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