# Wilson's Theorem/Examples/10 does not divide (n-1)!+1

## Example of Use of Wilson's Theorem

For all $n \in \Z_{>0}$, $10$ is not a divisor of $\paren {n - 1}! + 1$.

## Proof

For the first few $n$ we see:

 $\, \ds 10 \nmid \,$ $\ds 2$ $=$ $\ds \paren {1 - 1}! + 1$ where $\nmid$ denotes non-divisibility $\ds$ $=$ $\ds \paren {2 - 1}! + 1$ $\, \ds 10 \nmid \,$ $\ds 3$ $=$ $\ds \paren {3 - 1}! + 1$ $\, \ds 10 \nmid \,$ $\ds 7$ $=$ $\ds \paren {4 - 1}! + 1$ $\, \ds 10 \nmid \,$ $\ds 25$ $=$ $\ds \paren {5 - 1}! + 1$

Now consider $n > 5$.

We have that:

 $\ds 2$ $\divides$ $\ds \paren {n - 1}!$ where $\divides$ denotes divisibility $\ds 5$ $\divides$ $\ds \paren {n - 1}!$ $\ds \leadsto \ \$ $\ds 10$ $\divides$ $\ds \paren {n - 1}!$ as $10 = \lcm \set {2, 5}$

Hence $10 \nmid \paren {n - 1}! + 1$.

$\blacksquare$