Cycle length of 1/p

coffeeking · 2009-09-07 17:44:00

Hi, I am currently taking a course on number theory and was doing some revision on my own. There is one part of the topic that my lecturer has only briefly touches on but has however, sparked my interests.

It's mentioned that for some recurring decimal like 1/23, 1/17 has the property that it will follow the same cycle for any integer multiple of it eg:

and

however recurring decimal like 1/13, 1/37 does not exhibit this trait. May I know what the reasoning behind this is? I really keen to know.

Thanks for your time and I will appreciate if anyone could explain this to me or at least point out on where I should research on to find out the answer.

bobbym · 2009-09-07 18:37:34

Hi coffeeking;

I think there is theorem to start you off. For any integer n, 1/n must terminate or repeat digits in (n-1) or less numbers.

I'm reading this page and it is informative.

http://en.wikipedia.org/wiki/Repeating_decimal

Using the multiplicative order you can predict the length of the cycle:

e.g.

so 1/17 has a period of 16 or

so 1/13 has a period of 6
This technique is loosely explained here:

http://mathworld.wolfram.com/MultiplicativeOrder.html

Last edited by bobbym (2009-09-07 18:56:40)

coffeeking · 2009-09-09 03:20:17

Hi bobbym thanks for replying . I guess the link that you gave above have cleared all my doubts and concepts I interested in. I appreciate your effort .

bobbym · 2009-09-09 04:02:25

Hi coffeeking;

Glad to do it. I learnt a lot there too.

Math Is Fun Forum

#1 2009-09-07 17:44:00

Cycle length of 1/p

#2 2009-09-07 18:37:34

Re: Cycle length of 1/p

#3 2009-09-09 03:20:17

Re: Cycle length of 1/p

#4 2009-09-09 04:02:25

Re: Cycle length of 1/p

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