Try this Remainder problem from Duke Math Meet 2009 Problem 9. This problem is from the Team Round of the meet.
Problem: Duke Math Meet 2009 Problem 9
What is the remainder when is divided by 13 ?
By Fermat's Little Theorem
Now if we can find out we can find the answer.
By Euler's theorem
Finally, if we can find we are done.
Thus (as we have previously computed )
Thus . (since we have previously computed )
Thus . But . Thus answer is 5.