Orders of Units in Modular Arithmetic

If a mod m is a unit then aφ(m) ≡ 1 mod m by Euler’s theorem. (When m = p is prime, then this becomes ap−1 ≡ 1 mod p for a 6≡ 0 mod p, which is Fermat’s little theorem.) Depending on a, it might happen that an ≡ 1 mod m for an n that is smaller than φ(m). Example 1.1. Take m = 11. Then a10 ≡ 1 mod 11 for all a 6≡ 0 mod 11. When a = 2 no exponent smaller than 10 works, but when a = 3 we can use 5 as an exponent: 35 ≡ 1 mod 11.