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prove number divides sum of a^a

Source: Mongolian MO 2007 Teachers P6

April 8, 2021
number theory

Problem Statement

Let n=p1α1⋯psαs≥2n=p_1^{\alpha_1}\cdots p_s^{\alpha_s}\ge2. If for any α∈N\alpha\in\mathbb N, pi−1∤αp_i-1\nmid\alpha, where i=1,2,…,si=1,2,\ldots,s, prove that n∣∑α∈Zn∗ααn\mid\sum_{\alpha\in\mathbb Z^*_n}\alpha^{\alpha} where Zn∗={a∈Zn:gcd⁡(a,n)=1}\mathbb Z^*_n=\{a\in\mathbb Z_n:\gcd(a,n)=1\}.
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