Composite number question?

Question

If anyone could explain how the following is done, it would be very much appreciated!

A composite number is considered a Carmichael number if the congruence a^(m-1) ≡ 1 (mod m) is true for every number a with gcd(a,m)=1.

Verify that m = 561 = 3 * 11 * 17 is a Carmichael number.

Answer 1

Note that m-1 = 560 is divisible by 2, 10 and 16.
Let (a,m) = 1 and note that we have
a^(m-1) = a^(2*280) = 1^280 = 1(mod 3),
by Fermat's little theorem
Similarly,
a^(m-1) = a^(10*56) = 1^56 = 1(mod 11)
and
a^(m-1) = a^(16*35) = 1^35 = 1(mod 17)
So, by the Chinese remainder theorem,
a^(m-1) = 1(mod m)