# Carmichael Number/Examples/1105

## Example of Carmichael Number

$1105$ is a Carmichael number:

$\forall a \in \Z: a \perp 1105: a^{1105} \equiv a \pmod {1105}$

while $1105$ is composite.

## Proof

We have that:

$1105 = 5 \times 13 \times 17$

and so:

 $\displaystyle 5^2$ $\nmid$ $\displaystyle 1105$ $\displaystyle 13^2$ $\nmid$ $\displaystyle 1105$ $\displaystyle 17^2$ $\nmid$ $\displaystyle 1105$

We also have that:

 $\displaystyle 1104$ $=$ $\displaystyle 276 \times 4$ $\displaystyle$ $=$ $\displaystyle 92 \times 12$ $\displaystyle$ $=$ $\displaystyle 69 \times 16$

The result follows by Korselt's Theorem.

$\blacksquare$