Why does $a^b \bmod c=(a \bmod c)^b \bmod c$ This may be a very basic number theory question, but I don't understand:
Why does $$a^b \bmod c = (a \bmod c)^b \bmod c$$

If you are comfortable with $x\equiv y\bmod n\Rightarrow xz\equiv yz \;(\bmod n)$, then 
$a\equiv (a \bmod c)\; (\bmod c)$
$a^b\equiv (a\bmod c)^b\; (\bmod c)$
$a^b \,\bmod c=(a\bmod c)^b\bmod c$.
