2)

power product at k=6 = 22

product=22

a^6 mod 21=1

all positive k<=6: [[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]]

first positive return to1 occurs at k=6

multiply unit 1 successively by 2

orbit=[1, 2, 4, 8, 16, 11]; length=6

multiply unit 5 successively by 2

orbit=[5, 10, 20, 19, 17, 13]; length=6

complete disjoint unit orbits=[[1, 2, 4, 8, 16, 11], [5, 10, 20, 19, 17, 13]]

phi=12=2·6; remainder=0

Reverse checks

• S019 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S018 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• 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 — Recompute power product at k=6 with exact rational arithmetic, not rounded decimals.
• 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

(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)

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.