I want to show that the elements of $\displaystyle \mathbb{Z}_n$ under multiplication that have inverses are also the generators of $\displaystyle \mathbb{Z}_n$ under addition.

My answer so far: The only integers with multiplicative inverses are -1 and 1; and these are generators of ($\displaystyle \mathbb{Z}_n, +_{n}$).

I feel like this answer is too simple though... Can I show this more rigorously? Besides saying that (obviously) the gcd(1,n)=1, and $\displaystyle \phi(x)$=(1)x, $\displaystyle \phi(x)$=(-1)x, generates $\displaystyle \mathbb{Z}_n$? Or am I interpreting this question completely wrong?