Sum of Terminating Decimals

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?

• Where did you find this question, because I think, that the sum diverges. Nov 23, 2017 at 23:31
• Alcumus from AoPS Nov 23, 2017 at 23:33

1 Answer

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 $a,b$.

• I'm aware of this...How can I apply this? Nov 23, 2017 at 23:37
• It's nice thing I learnt today. Can you please give me a link so that I can understand the reason behind this fact? Nov 23, 2017 at 23:39
• 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 $5$. Nov 23, 2017 at 23:39
• @JaideepKhare Try this: en.wikipedia.org/wiki/Repeating_decimal Nov 23, 2017 at 23:40
• Ahhh... I see...So I would carry on finding the numbers...1/5 , 1/25, 1/125, and such Nov 23, 2017 at 23:40