Carmichael numbers (statement)

Definition. A composite $n$ such that $a^{n-1}\equiv1\pmod n$ for all $a$ coprime to $n$. (Fermat's little theorem fails to detect compositeness of Carmichael numbers.)

561 = 3·11·17. For $a$ coprime to $561$:
- mod $3$: $a^2\equiv1$, so $a^{560}=(a^2)^{280}\equiv1$.
- mod $11$: $a^{10}\equiv1$, so $a^{560}=(a^{10})^{56}\equiv1$.
- mod $17$: $a^{16}\equiv1$, so $a^{560}=(a^{16})^{35}\equiv1$.

CRT. Since $3,11,17$ are pairwise coprime and each divides $a^{560}-1$, their product $561$ divides $a^{560}-1$. So $a^{560}\equiv1\pmod{561}$.

Conclusion. $561$ is a Carmichael number.