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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?

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  • $\begingroup$ The last two sentences in the highlighted problem does not make sense. Can you reply a translation? $\endgroup$ – Alberto Takase Mar 8 at 2:35
  • $\begingroup$ I wanted to say that there is a base $b$ such that the number of elements in set Q that divide itself in base b is maximum. sorry for my poor english. $\endgroup$ – weareallin Mar 8 at 2:40
  • $\begingroup$ I see. So if zero was in the list, then that would be the number (since every number divides zero). $\endgroup$ – Alberto Takase Mar 8 at 2:41
  • $\begingroup$ Yeah, but the base should be nonzero. I should add that. $\endgroup$ – weareallin Mar 8 at 2:43

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