Constructing a holomorphic function with some specific points zero/nonzero

Question

Given $n \in \mathbb{Z}$, is it possible to construct a holomorphic function $f : \mathbb{C} \rightarrow \mathbb{C}$ such that $f(n) \neq 0$, but for any integer $m \neq n$ we have $f(m)=0$?

This is actually a homework problem in algebra which I reduced to this statement (in case it is correct).

Answer 1

Here's a hint: $\dfrac{\sin z}{z}$.