# $$n$$ to the $$n^{th}$$

Number Theory Level 5

Let $$k=\displaystyle\sum_{n=1}^{10100}{n^n} \pmod{101}$$ where $$0\le k\le 100$$

$$k+50$$ people sit in a circle. They are numbered $$1$$ through $$k+50$$. Starting with person $$2$$, every other person leaves. So $$2,4,6$$ etc leave, and it loops back to the beginning when you get to $$k+50$$ because it's a circle. What is the number on the last person to remain after that process?

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