Competition problem — Number theory

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Number theory Difficulty 4.9 AIME Find the answer

Determine the number of integers $2 \leq n \leq 2016$ such that $n^{n}-1$ is divisible by $2,3,5,7$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Only $n \equiv 1(\bmod 210)$ work. Proof: we require $\operatorname{gcd}(n, 210)=1$ . Note that $\forall p \leq 7$ the order of $n$ $(\bmod p)$ divides $p-1$ , hence is relatively prime to any $p \leq 7$ . So $n^{n} \equiv 1(\bmod p) \Longleftrightarrow n \equiv 1(\bmod p)$ for each of these $p$ .

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