I studying for a discrete mathematics exam and am stuck on this question:
Find the value of the unique integer $x$ satisfying $0 \le x < 17 $ for which: $$ 4^{1024000000002} ≡ x \pmod{17} $$
I have been reading up on how to solve similar problems but none that look similar to this. Can anyone help? Thank you very much.
Hint: Note that $4^2 = -1 \mod 17$, so
$$4^{1024000000002} = (-1)^{1024000000002/2} \mod 17$$
This is because
$$4^{1024000000002} = (4^2)^{1024000000002/2} = 16^{1024000000002/2} = (-1)^{1024000000002 / 2} \mod 17$$