Carmichael Number/Examples/561

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Example of Carmichael Number

$561$ is a Carmichael number:

$\forall a \in \Z: a \perp 561: a^{561} \equiv a \pmod {561}$

while $561$ is composite.


Proof

We have that:

$561 = 3 \times 11 \times 17$

and so:

\(\displaystyle 3^2\) \(\nmid\) \(\displaystyle 561\)
\(\displaystyle 11^2\) \(\nmid\) \(\displaystyle 561\)
\(\displaystyle 17^2\) \(\nmid\) \(\displaystyle 561\)


We also have that:

\(\displaystyle 560\) \(=\) \(\displaystyle 280 \times 2\)
\(\displaystyle \) \(=\) \(\displaystyle 56 \times 10\)
\(\displaystyle \) \(=\) \(\displaystyle 35 \times 16\)

The result follows by Korselt's Theorem.

$\blacksquare$