Assume $d|n$ means that integer $d$ divides integer $n$. Define $\chi^k$ to be the $k$-th element in a set. Consider the set $\mathcal{Q} =\{1, 2, 3, \dots, 2019\}$. For a base $b$ such that $1<b<2019$, the amount of elements $\chi^k$ in the set $\mathcal{Q}$ such that $\chi^k | {\chi^k}_b$, is maximum. If $d_b$ means $d$ in base $b$, find $b$.
This is a problem I made.
I started by thinking about how we can turn something in base $10$ into base $b$. We have $b^{e_1} + b^{e_2} + b^{e_3} +\cdots +b^{e_n} = \chi$, where $e_n$ is an exponent. We know that $b>10$, as it is necessary. I have no idea where to go from here, as there is nothing I can do. Someone help me out?