Prove that the highest common factor of
$\displaystyle q^{n (n-1)/2}$ and $\displaystyle \prod_{1}^n (q^i -1)$ = 1, where $\displaystyle q = p^m$ for some prime p and natural number m.
If the hcf you mentioned is the same as greatest common divisor (I don't what's the difference between them, if there is!!) then you can notice that any factor of q is power of p then p divide the hcf if it isn't equal to 1. Therefore
$\displaystyle p | \prod_i(q^i-1) $
which implies, for some $\displaystyle i,\ p | (q^i-1)$ (property of prime elements) which is impossible. So the hcf is 1.