# Numbers of Zeroes that Factorial does not end with

## Theorem

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $n!$ denote the factorial of $n$.

Let $n!$ be expressed in decimal notation.

Then $n!$ cannot end in the following numbers of zeroes:

$5, 11, 17, 23, 29, 30, 36, 42, \ldots$