If $k = \frac{1}{1+2x}$, where $x$ is an integer greater than $1$ and $k$ can be represented as a terminating decimal, find the sum of all possible values of $k$.

I know that there are a lot of values for k, because there is an infinite amount of terminating decimals. However, I need to find the sum of them. I tried listing most of them, but I got nowhere. Can anyone guide to me to find the solution for this problem?

  • $\begingroup$ Where did you find this question, because I think, that the sum diverges. $\endgroup$ – Jaideep Khare Nov 23 '17 at 23:31
  • $\begingroup$ Alcumus from AoPS $\endgroup$ – A Piercing Arrow Nov 23 '17 at 23:33

HINT: The decimal expansion of $$\frac{1}{n}$$ where $n$ is an integer only terminates iff all prime factors of $n$ are also prime factors of $10$ - that is, if $$n=2^a\cdot 5^b$$ for some positive integers $a,b$.

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  • $\begingroup$ I'm aware of this...How can I apply this? $\endgroup$ – A Piercing Arrow Nov 23 '17 at 23:37
  • $\begingroup$ It's nice thing I learnt today. Can you please give me a link so that I can understand the reason behind this fact? $\endgroup$ – Jaideep Khare Nov 23 '17 at 23:39
  • $\begingroup$ This implies that $1+2x$ is divisible only by the primes $2$ and $5$, and since $x$ is an integer, $1+2x$ cannot be even. Thus $1+2x$ is a power of $5$. $\endgroup$ – Franklin Pezzuti Dyer Nov 23 '17 at 23:39
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    $\begingroup$ @JaideepKhare Try this: en.wikipedia.org/wiki/Repeating_decimal $\endgroup$ – Franklin Pezzuti Dyer Nov 23 '17 at 23:40
  • $\begingroup$ Ahhh... I see...So I would carry on finding the numbers...1/5 , 1/25, 1/125, and such $\endgroup$ – A Piercing Arrow Nov 23 '17 at 23:40

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