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Part of 2010 Federal Competition For Advanced Students, Part 1

Problems(1)

m does not divide f(n) - Austria 2010

Source:

5/13/2010
Let f(n)=∑k=02010nkf(n)=\sum_{k=0}^{2010}n^k. Show that for any integer mm satisfying 2⩽m⩽20102\leqslant m\leqslant 2010, there exists no natural number nn such that f(n)f(n) is divisible by mm.
(41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 1)
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