Divisibility by 3

How can we tell that an integer is divisible by 3? If the sum of its digits is divisible by 3. We get the sum of the digits of 126 by adding up all the digits in the number:

$$1+2+6=9$$

Since 9 is clearly divisible by 3, 126 is divisible by 3 too. The important point is that we can repeat this procedure as many times as we need. For example, is the number 1,962,963 divisible by 3? First we find the sum of its digits:

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

Is 36 divisible by 3? Probably, but if you want to be sure, you can simply take the sum of the digits again:

$$3+6=9$$

We are back at 9, so 36 is divisible by 3, and therefore the original number 1,962,963 is divisible by 3 as well. And for an example of a number that is not divisible by 3: the number 25 has a digit sum of 7, which is not divisible by 3. So 25 is not divisible by 3 either.

Interesting properties

• If you multiply any number that is not divisible by 3 by 10, you get another number that is not divisible by 3. We have seen that 25 is not divisible by 3. The number 250 has a digit sum of 7 again, so it is not divisible by 3 either. The same goes for 2,500 or 25,000 and so on.
• If you multiply any number that is divisible by 3 by 10, you get another number that is divisible by 3, because the digit sum does not change. The number 126 is divisible by 3, and so are 1,260 and 12,600.