HMMT 2015 T3

Let $z=a+b i$ be a complex number with integer real and imaginary parts $a, b \in \mathbb{Z}$, where $i=\sqrt{-1}$ (i.e. $z$ is a Gaussian integer). If $p$ is an odd prime number, show that the real part of $z^{p}-z$ is an integer divisible by $p$.