Suppose $x,y,z$ are integers and $x \neq 0 $ if $x$ does not divide $yz$ then $x$ does not divide $y$ and $x$ does not divide $z$.
So far I have:
Suppose it is false that $x$ does not divide $y$ and $x$ does not divide $z$. Then by De Morgan's law, $x|y$ or $x|z$.
Suppose $x|y$ then $y = xk$, where $k$ is an integer.
I'm a bit unsure of where to go from here.
Best Answer
If $x\mid y$, then $y = kx$ for some integer $k$.
If $x \mid z$, then $z = jx$ for some integer $j$.
$$\implies yz = (kz)x \lor yz=(jy)x \implies x\mid yz$$ by the definition of divisibility.