Four positive integers from three pairwise products

Positive integers $A,B,C,D$ satisfy $AB=16$, $BC=14$ and $CD=63$. Find $A+B+C+D$.

$26$. $B$ divides both $16$ and $14$, so $B\in\{1,2\}$; $B=1$ would make $C=14$, which does not divide $63$. So $B=2$, $A=8$, $C=7$, $D=9$, summing to $26$.