What is the significance of the repeating pattern in a circle of numbers?

[soeren]

hi,

i've got a question (don't know, if this is the right forum for this.. but i think, it isn't really a brain-teaser)

My biology teacher told me this and said, that he doesn't know anything about the reason for them.
It isn't a homework or something like that, but I and my friends are very interested in solving the "problem".. or even being able to explain it to oneself.

I've got the Numbers written in this circle:

$$\begin{matrix}& 7 & 1 & \\ 5 & & & 4 \\ & 8 & 2 & \end{matrix}$$


When i read it to the right, I get for example the number 285714

When i multiplicate this with 2, i get ..
The amazing thing of them is, that 285714 is readable in that circle, too.

That works with the factor 3; 4; 5, 6; too.

142857 * 2 = 285714
142857 * 3 = 428571
142857 * 4 = 571428
142857 * 5 = 714285
...

I tried to explain this to myself, but i didn't really understand it.

Can one of you please tell me why this works with that numbers?
Is contingency or _must_ it be like that?

thnx for help!

greets

Soeren

ps: sorry for my bad english, but this is not my mother-language...

 

[robert Ihnot]

hi,

142857 * 2 = 285714
142857 * 3 = 428571
142857 * 4 = 571428
142857 * 5 = 714285
QUOTE]

what you seem to be talking about are cyclic numbers. In the case above 1/7 repeats in blocks of six digits, which same digits are also found in 2/7, 3/7...6/7.

This will occur when the reciprocial of a prime has a repeating block of p-1 digits. The next cases are: 17, 19, 23.

There is an article on this: http://mathworld.wolfram.com/CyclicNumber.html

 

[soeren]

very thanks! That explains it very good. :-)

greets

soeren