TITLE: Divisibility, LCM and GCD for fractions (rationals)
QUESTION [7 upvotes]: how do i find lcm of two fractions?  For example:
$\frac{2}{3}$ and $\frac{5}{8}$

REPLY [9 votes]: A notion of gcd and lcm for rationals arises naturally by extending the divisibility relation from integers to rationals, i.e. for rationals $\rm\:r,s,\:$ we define $\rm\:r\:$ divides $\rm\:s,\:$  if $\rm\ s/r\:$ is an integer, $ $ in symbols $\rm\:r\:|\:s\:$ $\!\iff\!$ $\rm\:s/r\in\mathbb Z\: $  [such divisibility relations induced by subrings are discussed further here].
Then, as proved here, we have these dual formulas for $\,\rm\color{#c00}{reduced}\,$ fractions $\rm \,a/b,\ c/d$
$$\rm\ gcd\left(\frac{a}b,\frac{c}d\right) = \frac{gcd(a,c)}{lcm(b,d)}\ \ \ if\ \ \ \color{#c00}{\gcd(a,b) = 1 = \gcd(c,d)}$$
$$\rm\ lcm\left(\frac{a}b,\frac{c}d\right) = \frac{lcm(a,c)}{gcd(b,d)}\ \ \ if\ \ \ \color{#c00}{\gcd(a,b) = 1 = \gcd(c,d)}$$
Some of these ideas date to Euclid, who computed the greatest common measure of line segments, by anthyphairesis (continually subtract the smaller from the larger), i.e. the subtractive form of the Euclidean algorithm. Euclid's method's become clearer when one learns about fractional ideals. The above methods work much more generally since they do not require the existence of a Euclidean (division) algorithm but, rather, only the existence of (certain) gcds.