How to Tell If a Number is Divisible by 2 or 3
Learn how to test if a number is divisible by 2 or 3, and find out why each of these divisibility tests actually work.
There are many types of problems in life that require dividing one number by another. But sometimes, we don’t actually need to do the division—we just need to know if it’s even possible to do the division in the first place. In these situations, you can save yourself time and trouble by learning a few quick and dirty tips. So today we’re kicking things off by looking at how to quickly tell if a number is divisible by 2 or 3.
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What Does it Mean for a Number to be Divisible?
Before we get to testing for divisibility, let’s first make sure we understand what divisibility actually means. As you know, whenever you multiply two numbers together, you get another number. So, in a way, you can think of the process of multiplying numbers as a means of building new numbers. For example, in 3 x 7 = 21 the product of the numbers 3 and 7 “build” the number 21.
Whenever a number can be built from other numbers like this, it obviously must be true that we can also divide the number by one of those building numbers to get the other building number. In other words, in this case, we can divide 21 by 7 to get 3, or we can divide 21 by 3 to get 7. When a number can be divided evenly by another number like this, we say that the number is “divisible by” that number. So, in this case, 21 is evenly divisible by both 3 and 7 since 21/3 is exactly equal to the whole number 7 and 21/7 is exactly equal to the whole number 3.
How to Tell if a Number is Divisible by 2
Now that we understand what we’re trying to test for, let’s turn our attention to actually doing the testing. Up first, let’s talk about how to quickly tell whether or not a number is divisible by 2. The quick and dirty tip is that all even numbers are divisible by 2. That means that any number whose final digit is a 0, 2, 4, 6, or 8 must be divisible by 2. So now right away—without doing any division—we can tell that the number 1,018 (which is even) must be divisible by 2, while the number 1,033 (which is odd) is not.
Why does this work? Well, in this case the answer is pretty intuitive; but let’s talk about it a little to help prepare us to understand the divisibility tests for bigger integers that we’ll soon see. The important thing to notice is that when we divide an even number by 2, the remainder is always 0; and when we divide an odd number by 2, the remainder is always 1. If you think about this for a minute, you’ll see that we can use that information to conclude that a number is only divisible by 2 if it is even—since it won’t have a remainder.
For the math whizzes out there, you’ll notice that we could just as well have summed this all up using the connection between modular arithmetic and remainders that we talked about in the past few weeks to say that a number is only divisible by 2 if that number modulo 2 is congruent to 0.
How to Tell if a Number is Divisible by 3
Let’s now talk about how to test if a number is divisible by 3. The quick and dirty tip to check for divisibility by 3 is to see if the sum of all the digits in the number is divisible by 3. If so, the number itself must also be divisible by 3. For example, is 1,529 divisible by 3? Well, the sum of the digits of 1,529 is 1+5+2+9=17. Since 17 is not divisible by 3, we can conclude that 1,529 is also not divisible by 3. How about 1,530? Well, this time the sum is 1+5+3+0=9. Since 9 is divisible by 3, we know that 1,530 is divisible by 3 too.
Why Does the Divisibility by 3 Test Work?
But why does this work? Well, to find out let’s take a look at the number 111. First, since 1+1+1=3 is divisible by 3, we can immediately tell that 111 must be divisible by 3 too. To help us see why this works, let’s write 111 in a funny way (you’ll see why in a minute):
111 = 100 + 10 + 1
Now, let’s do one more thing that’s even a little weirder. Let’s use the fact that 10 = 9+1 and that 100 = 99+1 to write this sum for 111 as
111 = 99+1 + 9+1 + 1
What does this do for us? Well, believe it or not, writing 111 like this is extremely useful since it allows us to look at the problem 111/3 in a very interesting way.
In particular, if you think about it for a minute, you’ll see that any number of 9s or any number of 99s can always be evenly divided by 3. So since we know that the 9s and 99s are divisible by 3, the only question is whether the sum of everything that’s not a 9 or a 99 is divisible by 3 too. If it is, then since all the parts are divisible by 3, the whole thing must be divisible by 3! And notice that the only things in the sum 111 = 99+1 + 9+1 + 1 that aren’t 9s or 99s are the three 1s—precisely the digits of the number 111.
Try breaking apart some other numbers like this for yourself and you’ll see that they can all be written as some number of 9s, plus some number of 99s, plus some number of 999s (for larger numbers), and so on, plus some other stuff that will always be the digits of the original number. That’s precisely where the quick and dirty tip comes from!
How Modular Arithmetic Can Explain Divisibility by 3
[[AdMiddle]Once again, for the math superstars out there, notice that we could have written this whole explanation using the language of modular arithmetic that we talked about in previous articles. In particular, 111 is only divisible by 3 if 111 mod 3 is congruent to 0. And since going around a modulus 3 clock any number of 9s, 99s, 999s, and so on always leaves you exactly where you started, the only thing that matters is whether the sum of the digits in the original number also leave you where you started. If so, then the number is divisible by 3. Who knew modular arithmetic was so useful!
Okay, that’s all the time we have for today. But before we finish up, here are a few practice problems to help you test your divisibility testing skills:
Is 213 divisible by 2? ____ (Yes/No) By 3? ____ (Yes/No)
Is 2,023,182 divisible by 2? ____ (Yes/No) By 3? ____ (Yes/No)
Is 1,109 divisible by 2? ____ (Yes/No) By 3? ____ (Yes/No)
You can find the answers at the very end of this article. After checking them, feel free to leave a comment at the bottom of the page and let me know how you did.
If you have questions about how to solve these practice problems or any other math questions, please email them to me at firstname.lastname@example.org, send them via Twitter, or become a fan of the Math Dude on Facebook and get help from me and the other math fans there.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading math fans!
Practice Problem Solutions
1. Is 213 divisible by 2? No. By 3? Yes.
2. Is 2,023,182 divisible by 2? Yes. By 3? Yes.
3. Is 1,109 divisible by 2? No. By 3? No.