Any $\mathbb{Z}$-module $P$ is projective if it is a direct summand in some free $\mathbb{Z}$-module $M$.
Now, if $M$ is some free $\mathbb{Z}$-module, then $M \cong \bigoplus_{a \in A} \mathbb{Z}$ for some basis $A \subset M$. Now, we note that each summand has an countable infinite number of different elements, while $P$ is finite. I believe this constitutes a contradiction? I am trying to think of a way to argue it more precisely though, any suggestions? Actually, I believe the contradiction is that the direct sum (with $A$ as a summand) will have a non-zero element with finite order, while $M \cong \bigoplus_{a \in A} \mathbb{Z}$ has no non-zero element with finite order.
Any suggestions?