Carmichael Number/Examples/561

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:

\(\ds 3^2\)

\(\nmid\)

\(\ds 561\)

\(\ds 11^2\)

\(\nmid\)

\(\ds 561\)

\(\ds 17^2\)

\(\nmid\)

\(\ds 561\)

We also have that:

\(\ds 560\)

\(=\)

\(\ds 280 \times 2\)

\(\ds \)

\(=\)

\(\ds 56 \times 10\)

\(\ds \)

\(=\)

\(\ds 35 \times 16\)

The result follows by Korselt's Theorem.

$\blacksquare$