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### Primes and Repeating Unit Numbers

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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:

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/
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Associated Topics:
Elementary Division
Elementary Prime Numbers
High School Number Theory
Middle School Division