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Divisibility Rules

Date: 29 Apr 1995 15:59:03 -0400
From: Dick Dovenberg
Subject: Rule of 7

Dear Doctor Math:

We read your rule of divisibility for seven and were quite impressed.  
We would like to know the rules for the other numbers.  Obviously, we 
understand the divisibility by 2, but we would like the other rules too.

Thank you.
Northeast Iowa teachers

Date: 2 May 1995 15:49:31 -0400
From: Dr. Ethan
Subject: Re: Rule of 7

Well, here are the ones that I can think of right now.  You may already know
them.  I will try to come up with some more later.

For three you can add up the digits and if that sum is divisible by three, 
then the number is as well.  Like 111111: the digits add to 6 so the whole 
number is divisible by three.  Another one is 87687687.  These digits
add up to 57, and 5 plus seven is,12 so the original number is divisible by

For four, you can just look at the last two digits.  100 is divisible by 4. 
1732782989264864826421834612 is divisible by four also, because 12 
is divisible by four.

For five, just look at the last digits.  If the last one is a five or a 
zero then you can divide.

For six, just check three and two.

For eight, just check the last three digits.

For nine, add the digits.  If they are divisible by nine then you are cool.
In fact this works for any power of three.

Ten, you probably know.

And we just got a really cool one.  My buddy Ken and I are goofing off while
we should be studying and we just got one for eleven.

Let's look at 352, which is divisible by 11, and the answer is 32.  Well, 3+2
is 5; pretty neat, eh.  Or another way to say this is that 35 -2 is 33.

Now look at 3531 this is also divisible by 11 and it is not a coincidence
that 353-1 is 352 and 11 * 321 is 3531.

Well, here is the generalization of that system.  I think that if
you look at how you multiply with eleven you will see how it works. I will
not justify it here; if you want to know why this works you can write back
and I'll prove it.

Lets look at the number 


first we want to find if it is divisible by 11, but on the way we are going to
save the numbers that we use.

In every step we will subtract the last digit from the other digits, then
save the subtracted amount in order.

        Start with      9418656  -5 = 9418651  SAVE 5
        Then            941865   -1 = 941864   SAVE 1
        Then            94186    -4 = 94182    SAVE 4
        Then            9418     -2 = 9416     SAVE 2
        Then            941      -6 = 935      SAVE 6
        Then            93       -5 = 88       SAVE 5
        Then            8        -8 = 0        SAVE 8

And now write the numbers we saved in reverse order and we have

8562415,  which times 11 is 9418656.

For more information on divisibility rules, including reasons why they work, see   

Dr. Ethan, The Geometry Forum
Associated Topics:
Middle School Division

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