 Solution

Choose nonzero $g$ of least degree with $fg=0$. If $b$ is its leading coefficient, then the top coefficient equation gives $a_{n}b=0$; minimality applied after removing the top term forces $bf=0$. In $(\mathbb{Z}/m\mathbb{Z})[X]$, a nonzero annihilator $r$ therefore satisfies $r{a}_i=0$ for every $i$; a prime divisor of $m/\gcd (r,m)$ divides all $a_i$ and $m$, so the displayed gcd exceeds one.