Divisible By Three

Want to figure out if something is divisible by two? Just check to see if the last digit is even or odd. If it’s 81, it’s not divisible by two, but if it’s 82, it is.

If you want to know if a number is divisible by four, it’s not quite so simple. Still, there’s a neat trick: see if the last two digits are divisible by four. If so, you have a number that’s divisible by four, like 100 or 23,424.

I remember learning the trick of fives very early on. If a number ends in five or zero, it’s divisible by five. No wonder I learned this trick so early on! It’s really, really simple to remember, like the even number trick for multiples of two.

These are all pretty intuitive for me. They use logic at a fundamental level, and you can grasp how they work pretty quickly.

Notice how we skipped over three?

That’s because three has one weird trick you can use, but it’s not so intuitive to me. Let’s see if you feel the same way.

I probably learned this trick in elementary or middle school, and subsequently used it when forced to do long division (do kids do this anymore? let me know in the comments).

It goes like this:

Add all the digits. If that sum is divisible by 3, then the number is divisible by 3.

Way less intuitive. Why should this work?

This turns out to have everything to do with the number of fingers a typical human being has.

You’re thinking: umm, ten is definitely not divisible by three. What’s your point?

I’m saying: three goes into ten three times, with one left over. The thing that seems eerie or uncanny is that this works for a hundred or a thousand, too. Divide 100,000 by 3 and you’re going to end up with 33,333 x 3, with one left over.

Let’s see if 729 is divisible by 3. 7 + 2 + 9 = 18, and 18 is divisible by 3. So, yes, using our shortcut, we can see that 729 is indeed divisible by 3.

Can we do it without woo, though?

We can. There are seven sets of 100 (700), plus two sets of ten (20), along with a single 9 at the end.

When you divide 700 by 3, you get 233, with a remainder of 1—or 233 1/3. This is the same remainder as when you divide 7 by 3, and that’s not a coincidence. Similarly, if you divide 20 by 3, you get a remainder of 2—or 6 2/3.

Each digit, then, stands alone without those powers of ten. In the case of 700, there are going to be seven leftover (remainder) ones. With 20, you’ll have two leftover ones.

That’s why you add the digits up. You’re really multiplying the remainder for each digit by the number of hundreds or tens, but that remainder is one—so you can just add them all up.

If you’ve enjoyed this, you might also enjoy this dive I did into special numbers: