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How to Tell If a Number is Divisible by 7, 8, or 9

Learn how to test if a number is divisible by 7, 8, or 9, and find out why each of these divisibility tests actually work.

By
Jason Marshall, PhD
April 8, 2011
Episode #058

divisionTwo weeks ago we looked at how to quickly test whether or not a number is divisible by 2 or 3, and last week we learned a few clever tricks that you can use to test whether or not a number is divisible by 4, 5, or 6. So what’s the logical next step for us? Well, today we’re going to finish up this series by learning how to test whether or not a number is divisible by 7, 8, or 9.

How to Tell if a Number is Divisible by 7

The quick and dirty tip to test a number for divisibility by 7 is a three steps process:

  1. Take the last digit of the number you’re testing and double it.

  2. Subtract this number from the rest of the digits in the original number.

  3. If this new number is either 0 or if it’s a number that’s divisible by 7, then you know that the original number is also divisible by 7. If you can’t easily tell yet if the new number is divisible by 7, go back to the first step with this new smaller number and try again.

For example, is the number 203 divisible by 7? Well, let’s use our three step process to find out:

  1. The last digit of 203 is 3, so double that is 3 x 2 = 6.

  2. Subtracting this new number, 6, from 20 (the remaining digits of the original number 203) gives 14.

  3. Since 14 is divisible by 7, we can immediately tell that the original number, 203, must also be divisible by 7.

Let’s try a larger number. Is 2,023 divisible by 7?

  1. The last digit of 2,023 is 3, so double that is 6.

  2. Subtracting 6 from 202 (the remaining digits from 2,023) gives us 202 – 6 = 196.

  3. Is 196 divisible by 7? I’m not sure. So let’s repeat the process using the new number 196. The final digit of 196 is 6, so twice that is 12. Subtracting this from 19 (the remaining digits of 196) leaves us with 19 – 12 = 7. Since 7 certainly is divisible by 7, we immediately know that the original number, 2,023, is divisible by 7 too!

Why Does the Divisibility by 7 Test Work?

Okay, that’s easy enough to do, but it’s definitely a little strange…how can that actually work? Well, that’s a fantastic question, but unfortunately the logic behind this trick is a bit too complex for me to explain here. So though I normally don’t do this, in this case I’m going to leave the explanation of the divisibility by 7 trick for a later date. In the meantime, if you’re interested in knowing more, you can check out the explanation found in the “Addendum” section at the bottom of this page.

How to Tell if a Number is Divisible by 8

Now that we know how to check for divisibility by all the numbers from 2 through 7, it’s time for us to check for divisibility by 8.

The quick and dirty tip for testing whether or not a number is divisible by 8 is to check to see whether the last three digits of the number are divisible by 8. If they are, then the entire number is divisible by 8 too. For example, is the number 1,523,424 divisible by 8? Well, the quick and dirty tip says that we can ignore all of the numbers here except for the last three: 424. All we have to do is figure out if this number is divisible by 8. As you can check (by long division or using a calculator), 424 / 8 = 53…so 424 is divisible by 8. Which means that the original number, 1,523,424, is divisible by 8 too!

Why Does the Divisibility by 8 Test Work?

But why does this work? How can we just ignore all those other digits? Well, since all the powers of 10 larger than 100—that is, 1,000, 10,000, 100,000, and so on—are all evenly divisible by 8 (for example, 1,000 / 8 = 125), the only thing that’s left to check is whether or not the part of the number that’s less than 1,000 is divisible by 8 too. And that’s exactly what the quick and dirty tip tells us to do.

You might be thinking that checking to see if the last three digits of a number are divisible by 8 is a lot of work…and that it’s not something you can always do in your head. You might also be wondering if there’s an easier method. But, unfortunately, the answer is “not really”…although there is a trick that can speed up the process and allow you to do the work in your head. If you’re interested, be sure to check out the “Web Article Bonus” section near the end of this article to read all about it.

How to Tell if a Number is Divisible by 9

The last topic for today is how to check if a number is divisible by 9. As it turns out, this test is very similar to the divisibility by 3 test that we talked about a few articles ago.

The quick and dirty tip for checking if a number is divisible by 9 is to add up the digits in the number and check if the resulting sum is divisible by 9. If it is, then the original number is divisible by 9 too. For example, is 1,278 divisible by 9? Well, first add up the digits of 1,278: 1+2+7+8=18. Since 18 is divisible by 9, the entire number is too!

Why Does the Divisibility by 9 Test Work?

But why does the divisibility by 9 test work? Again, the logic is very similar to the divisibility by 3 test that we talked about before. And instead of coming right out and explaining it, I’m going to let you try to work out the reasoning. So go back and look at how the explanation for the divisibility by 3 test worked, and see whether you can use that to help explain the divisibility by 9 test. If you get stuck, or if you just want to check your logic, I’ll post an explanation on the Math Dude’s Facebook page…be sure to check it out.

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 952 divisible by 7? ____ (Yes/No) By 8? ____ (Yes/No) By 9? ____ (Yes/No)

  2. Is 504 divisible by 7? ____ (Yes/No) By 8? ____ (Yes/No) By 9? ____ (Yes/No)

  3. Is 792 divisible by 7? ____ (Yes/No) By 8? ____ (Yes/No) By 9? ____ (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!

Web Article Bonus: How to Test a 3-digit Number for Divisibility by 8

As we saw earlier, the divisibility by 8 test requires you to either do long division or use a calculator to check whether or not the last three digits of the number you’re testing are themselves divisible by 8. Since that’s kind of a pain, and since it defeats the “figure out the answer in your head” goal of this series, here’s a bonus quick and dirty tip that you can use to check if a 3-digit number is divisible by 8:

  • If the first digit of the 3-digit number is even, then the entire number is divisible by 8 if the last two digits are divisible by 8.

  • If the first digit of the number is odd, then subtract the number 4 from the last two digits and check to see if this new number is divisible by 8. If it is, then the entire number is too.

For example, we can immediately tell that the number 658 is not divisible by 8. How? Well, since the first digit, 6, is even, all we have to do is check if the last two digits, 58, are divisible by 8. Since they’re not, we know that the entire number is not divisible by 8. On the other hand, the first digit of the number 344 is odd. Which means that we can test to see if 344 is divisible by 8 by subtracting 4 from its last two digits, 44 – 4 = 40, and then checking to see if this new number is divisible by 8. Since 40 is divisible by 8, we immediately know that 344 is divisible by 8 too.

Practice Problem Solutions

  1. Is 952 divisible by 7? Yes. By 8? Yes. By 9? No.

  2. Is 504 divisible by 7? Yes. By 8? Yes. By 9? Yes.

  3. Is 792 divisible by 7? No. By 8? Yes. By 9? Yes.

Division image courtesy of Shutterstock

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