# Divisible By Three

This is probably something I learned in grade school, but must have forgot along the way. Math is all about tricks or rules of thumb such as this.

As John von Neumann once said

…in mathematics you don’t understand things. You just get used to them.

Here is a quick way to determine if a number is divisible by 3.

Take any integer, say $$12345$$. Sum the individual numbers. If that sum is divisible by 3, then the original chosen number is divisible by 3.

So, in this case, $$12345$$ is divisible by 3. Let us check that in our calculator

The question is why does this work? We can decompose an integer like so

equivalently

multiply that out

for the whole term to be divisible by 3, each term has to be evenly divisible by 3. Each term that is 9, 99, 999 etc. is obviously divisible by 3. Therefore we are left with

which must be divisible by 3, to imply the original number is divisible by 3. The cool thing it works at the second level as well

And $$6$$ is obviously divisible by 3. Voila!