Prove that if $x =\frac{p}{q} \in (0, 1]$ is a rational number, $q > 1$, then the period $P$ of repeating digits in the decimal representation of $x$ is in fact less than or equal to $q − 1$.
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I have been, I don't see how I can link the number of repeating digits to the denominator at all. All I can see is that p must be less than q which doesn't get me anywhere. – Nykis Sep 21 '18 at 08:36
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Do you know how to do long division? Can you use it to compute decimal expansions, e.g. of 1/7 or 1/13, without using a calculator? – Mees de Vries Sep 21 '18 at 08:37
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yes I do know how to do long division – Nykis Sep 21 '18 at 08:38
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You should try some examples, like computing 1/3, 1/7, 1/9, 1/11, 1/13... – Mees de Vries Sep 21 '18 at 08:39
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If you have any knowledge of modular arithmetic, then you can do it. But it is important that you enjoy proving it yourself.
Hint 1: (follows from the Division algorithm) when $a,b \in \mathbb N,$ dividing $a$ by $b$ always gives a remainder $r$ such that $r \in \{0,1,...,(b-1)\}.$ This set is infact known as the residue system of $b$.
Hint2: The Pigeon Hole Principle (which is pretty much common sense).
Please let me know whether you could prove it yourself.
Soham
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I've gotten as far writing down that the remainder x is either has finite decimals or recurring decimals – Nykis Sep 21 '18 at 13:05