## A condition for divisibility by three

Posted by peeterjoot on November 26, 2012

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# Motivation

My daughter told me that any number that any number who’s digits add up to a multiple of three are divisible by three. I’d never heard of such a thing, and was suprised by it. In the spirit of a true geek dad, I had to figure out why it works.

# Guts

Let’s represent our number by a sum of digits

and the condition for digits adding up to a multiple of three (say times ) is

We can pull that out of the sum, and sure enough, the remainder (in base 10) is divisible by three

Aurora also mentioned that this works for too, which we can see by inspection. Pretty cool (just like my daughter).

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