last non-zero digit of factorial

We will show how to compute the last non-zero digit of the factorial of a number from its digits without having to compute the factorial itself.

Let $L(n)$ denote the last non-zero digit of $n$ in base 10. We note some basic properties of $L$ which can easily be checked:

• For all $n$, we have $L(10n)=L(n)$.

• If $L(m)\ne 5$ and $L(n)\ne 5$, then $L(mn)=L(L(m)L(n))$.

We also tabulate the values of $L(n!)$ for small values of $n$:

$L(0!)$ $=$ $1$

$L(1!)$ $=$ $1$

$L(2!)$ $=$ $2$

$L(3!)$ $=$ $6$

$L(4!)$ $=$ $4$

$L(5!)$ $=$ $2$

$L(6!)$ $=$ $2$

$L(7!)$ $=$ $4$

$L(8!)$ $=$ $2$

$L(9!)$ $=$ $8$

$L(10!)$ $=$ $8$

Next, we make two less trivial observations:

For all positive integers $n$, we have $L(n!)\ne 5$.

Title	last non-zero digit of factorial
Canonical name	LastNonzeroDigitOfFactorial
Date of creation	2013-03-24 0:23:36
Last modified on	2013-03-24 0:23:36
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	4
Author	rspuzio (6075)
Entry type	Definition