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.