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?