Theorem: If $w,x,y,$ and $z$ are integers where $w$ divides $x$ and $y$ divides $z,$ then $wy$ divides $xz.$ For each "proof" of the theorem, explain where the proof uses invalid reasoning or skips essential steps. $($a$)$ Proof. Let $w,x,y,z$ be integers such that $w$ divides $x$ and $y$ divides $z.$ Since, by assumption, $w$ divides $x,$ then $x=kw$ for