Numbers of Zeroes that Factorial does not end with

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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:

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

This sequence is A000966 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof


Sources