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Divisibility by 7

Divisibility by 7 is surprisingly simple, although certainly not trivial.
To decide the divisibility of a 3-digit number abc, do as follows: take ab and subtract c x 2. If the result is divisible by 7, then abc is divisible by 7.
Example: 567. take 56 and subtract 7 x 2 and you get 56 - 14 = 42, and 42 is divisible by 7. Example: 871: 87 - 1 x 2 = 85. 85 is __not__ divisible by 7.
What if your number is more than 3 digits? Then you use a rule to make it 3 digits. This is best illustrated by an example.
Example: 1092. Split the number into 3-digit numbers, in this case 1 and 092. Then you subtract 092 - 1 = 91, which is divisible by 7.
Example: 2,890,034. You have three numbers this time: 2, 890, 034. You do 34 - 890 + 2 = -854. Remove the sign and continue with 854: 85 - 4 x 2 = 85 - 8 = 77, which is divisible by 7. Hence the number 2,890,034 is divisible by 7. Notice here that you do 34 - 890 + 2, __not__ the other way round 34 + 890 - 2. So you start with a subtraction, then an addition, then a subtraction again and so on, if the number you begin with is sufficiently large.

If you want to know the first real rule for divisibility by 7 of the History of Number Theory watch this video:

ReplyDeletehttp://www.youtube.com/watch?v=ZUozMuPE1RA

It works quickly with numbers of any magnitude.

Hi Silvio! Looks interesting! I will have a closer look at your video later.

ReplyDeleteHi again Silvio! I had a thorough look at your video now. Your rule works. Nice of you to tell about it. Thank you very much and please let me know if you make more numerical videos!

ReplyDeleteThank you! I created various videos about divisibility by 7, 11 and 13. To watch my last video, Google: "Talmud & Pascal" Youtube that explains how Mathematicians might have created a real rule for divisibility by 7 since the beginning of the first millennium. There is also a video that compares the procedure mentioned above and one of my rules.

ReplyDeleteHi Silvio! I found the video and will watch it tomorrow. Thanks for letting me know!

ReplyDeleteHi again Silvio! I've watched the video now, with lots of pausing to really see the slides and think about them. I've also looked at some other methods in other clips. It seems your method is very efficient.

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ReplyDeletePerhaps you would like to solve a puzzle that I created based on one of my rules. Nobody has been able to solve it until now. If you wish to try a solution, this is the link: www.youtube.com/watch?v=fNKZENi4Rjk

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