Question 14
Let $p$ be a prime number and let $a$, $b$ and $c$ be integers divisible by $p$, such that the polynomial $$ q(x) = x^{3} + a x^{2} + b x + c $$ has at least two different integer roots. Prove that $p^{2}$ divides $b$ and $p^{3}$ divides $c$.