Why does $561$ divides $3^{561}-3$ if it doesn't divide $3$ nor $3^{560} -1$ $561$ is a Carmicheal number.
I was asked in an exercise to prove that it is so by proving that  $561 | a^{561}-a$ for any integer a.
Now, if $561 | 3^{561}-3$    then $561|3*(3^{560}-1)$
But since $\gcd(561,3) > 1$  then it is false that $561 | 3^{560}-1$ (Isn't it?)
Then $561 | 3$ which is false.
Then by counterexample it would be false that $561 | a^{561}-a$ for any integer a.
What I am missing in the argument?
Thanks.

You have $561 = 3 \cdot 187$, so $561$ divides $3x$ iff $187$ divides $x$. You don't need another factor of 3.
