MathNet 2008-part-01 — 165 problems (EN–VI) Question 24

Let $m$ and $n$ be integers such that the polynomial $P(x) = x^3 + m x + n$ has the following property: if $x$ and $y$ are integers and $107$ divides $P(x) - P(y)$, then $107$ divides $x - y$. Prove that $107$ divides $m$.