Problem

Source: MOP 2006 Homework - Black Group

Tags: number theory unsolved, number theory



Let $c$ be a fixed positive integer, and let ${a_n}^{\inf}_{n=1}$ be a sequence of positive integers such that $a_n < a_{n+1} < a_n+c$ for every positive integer $n$. Let $s$ denote the infinite string of digits obtained by writing the terms in the sequence consecutively from left to right, starting from the first term. For every positive integer $k$, let $s_k$ denote the number whose decimal representation is identical to the $k$ most left digits of $s$. Prove that for every positive integer $m$ there exists a positive integer $k$ such that $s_k$ is divisible by $m$.