# 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

- 1953:
*Problems and Questions*(*Math. Mag.***Vol. 27**,*no. 1*: 53) www.jstor.org/stable/3029408

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$