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Shortcut: Divisible by 7 ?

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Shortcut: Divisible by 7 ? [#permalink] New post 21 May 2010, 09:17
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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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Re: Shortcut: Divisible by 7 ? [#permalink] New post 21 May 2010, 10:16
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. :-D
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Re: Shortcut: Divisible by 7 ? [#permalink] New post 21 May 2010, 10:24
Here's another one:

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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Re: Shortcut: Divisible by 7 ? [#permalink] New post 21 May 2010, 10:32
dalmba wrote:
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. :-D

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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Re: Shortcut: Divisible by 7 ? [#permalink] New post 21 May 2010, 10:43
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.

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Re: Shortcut: Divisible by 7 ? [#permalink] New post 27 May 2010, 07:00
Its absolutely not required to remember this formula. Just divide as you do normal division. It will be much faster.
Re: Shortcut: Divisible by 7 ?   [#permalink] 27 May 2010, 07:00
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