Finding the Last Non-Zero Digit of a Repeated Factorial Expression

[MAXIM LI]

Homework Statement

$$(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$$?

Relevant Equations

$$(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$$?

Without using computer programs, can we find the last non-zero digit of $$(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$$?

What I know is that the last non-zero digit of $2018!$ is $4$, but I do not know what to do with that $4$.

Is it useful that $!$ occurs $1009$ times where $1009$ is half of $2018$? If that is useful, then what if $1009$ was another value, say $1234$?

Any help will be appreciated. THANKS!

 

[PeroK]

What I know is that the last non-zero digit of $2018!$ is $4$, but I do not know what to do with that $4$.

How do you calculate that?