This is a made up problem, enjoy it!Show that if $\displaystyle t\in \mathbb{Z}^+$ there is $\displaystyle S \subseteq \mathbb{Z}^+$ ( with $\displaystyle 0<|S|<+\infty$ ) such that:

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

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