What is the last non - zero digit in (20!)!? (That is, factorial of 20 factorial).

In a way that everyone can understand:

10! = ...800
Since the last digit is all that matters, 20! would essentially be multiplying the last non-zero digit in 10! by itself. In the same way, 30! would cube the last nonzero digit, and 40! raise it to the fourth power.

8, (the last nonzero digit in 10!) when raised to a power, the last digit goes in a pattern.
8^1=8
8^2=...4
8^3=...2
8^4=...6
8^5=...8
And it repeats endlessly in a pattern every fourth power (making x!'s last nonzero digit repeat every 40).

So, 20! is a really high number and it has 4 zeroes at the end because of its 4 factors of 5. To take (20!)!, we have to know the mod 40 of 20!. Mod 40 of 20! is 0 because it is a multiple of 10000. Therefore, the last nonzero digit, according to the pattern shown, is a 6. Note that for (x!)!, the answer is always six when x is a whole number 15 or greater.

Posted by Tristan on 2003-12-22 01:51:56