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:
\(\ds 3^2\) | \(\nmid\) | \(\ds 561\) | ||||||||||||
\(\ds 11^2\) | \(\nmid\) | \(\ds 561\) | ||||||||||||
\(\ds 17^2\) | \(\nmid\) | \(\ds 561\) |
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.
$\blacksquare$