Carmichael Number/Examples/1105

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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.

$\blacksquare$