# Base $$n$$ divisibility

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 $$m$$?

×