Proving that the additive integer is prime...

I've run into this question and I feel like im on the brim of solving it but i cant come up with final undisputable proof:

"Prove that if p is a prime number and a is an integer such that p does not divide a, then the additive order of a modulo p is equal to p."

I know that it involves proving 2 cases, one where the additive order is greater or equal to p, and one where it is less than or equal to p. Both of these can be done by contradiction.

I have only written simple proofs before and I'm wondering if someone could show me how this problem is done.

Thank you.