Primes and Repeating Unit Numbers

Date: 12/09/98 at 12:52:30
From: Nichol
Subject: Prime repeating unit number

Hi. I was wondering how I would prove this statement:

For every prime number there exists a repeated unit number that is a 
multiple of that prime. 

Thank you, 
Nicho

Date: 12/10/98 at 15:04:18
From: Doctor Terrel
Subject: Re: Prime repeating unit number

Hi Nicho,

I like your problem. Because, believe it or not, today I did an 
activity with a 5th grade class that was based on your very question.

First, though you have to eliminate the primes 2 and 5 from 
consideration, right? But after that, all is okay.

As to a fancy proof, I don't know if what I can say here would be what 
you want, but it should start you on your way.

Recall that when dividing any integer M by any integer N [prime or 
not], the number of remainders obtained are limited to those values 
less than N. That is to say, for a divisor of N, the maximum number of 
remainders cannot exceed N-1. When a remainder reappears, as in the 
division of 1 by 7, you begin a cycle all over again. [This is the 
basis of repeating decimals, repetends, and all that.]

Now when you are dividing a repeating unit number, like 111..., by a 
prime, like 7, eventually one of the remainders will be paired with a 1 
that you "bring down" (in the elementary school algorithm). When the 
number/remainder is 2, we have "21", which is "7 x 3".  Hence, the 
division "comes out even," and your statement is proved for the prime 
7.  [111,111 / 7 = 15,873.]

Since all primes greater than 5 must end in 1, 3, 7, or 9, there will 
always be something which when multiplied by another integer that will 
end in a 1. This assures that if one is patient enough the division 
comes out even.

To see some more on this, I invite you to go to my website at:

   http://www.geocities.com/CapeCanaveral/Launchpad/8202/   

and then look at numbers 52 and 66.

If you need more help on this, please feel free to write back.

- Doctor Terrel, The Math Forum
  http://mathforum.org/dr.math/