order of an element

In a group (a special set with an operation on it like addition or multiplication), elements have orders.  Usually, on these pages, the group is the set of non-zero remainders modulo a prime and the order of a modulo p then is the least positive integer n such that aⁿ ≡ 1 (mod p).

For example, let us use a=3 and p=7.  Look at the powers of 3 modulo 7:

3¹ ≡ 3, 3² ≡ 2, 3³ ≡ 6, 3⁴ ≡ 4, 3⁵ ≡ 5, 3⁶ ≡ 1

The order of 3 modulo 7 is 6.  The order of 2 modulo 7 is 3.  The order of 6 modulo 7 is 2.

When working modulo a prime, the set of non-zero remainders form a multiplicative group.  This is not true modulo a composite.  Fermat's Theorem tells us the order of a non-zero element modulo a prime divides the prime minus one.  Euler's theorem gives us a similar result for composites.