MathNet 2021 — Part 10 Question 70

Let $f(X)$ be a polynomial with integer coefficients. Suppose that $p\mid f(n)$ implies $p^2\mid f(n)$ for every integer $n$ and every prime number $p$. Show that $(X-k)\mid f(X)$ implies $(X-k)^2\mid f(X)$ for every integer $k$.