A number is a multiple of 3 if you can add up the digits and the sum is also a multiple of 3. But why does it work that way? And do similar phenomena appear in other base number systems?

Question

A number is a multiple of 3 if you can add up the digits and the sum is also a multiple of 3. But why does it work that way? And do similar phenomena appear in other base number systems?

In: 17

Answer 1

It’s also works for multiples of 9, which is one less than our base. 3 is the square root of 9, so also works.

This generalizes to every base n, for n-1, and also any ~~root~~ factor of (n-1). In base-6, the trick works for mutiples of 5. In base-17, it would work for multiples of 16, but also multiples of 2, 4 or 8.