Dividing and Divisibility by Seven

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There are quick and easy ways to determine whether a given number is divisible by 2, 3, 4, 5, 6, 9, and 10. 8 isn't so bad, either. I cover all of the important ones in the Number Properties section of my Total GMAT Math.

That leaves 7. All too often, that's the number the GMAT forces you to work with; or, you have to figure out whether a number like 91 is prime, which means you need to know whether it is divisible by 7.

For Seven, Ignore Divisibility Tricks

There are several tests you can use to see if a number is divisible by 7. Don't use them. Don't get me wrong: Many of them are ingenious, and if you understand them, you're several steps ahead of the average GMAT test-taker.

However, applying them takes a long time. If you needed to know whether 82,933 was divisible by 7, those methods would come in handy. But if you've spent much time with GMAT practice questions, you know that's not the kind of thing you'll have to work out on test day.

A Better Way - Not Just For Seven

For any number, you should be able to quickly figure out some multiples. For instance, if you need multiples of 12, you could come up with 36, 48, 60, 120, 144, 240, and 1,200. The more you practice, the more multiples you can generate.

For seven, there are plenty of "easy" multiples as well: 21, 28, 35, 70, 140, 350, and more.

If you need to test a number for divisibility by 7, use what I call the "nearest neighbor" method. Let's go back to 91 for a moment. To test 91 for divisibility by 7, find the nearest "easy" multiple. In this case, it may be 70. (If you've been practicing GMAT math for a long time, you should be able to come up with 84 off the top of your head. That would make it easier.)

Assuming that you know 70 is a multiple of 7, you can count by 7's to generate more multiples of 7:


There it is. Since 91 appears in that list, it is divisible by 7. Along the same lines, the list tells us that all sorts of numbers--73, 81, 86, 93, and many more--are not multiples of 7.

Not Just Divisibility - Division

Now you know how to test a number such as 91 for divisibility by 7. But let's set a harder task for ourselves. What if you need to calculate 196 divided by 7?

Again, start by finding the nearest "easy" multiple. You can probably come up with 140 and 210. 210 is higher than 196, but it's closer, so let's use that. Because it's an "easy" multiple, it's not too difficult to figure out what 210 divided by 7 is. Since 21 divided by 7 is 3, 210 divided by 7 is 30.

To find 196 divided by 7, count down from 210 by sevens, keeping track of how many sevens you're counting:

210 (30 sevens)
203 (29 sevens)
196 (28 sevens)

There's our answer. It even works for non-multiples. If we needed to find 198 divided by 7, we could find that 196 is 28 sevens, and realize that since 198 is two more, 198 divided by 7 is 28 2/7.

No matter how comfortable you feel with this method after reading about it, it's crucially important to practice regularly. You'll get more comfortable with it, and--most importantly--you'll get faster.



About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

Total GMAT Math

The comprehensive guide to the GMAT Quant section. It's "far and away the best study material available," including over 300 realistic practice questions and more than 500 exercises!
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