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How to Tell If a Number is Divisible by 4, 5, or 6?

Learn how to test if a number is divisible by 4, 5, or 6, and find out why each of these divisibility tests actually work.

By
Jason Marshall, PhD,
March 31, 2011
Episode #057

How to Tell If a Number is Divisible by 4, 5, or 6?

In the last article, we learned two quick and dirty tips that you can use to quickly test whether or not a number is divisible by 2 or 3. Why would you need to do that?

Well, there are many reasons—but the most obvious is that there’s no point in you spending a bunch of time trying to divide a number into 2 or 3 even pieces if that isn’t actually possible! So, with that in mind, today we’re going to continue on from where we left things last time and learn three more tips that you can use to test if numbers are divisible by 4, 5, and 6.

How to Tell if a Number is Divisible by 4

The quick and dirty tip to test whether or not a number is divisible by 4 is to check to see if the number that’s made from the final two digits of the original number is itself divisible by 4. If it is, then the entire number is divisible by 4 too.

For example, is 19,233 divisible by 4? Well, all we have to do is check whether or not the number 33 (which is the number made from the last two digits of 19,233) is evenly divisible by 4. Since 4x8=32 and 4x9=36, we see that 33 is not divisible by 4. Therefore, 19,233 is also not divisible by 4.

On the other hand, the number that’s one less than 19,233—namely 19,232—is indeed divisible by 4 since the number made from its final two digits, 32, can be divided by 4.

But why does this work? How is it possible that we only need to worry about the final two digits of a number when testing whether or not it’s divisible by 4? Well, the fact that a number is divisible by 4 simply means that the remainder when it’s divided by 4 is zero. If you think about it, you’ll see that dividing the numbers 100, 1,000, 10,000, and any other higher power of 10 by 4 always gives a remainder of zero. Which means, of course, that all of these numbers are divisible by 4. And not only that, but it also means that any number of 100s, 1,000s, 10,000s, and so on is divisible by 4. So the only thing left to check when testing for divisibility by 4 is the part of the number that’s less than 100—in other words, the final two digits. If they’re divisible by 4, then the whole thing must be divisible by 4 too!

How to Tell if a Number is Divisible by 5

You’ll be happy to know that checking for divisibility by 5 is really easy. The quick and dirty tip is that for a number to be divisible by 5, it must end with either a 0 or a 5. For example, the numbers 5, 10, 15, 20, and so on up to 1,005, 1,010, and on and on forever, are all divisible by 5 since they all end in either a 0 or 5. On the other hand, the numbers 7 and 3,111,428 are not divisible by 5 since they do not end in either a 0 or 5.

Why does this work? Well, the reason is as easy as the tip: It’s simply that no matter what you multiply the number 5 by, the result is always a number that ends in 0 or 5. That’s all there is to it!

How to Tell if a Number is Divisible by 6

Okay, it’s now time for us to talk about our last divisibility test for today. The quick and dirty tip for testing whether or not a number is divisible by 6 is to check if it’s divisible by both 2 and 3. If it is, then it’s also divisible by 6. As you’ll recall, in the last article we found that:

  • A number is divisible by 2 only if it’s even.

  • A number is divisible by 3 only if its digits add up to a number that’s divisible by 3.

So, for a number to be divisible by 6, it must satisfy two conditions:

  1. It must be even.

  2. The sum of its digits must be divisible by 3.

For example: Is the number 202 divisible by 6? Well, it’s even—so we’re okay there. But its digits add up to 4…and 4 is not divisible by 3. So we’ve found that 202 is not divisible by 6. How about 402? Again, the number is even. And this time, the digits add up to 6—which is indeed divisible by 3. So 402 must be divisible by 6. In fact, as you can check, 402 / 6 = 67.

Why does this test work? Well, if you think about it, you’ll see that if a number is divisible by both 3 and 2, then it must also be divisible by 3 x 2. In other words, it must also be divisible by 6!

Practice Problems

Okay, that’s all the math we have time for today. But before we finish up, here are a few practice problems for you test your divisibility testing skills on:

  1. Is 212 divisible by 4? ____ (Yes/No) By 5? ____ (Yes/No) By 6? ____ (Yes/No)

  2. Is 4,125 divisible by 4? ____ (Yes/No) By 5? ____ (Yes/No) By 6? ____ (Yes/No)

  3. Is 1,128 divisible by 4? ____ (Yes/No) By 5? ____ (Yes/No) By 6? ____ (Yes/No)

You can find the answers at the very end of the article. After checking them, feel free to leave a comment at the bottom of the page and let me know how you did.

Wrap Up

If you have questions about how to solve these practice problems or any other math questions, please email them to me at mathdude@quickanddirtytips.com, 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 212 divisible by 4? Yes. By 5? No. By 6? No.

  2. Is 4,125 divisible by 4? No. By 5? Yes. By 6? No.

  3. Is 1,128 divisible by 4? Yes. By 5? No. By 6? Yes.

Boy Calculating image from Shutterstock

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