KöMaL 2009 A492

For every finite, nonempty set $H$ of positive integers, denote by $\mathrm{gcd}(H)$ the greatest common divisor of the elements in $H$. Show that if $A$ is a finite, nonempty set of positive integers then the following inequality holds: $\sum_{\emptyset\ne H\subseteq A} {(-2)}^{|H|-1} \mathrm{gcd}(H) > 0.$