Is there a branch of mathematics that studies the factors of rational numbers? I am imagining that defining this would work pretty much the same way as defining the factors x of an integer n:
$\{x \mid n\mod x\ = 0\}$
but maybe as something more like
$\{x^{-1} \mid n\mod x\ = 0\}$
giving the factors of $n^{-1}$.
Just like every integer $n\in \mathbb{Z}$ has a unique representation as a product of primes (up to multiplication by $-1$), you could say the same for every rational number, accept now we allow negative powers in the "product of primes" representation.
It seems you may be asking if we can extend the theory of divisibility from integers to rationals. One general way to do this is as follows. Consider the natural extension of the divisibility relation from integers to rationals: 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.\: $ [divisibility relations induced by subrings are discussed further here]
Then we can obtain lcm and gcd of rationals by scaling the gcd arguments by a factor that yields a known gcd (of integers), then performing the inverse scaling back to rationals.
Even in more general number systems (integral domains), where gcds need not always exist, this scaling method still works to compute gcds from the value of a known scaled gcd, namely
$\rm{\bf Lemma}\ \ \ gcd(a,b)\ =\ gcd(ac,bc)/c\ \ \ if \ \ \ gcd(ac,bc)\ $ exists $\rm\quad$
Therefore $\rm\ \ gcd(a,b)\, c = gcd(ac,bc) \ \ \ \ \ if\ \ \ \ gcd(ac,bc)\ $ exists $\quad$ (GCD distributive law)
The reverse direction fails, i.e. $\rm\:gcd(a,b)\:$ exists does not generally imply that $\rm\:gcd(ac,bc)\:$ exists. $\ $ For a counterexample see my post here, which includes further discussion and references.
Generally, as proved here, we have these dual formulas for $\rm\color{#c00}{reduced}$ fractions
$$\rm\ gcd\left(\frac{a}b,\frac{c}d\right) = \frac{gcd(a,c)}{lcm(b,d)}\ \ \ if\ \ \ \color{#c00}{\gcd(a,b) = 1 = \gcd(c,d)}$$
$$\rm\ lcm\left(\frac{a}b,\frac{c}d\right) = \frac{lcm(a,c)}{gcd(b,d)}\ \ \ if\ \ \ \color{#c00}{\gcd(a,b) = 1 = \gcd(c,d)}$$
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. 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.
gcds and lcms of rationals (fractions) are disscussed in a few prior posts here, e.g. see my post here, which gives a direct proof of the gcd * lcm formula using universal gcd laws.
If $a = \prod_{p \text{ prime}} p^{n_p}$ and $b = \prod_{p \text{ prime}} p^{m_p}$ then $a/b =\prod_{p \text{ prime}} p^{n_p -m_p}$.
