Associated Topics || Dr. Math Home || Search Dr. Math

### 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!
```
Associated Topics:
Middle School Division