Four Integers from Pair Products

$B$ divides both $AB=16$ and $BC=14$, so $B \mid \gcd(16,14)=2$. Likewise $C$ divides both $BC=14$ and $CD=63$, so $C \mid \gcd(14,63)=7$.

Hence $B\le 2$ and $C\le 7$. Since $BC=14=2\cdot 7$, both bounds must be attained: $B=2$ and $C=7$.

Then $A=16/2=8$ and $D=63/7=9$. The quadruple $(8,2,7,9)$ satisfies all three equations, and it is the only solution. Therefore