In base 10, you can determine the divisibility by 3 or 9 simply by adding up all the digits in the number; if the results are divisible by 3 or 9, then the numbers are divisible by 3 or 9, respectively.

What is the smallest base $n$ such that we can do the same trick for all the numbers from 2 to 6?

In other words, what is the smallest integer $n > 1$ such that for any number $x$ written in base $n$ we can determine the divisibility by all integers $m$ $(2 \leq m \leq 6),$ by adding up all the digits of $x$ and, if the result divides by $m$, concluding that $x$ is divisible by $m?$