1
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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
2
$\begingroup$

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 $a,b$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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
  • 1
    $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.