You are given $k$ positive integers $a_1, a_2, \ldots, a_k$. Prove that there exists a positive integer $N$, such that sums of digits of numbers $Na_1, Na_2, \ldots, Na_k$ differ in less than $\frac{2021}{2020}$ times. That is, if $S_{max}$ is the largest sum of digits of any of these numbers, and $S_{min}$ is the smallest, then $$\frac{S_{max}}{S_{min}} < \frac{2021}{2020}$$ Proposed by Arsenii Nikolaiev