Orders in Modular Arithmetic

Example 1.1. Let m = 7. The following table shows that the first time a nonzero number mod 7 has a power congruent to 1 varies. While Fermat’s little theorem tells us that a6 ≡ 1 mod 7 for a 6≡ 0 mod 7, we see in the table that the exponent 6 can be replaced by a smaller positive exponent for 1, 2, 4, and 6. k 1 2 3 4 5 6 1k mod 7 1 2k mod 7 2 4 1 3k mod 7 3 2 6 4 5 1 4k mod 7 4 2 1 5k mod 7 5 4 6 2 3 1 6k mod 7 6 1 Example 1.2. Since φ(15) = 8, if (a, 15) = 1 then a8 ≡ 1 mod 15. But in fact, as the table below shows, the 8th power is always higher than necessary: the first, second, or fourth power of each number relatively prime to 15 is congruent to 1 mod 15. k 1 2 3 4 1k mod 15 1 2k mod 15 2 4 8 1 4k mod 15 4 1 7k mod 15 7 4 13 1 8k mod 15 8 4 2 1 11k mod 15 11 1 13k mod 15 13 4 7 1 14k mod 15 14 1 Definition 1.3. If (a,m) = 1 then the order of a mod m is the least n ≥ 1 such that an ≡ 1 mod m.