"How many incongruent solutions does $x^3 \equiv 1$ have modulo 59? What about for mod 61? Now consider, $x^3 \equiv 8 \mod 59$ and $\mod61$? How about the congruences $y^5 \equiv 1$ and $y^5 \equiv 32$ mod 59 and 61? Explain your reasoning."
First of all, I'm not 100% sure what an incongruent solution is, but I'm assuming it just means the solutions you get are unique up to congruence mod 59 or 61 depending on which case you're looking at.
Looking at $x^3 \equiv 1$ (mod 61) I think this should automatically have exactly three solutions from something we proved in class since $3\mid p-1$ which is $60$ in this case. But on the others, I'm not sure where to begin or how to work this out. Any help with how to set things up would be greatly appreciated.