TITLE: How to solve the congruence $x^{59} \equiv 604 \pmod{2013}$?
QUESTION [5 upvotes]: $$x^{59} \equiv 604 \pmod{2013}$$
Could somebody give me any clue?  I have no idea how to start.
I see that $59$ is prime.

REPLY [3 votes]: As $(604,2013)=1,x$ must be co-prime to $2013$
As $2013=3 \cdot 11 \cdot 61,$  using Carmichael Function, 
$$x^{60}\equiv1\pmod{2013}$$
$$x^{59}\equiv604\pmod{2013}\implies x^{60}\equiv604x$$
So, we have $$604x\equiv1\pmod{2013}$$
Can you find $x$ using Euclid's Algorithm?