IMO 1983 - Shortlisted problems Question 24

(USS 2) Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n\ge n_0$, $d_{n+T}=d_n$.