Using Fermat's little theorem to find remainders.

Question

I don't understand how to use Fermat's little theorem to find remainders e.g if we are asked to find remainder of 50^50 on division by 13, what is a and what is p in the formula? What is going on?? Are we using congruence classes when using Fermat's little theorem? Can someone please give me a step by step explanation from the beginning, I'm just not understanding this stuff.

Answer 1

$50^{50} = (50^{12})^4\cdot 50^2$. Now apply Fermat little theorem: $50^{12} \equiv 1 \pmod {13}$. So $(50^{12})^4 \equiv 1^4 \equiv 1 \pmod {13}$, and $50^2 \equiv 4 \pmod {13}$. So: $50^{50} \equiv 4 \pmod {13}$