How to find all polynomials that map integers to integers?

Also $n\mapsto \binom{n}k$ is integer valued for each $k$, and so is any integer linear combination of these: $\sum_{k=0}^m a_k\binom{n}k$. Indeed these are all the integer-valued polynomials. The key to proving this is to note that if $f$ is integer-valued, then $n\mapsto f(n+1)-f(n)$ is also integer-valued.

Hint. A polynomial $P(x)$ always takes integer value iff