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Every positive integer divides some number of a sequence

Source: 1st Romanian Master in Mathematics (RMIM) 2008, Bucharest, Problem 3

February 9, 2008
floor functionmodular arithmeticfunctionnumber theorynumber theory proposed

Problem Statement

Let a>1 a>1 be a positive integer. Prove that every non-zero positive integer N N has a multiple in the sequence (an)n≥1 (a_n)_{n\ge1}, a_n\equal{}\left\lfloor\frac{a^n}n\right\rfloor.
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