How to show that direct product of $\mathbb Z\times\mathbb Z\times\mathbb Z\times...$ is not projective as a $\mathbb Z$ module?

Question

How to show that direct product of $\mathbb Z\times\mathbb Z\times\mathbb Z\times...$ is not projective as a $\mathbb Z$ module?

I know that $\mathbb Z$ is a free $\mathbb Z$ module since it has $\{1\}$ as a basis.

An $R-$ module $P$ is said to projective if $P$ is a direct summand of a free $R-$ module .Also an $R-$ module is free if it is isomorphic to a direct sum of copies of the underlying ring.

However I cant use these facts to arrive at my proof.How to do it?