New Rule for Divisibility by 7

Date: 11/01/97 at 18:15:06
From: Anonymous
Subject: A new seven's divisibility rule

Hi, Dr. Math -

Now since you're a big "math expert," you should know the 7's
divisibility rule. You know, double the one's digit and subtract that 
from the first integer excluding the one's digit. Well I have another 
7's rule which has never been created before and I have made up ALL BY 
MYSELF. You can look in any book, but you won't find it because I made 
it up. 

Here it is: Multiply the integer excluding the one's digit by 3, and 
then add the one's digit. If the resulting integer is divisible by 7, 
the first integer is divisible by 7, and the resulting integer will 
be either positive or negative depending on the first integer - i.e., 
07 = 0 x 3, 0 + 7 = 7  and -07 = 0 x 3, 0 + -7 = -7. The other 
divisibility rule doesn't have that happen in some cases- i.e., 
7 = 7-14 = -7.

Last year in school, I presented it to a classroom of students and 
they all tried to prove it wrong. They failed. So officially, the new 
divisibility rule was named The Peterson Rule, but it was only 
official in our school. I was hoping that you'd have connections to 
professors, so that's why I'm e-mailing you. I'd appreciate it if you 
could pass the word on so it could be in books one day. This would 
really, really, really mean a lot to me. Thanks.

Oh, and by the way, I'm 12 years old, my name's Torin Peterson, and 
it's not a joke.

Date: 11/05/97 at 11:15:46
From: Doctor Terrel
Subject: Re: A new seven's divisibility rule

Dear Torin,

Of course, it's not a joke. I believe you, really. You've just done a 
very good job of what so few students do these days: THINK for 
themselves. I congratulate you. You remind me of myself when I was 
younger. One time I did some independent thinking and found some nice 
results. But then later I saw them in a book; my ideas were really 
old. However, and this the most important thing, I did it by myself 
first. It's not important that other people had found it out before 
me, but that I did my own thinking first.

So as to your "new" rule, well, I knew it too. Some years ago I was 
playing around with divisibility rules for 7 and other numbers (like 
13, 17, 19, primes mostly). I discovered that each had a "subtract" 
rule, but they also had an "add" rule. The subtract rules seemed to be 
the more famous and are in the books, but the add ones work just as 
well. Maybe you should write your own math book - really, I'm not 
joking - and include your new rule in it. In fact, because you've 
written it to Dr. Math, it's now in our archives.  So you've taken the 
first step.

By the way, can you find rules for 13, 17, 19, etc., both subtract and 
add styles?  That would be a good project for you and impress all your 
friends. 

Write again to report your discoveries.  Good luck, friend.

-Doctor Terrel,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 11/10/97 at 20:35:28
From: Anonymous
Subject: Re: A new seven's divisibility rule

Hi, Dr. Terrel, it's Torin again. 

Well, I'm working on the 13, 17, and 19 rules now, and I've found the 
addition rule for 19. The rule is: 

  1. Make the ones digit an independent integer and the rest of the 
     digits an indepentent integer, i.e., 234 > 23,4
  2. Double the old ones digit. 
  3. Add that to the other integer.
  4. If it is divisible by 19, the starting integer is divisible by 
     19. You can keep going if you can't tell.
  5. Example: 570 - 1. 57,0	 2. 0(2) = 0  3. 57+0=57  4. It is, and 
     57 = 5 + 14 = 19

I hope  I'll be getting more in the near future, but I just wanted to
report in. Thanks for knowing I wasn't kidding and giving me new 
challenges to work on. I appreciate it. See ya!