Fermat's little theorem This is a very interesting word problem that I came across in an old textbook of mine. So I mused over this problem for a while and tried to look at the different ways to approach it but unfortunately I was confused by the problem and I don't really understand how to do it either, hence I am unable to show my own working or opinion. The textbook gave a hint about using Fermat's little theorem but I don't really understand it and I'm really not sure about how to approach it. Any guidance hints or help would be truly greatly appreciated. Thanks in advance :) So anyway, here the problem goes: (It is composed of three parts)

$a)$ Determine the remainder when $2^{2017}+1 $ is divided by $17$.
$b)$ Prove that $30^{99} + 61^{100}$ is divisible by $31$.
$c)$ It is known that numbers $p$ and $8p^2 + 1 $ are primes. Find $p$.


Hints:
a) $2016$ is divisible by $16$. Now use Fermat's Theorem. 
b) $30\equiv -1\pmod{31}$ and $61\equiv -1\pmod{31}$. 
c) If the prime $p$ is not equal to $3$ then $p^2\equiv 1\pmod{3}$.
