Divisibility by three

How do we find out that a whole number is divisible by three? If its digit sum is divisible by three. We get the sum of the digits of 126 by adding up all the digits, all the digits that are in the number:

$$1 + 2 + 6 = 9$$

And since the number 9 is obviously divisible by three, the number 126 is divisible by three. The important thing is that we can repeat this process as many times as necessary. For example, is the number 1 962 963 divisible by three? First, we do the first digit addition:

$$1+9+6+2+9+6+3=36$$

Is the number 36 divisible by three? I guess so, but if you wanted to be sure, you could just do another digit addition:

$$3+ 6=9$$

And we're back to nine, so 36 is divisible by three, and therefore the original number 1 962 963 is divisible by three. To give you an example of a number that is not divisible by three: the number 25 has a digit sum of 7, which is not divisible by three. So the number 25 is not divisible by three either.

Interesting properties

• If you multiply any number that is not divisible by three by ten, you get a number that is not divisible by three. We have seen that the number 25 is not divisible by three. The number 250 again has a digit sum of 7 and is therefore also not divisible by three. Just like 2500 or 25,000, etc.
• If you multiply any number that is divisible by three by ten, you get a number that is divisible by three again, because the sum of the digits does not change. The number 126 is divisible by three, just like 1260 or 12,600.