Given an arbitrary rational number $p$, is there a way to find out the denominator of its simplest form? In other words, can we formulaically (non-algorithmically) find the lowest integer denominator of all the ways $p$ can be represented?
Suppose $p$ = $\frac{n}{d}$ is the lowest form. Then, I am looking for an $f(x)$ such that $f(p)$ would have produced $d$. As an extension, can there also be a $g$ such that $g(p)$ would have been $n$?
I played around with inverses, proportions, and integer division, but I can't seem to figure out [1] if this can be done and if so, [2] how.
Given an arbitrary rational number p
How are you "given" $\,p\,$? If it's as a ratio of two integers, then just cancel out the $\,\gcd\,$. Otherwise you'll need to elaborate some more on what the question really means to ask. $\endgroup$the fact that p∈Q
How is that "fact" established? What is $\,p\,$, and in what form do you "receive" it? $\endgroup$