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.

Since

Thus (as we have previously computed )

Thus . (since we have previously computed )

Thus . But . Thus answer is 5.

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