Zn[X] and primality

Mathsgirl

1. The problem statement, all variables and given/known data

Hi,

I can't figure out this problem:
Let a in Zn be a unit. Show that (X+a)n - (Xn + a) = 0 in Zn[X] if and only if n is prime.

Thanks for any help :)

2. Relevant equations

Binomial expansion

3. The attempt at a solution

There is a hint which says to show that if n is not prime then n doesn't divide nCp if p is a prime divisor of n. I'm not sure how to show this or how to use it in the main proof.

If I assume the LHS is true I can expand it using the binomial expansion but I don't seem to get anything useful. I can factorise out an a.