Question about a theorem on primitive roots... Theorem 8.4 from David M. Burton, Elementry Numbrer Theory. 
Let $\gcd(a,n)=1$, and let $a_1,a_2,...,a_{\phi(n)}$ be the naturals numbers that less than $n$ and coprime to $n$. If $a$ is primitive root of $n$. then
$$a,a^2,...,a^{\phi(n)}$$
are congruent modulo $n$ to $a_1,a_2,...,a_{\phi(n)}$, in some order.
Proof:
From the fact that $\gcd(a,n)=1$ we deduce that $gcd(a^k,n)=1$. (The following statement I didn't understood) Therefore every $a^k$ is congruent modulo $n$ to $a_i$.
(The short proof continues)
Again:
 Why every $a^k$ is congruent modulo $n$ to $a_i$(from $a_1,a_2,...,a_{\phi(n)}$)?
Edit:
Solution to my problem:
"By the definition of reduced residue systems, it is sufficient to show that all these powers are coprime to $n$ and that no
two are congruent modulo $n$"
(Sad and Happy)

That part proves that $a, a^2, \ldots ,a^{\phi(n)}$ are indeed relatively prime to $n$. Since there are $\phi(n)$ of them, and since they are distinct (if not, then $a$ to some power less than $\phi(n)$ would be 1, contradicting primitivity), they must be the totatives, in some order.
