# Prove hcf(a,b) =1 where a,b are as follows...

$q^{n (n-1)/2}$ and $\prod_{1}^n (q^i -1)$ = 1, where $q = p^m$ for some prime p and natural number m.
$p | \prod_i(q^i-1)$
which implies, for some $i,\ p | (q^i-1)$ (property of prime elements) which is impossible. So the hcf is 1.