Find smallest positive integer $N$ such that $N \times r_{k}$ is a positive integer for a collection of positive rationals $\{r_{1},\ldots,r_{K}\}$ [closed]

Let $\{r_{1},\ldots,r_{K}\}$ be a collection of $K$ positive rational numbers.

How do I find the smallest positive integer $N$ such that $Nr_{k}$ is a positive integer for all $k$?

(Ideally, I am interested in a formula that is computationally cheap when $K$ is relatively large.)

• I was assuming you had the rationals as integers for numerator and denominator. If you have them as floats, how do you know they are rationals? In your example, the first two are $\frac {2431}{16000}, \frac {849}{32000}$. The LCM of the denominators is $32000$. That will make those two into integers. If your next rational were $\frac 47$ you would need to multiply by $224000$ to get all three to be integers. – Ross Millikan Jul 19 '18 at 21:32