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$n$ divides $a^{n-1}+a^{n-2}+\cdots+a+1$ (Romania TST 2009)

Source: Romania TST 5 2009, Problem

May 4, 2012
modular arithmeticnumber theory proposednumber theory

Problem Statement

Let aa and nn be two integers greater than 11. Prove that if nn divides (a−1)k(a-1)^k for some integer k≥2k\geq 2, then nn also divides an−1+an−2+⋯+a+1a^{n-1}+a^{n-2}+\cdots+a+1.
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