Question about the Chinese Remainder Theorem and the residue class ring ${\bf Z}/p\# {\bf Z}$

Question

In a question that I asked on MO, Terence Tao observed that:

The Chinese Remainder Theorem tells us that the residue class ring ${\bf Z}/p\# {\bf Z}$ is isomorphic (as a ring) to the product of of the finite fields ${\bf Z}/q {\bf Z}$ where $q$ ranges over the primes up to $p$.

Can someone explain how the Chinese Reminder Theorem can be used to show this isomorphism?