Problem

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

Tags: floor function, modular arithmetic, function, number theory, number theory proposed



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