Suppose that we have $\gcd(a,n) = 1$ where $a$ is primitive root of $n$ and $\{a_1, a_2 .. a_{\phi(n)} \}$ are numbers less than $n$ which are relative prime to it. Show that this set of numbers is congruent modulo $n$ to $\{a, a^2 ... a^{\phi(n)}\}$
I know that all powers upto $\phi(n)$ of primitive root are incongruent modulo $n$ to each other, but cannot prove this theorem. Hints are welcome!
Edit:
The definition of primitive root that I know is:
Primitive root is an integer $a$ so that $a^{\phi(n)} \equiv 1 (\mathrm{mod} \ n)$ and $a^{k} \not \equiv 1 (\mathrm{mod} \ n)$ for all $k < \phi(n)$