Geometric Mean problem

This is a made up problem, enjoy it!

Show that if $t\in \mathbb{Z}^+$ there is $S \subseteq \mathbb{Z}^+$ ( with $0<|S|<+\infty$ ) such that:

$\left(\displaystyle\prod_{x\in S}{\tfrac{x}{d(x)}}\right)^{\tfrac{1}{|S|}} = t$

Here $d(n)$ is the number of positive integers dividing $n$ .