I am looking for clever applications of Euler's Theorem. On browsing the internet, I see that nearly all the applications of the theorem asks for finding last few digits of a huge number. The only other application I could find was Euler's Proof of Fermat's theorem on sums of two squares. which uses Fermat's little theorem ( A weaker form of Euler's totient Theorem) and infinite descent.
This makes me wonder if we could find more uses for the theorem?
PS: I am interested in undergraduate level theorems/problems which could be solved using Euler's theorem (along with other technique, if necessary) Please mention suitable hints along with your problem.