Prove that there exists an integer n

Question

Let a,b,c,d be fixed integers with d not divisible by $5$. Assume that m is an integer for which $am^3+bm^2+cm+d$ is divisible by $5$. Prove that there exists an integer $n$ for which $dn^3+cn^2+bn+a$ is also divisible by $5$. How to solving this problem?

Answer 1

Hint: The number $m$ cannot be divisible by $5$. Let $n$ be the inverse of $m$ modulo $5$. Multiply each term of the first equation by $n^3$ (modulo $5$).