$(1)$ Under what conditions on $a,b,c,d$ is $\left[\begin{array}{c}c\\ d\end{array}\right]$ a multiple $m$ of $\left[\begin{array}{c}a\\ b\end{array}\right]?$ Start with the two equations $c=ma$ and $d=mb.$ By eliminating $m,$ find one equation connecting $a,b,c,d.$ You can assume no zeros in these numbers.