Define as the set of integers mod n: which is endowed with the operations and . It appears that this ring with identity is a field if and only if is prime. The key is whether or not elements of have inverses (they don't when is composite).

So I managed to prove that: if is not prime, then has no inverse mod n. Now I am having trouble going the opposite way: how do I show that if is prime, i.e. - , then for all , has an inverse mod n? Thanks!