Problem

Source: Iran MO 3rd round 2019 mid-terms - Number theory P1

Tags: number theory



Given a number $k\in \mathbb{N}$. $\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{1,2,\cdots,9\}$. For all $n\geq 0$ $$\left.\overline{a_{n}\cdots a_{1}a_{0}}+k \ \middle| \ \overline{b_{n}\cdots b_{1}b_{0}}+k \right. .$$Prove that there is a number $1\leq t \leq 9$ and $N\in \mathbb{N}$ such that $b_n=ta_n$ for all $n\geq N$. (Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$)