Math Help - primitive root and quadratic residue

1. primitive root and quadratic residue

Prove that a primitive root r modulo n cannot be a quadratic residue.

I've no idea how to prove it, pls help, thx very much!!!

2. I think the proof would be a lot easier if you $n$ was an odd prime. Anyway, if $r$ is a primitive root then,
$\{r,r^2,...,r^{n-1} \}$
Contains all the elements of $n$
But since $r$ is a quadradic residue.
That means $r^k$ is a quadradic residue.
Then all numbers $1\leq j\leq n$ are quadradic residues of $n$, which is surly not possible unless $n=1,2$.

Note: By odd primes the proof is extremely simple if $r$ is a quadradic residue then,
$r^{\frac{p-1}{2}}\equiv 1(\mbox{ mod }p)$
But that is not possible because the order of $r$ is $p-1$.