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We all know the rules for 2,3,4,5,6,8,9 and 10. But what is less talked about is the number 7. Here is the shortcut to see whether the number is divisible by 7 or not.

Step1: Take a number and multiply each of the digit beginning on the right hand side by 1,3,2,6,4,5 (repeat the pattern if the number is large enough)

Step 2: Add the product of the numbers. If the sum is divisible by 7 -- so is the number.

e.g. Check for number 133

3(1) + 3(3) + 1(2) = 14 --> which is divisible by 7. So, 133 is divisible by 7.

Another example: check for no. 2016

6(1) + 1(3) + 0(2) + 2(6) = 21 --> which is also divisible by 7. So, the no. 2016 is divisible by 7.
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That is totally weird. I can't be bothered to check if it's true (I assume it is)... I'd love to see the proof as to why it works. Not because I'm that big of a math geek, but because I'm sure it would boggle my mind even further.

1. Lop off the last digit of the number in question. For 2016, that would be 6.

2. Double that number. Double the 6 to get 12.

3. Subtract that number from the remaining digits of the number in question, as if they were a stand-alone number. So you don't subtract 12 from 2016, or from 2010. You subtract 12 from 201 to get 189.

4. Repeat steps 1-3 until you get to a number you recognize as either a multiple of 7 or not. If you get a multiple of 7 after step 3, the original number was a multiple of 7. If you get a non-multiple of 7 after step 3, the original number was not a multiple of 7.

Take 189, lop of the 9, double to get 18, and subtract 18 from the 18 we got when we took 9 off the end of 189. Result 0 is divisible by 7, so 2016 also was.

Another example: 13,587

1. Separate: 1358 and 7 2. Double the 7: 14 3. Subtract: 1358-14 = 1344 1. Separate: 134 and 4 2. Double the 4: 8 3. Subtract: 134-8 = 126 Recognize 126 as 14 less than 140, i.e. a multiple of 7.

Conclusion: 13,587 is a multiple of 7.
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Emily Sledge | Manhattan GMAT Instructor | St. Louis

That is totally weird. I can't be bothered to check if it's true (I assume it is)... I'd love to see the proof as to why it works. Not because I'm that big of a math geek, but because I'm sure it would boggle my mind even further.

I totally agree! Both of these methods are so wacky, unlike the more intuitive tests for divisibility by other single digit numbers.

I have to wonder, wouldn't simple long division by 7 be fastest?
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Emily Sledge | Manhattan GMAT Instructor | St. Louis

I have book for my kids that gives the rules for divisibility by 7 as: Multiply the final digit by 5 and then add the answer to the number preceding it. If the answer is divisible by 7 then the whole number is divisible by 7. (the book is called Speed Math for Kids by Bill Handley)

So for your example 13587. 1358 +35=1393. 1393 is divisible by 7 (=199.) so the larger number is.

Ultimately I do not see this speeding you up at all. It is absolutely slower AND more error prone for me. I would just divide it out. Not thinking this will be very relevant on the GMAT anyway.

If you found my comments helpful, please give kudos. ( only need a few more) Thanks, Skip

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Yes, no need to learn the divisibility rule for 7 as far as the GMAT is concerned. For every single computation I have seen on the GMAT, it would be faster to just divide by 7 than resorting to these rules.

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
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