Base nn 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 nn 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>1n > 1 such that for any number xx written in base nn we can determine the divisibility by all integers mm (2m6),(2 \leq m \leq 6), by adding up all the digits of xx and, if the result divides by mm, concluding that xx is divisible by m?m?


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