Proving or disproving $c-d|p(c)-p(d)$ where $p$ is a polynomial. I have $p(X)=\sum_{i=0}^{n}{a_iX^i}$, where $a_i\in\Bbb{Z}$. Let $c,d\in\Bbb{Z}$. 
Prove or disprove: $c-d|p(c)-p(d)$. 
I did some algebra but I can't think of a way to divide high power parts by $c-d$. I can't on the other hand find a counter example, and it does feel like a true statement. 
I would really appreciate your help here. 

It is true because $c-d \mid c^k-d^k, k \in \mathbb {N}$
