I need help getting started on a longer proof and this is the first part:
Show that if $\gcd(a, 3) = 1$, then $a^{560} = 1 \pmod 3$
Then we show the same thing with $11, 17, 561$. I have a feeling the same technique will be used to prove all of them, so if it can, all I really need to know is how to prove the first part.
The eventual goal is that we've shown $561$ is composite, but will pass the Fermat Primality Test.