The last digit of $n!$ for $n \ge 5$ is always $0$. What are the options for the last non-zero digit of $n!, n\ge 5$?

Question

I've found formulas online that use the greatest integer function, but they seem to answer my question for specific values of $n$. Is there an easier approach to find all values the last non-zero digit of a random $n$ can take? Is there another way to find these values (so not necessarily using the formulas with $\left\lfloor\cdots\right\rfloor$)?