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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

Page 1 of 4

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.

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