Hi I am trying to understand the proofs regarding number of primitive roots. I have tried many sources, but they all start basically the same without any justification. Maybe it is just me who is not smart enough...
The theorem goes like this: "If $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ incongruent primitive roots."
The first line of the proof is "If $a$ is a primitive root of $n$, then all primitive roots of $n$ must be of the form $a^k$ where $1 \leq k \leq \phi(n)$ according to theorem X.X", where theorem X.X is stating that the set $\lbrace a, a^2, \ldots , a^{\phi(n)} \rbrace $ is a complete set of reduced residues modulo $n$ when $a$ is a primitive root to $n$.
I do understand that $\lbrace a, a^2, \ldots , a^{\phi(n)} \rbrace $ is a complete set of reduced residues modulo $n$ when $a$ is a primitive root to $n$. But I do not understand what this has to do with the claim that any primitive root to $n$ must be of the form $a^k$ where $1 \leq k \leq \phi(n)$. I just can't see the connection, what am I missing?
Thanks in advance