I am completing a problem in hackerrank that counts the number of integers that can be divided by all elements of a set $A$ and divide all elements of another set $B$.
I completed the problem using a ugly brute force method then found the following solution.
From going through a few examples I can see that it works. I am not sure WHY though.
By the universal properties of $\rm gcd$ and $\rm lcm$ we have
$\qquad\begin{align} a_1,\ldots,a_j\mid n &\iff L:= {\rm lcm}(a_1,\ldots,a_j)\mid n\\[.5em] n\mid b_1,\ldots,b_k &\iff n\mid \gcd(b_1,\ldots,b_k) =: G \end{align}$
So both hold $\iff \ L\mid n\mid G\,\iff\, L\mid n,G \ $ & $\ n/L \mid G/L$
Therefore your method is correct: $ $ find all divisors $\,n/L\,$ of $\,G/L,\,$ then scale each by $L$ to get $n$.
If all elements of $A$ divide a number $c$, that means there is an integer $n$ with $c = n\cdot\mathrm{LCM}(A)$. Similarly, if $c$ divides every element of $B$, then there is an integer $m$ with $mc = \mathrm{GCD}(B)$. Thus, for each number $c$ that has the aforementioned properties, there is a unique pair of integers $(m,n)$ such that $mn\cdot\mathrm{LCM}(A)= \mathrm{GCD}(B)$, or equivalently that $mn = \mathrm{GCD}(B)/\mathrm{LCM}(A)$. Clearly there is one such pair for each divisor of $\mathrm{GCD}(B)/\mathrm{LCM}(A)$.
