If $x$ divides $x-z$, then $x$ divides $z$ 
For any integer x and z , if $x|(x-z)$ then $x|z$

My attempt: suppose $x|(x-z),$ let $y= x-z$
$x|y $  means there is any integer r such that $y=r*x$
So $ x-z=rx $, which equals $(x-z)/(x) =r $  this is where I get stuck. 

You con write:
$$\frac {x-z}{x}=k$$ with $k$ integer. This expression si equivalent: $$\frac {x}{x}-\frac {z}{x}=k$$ but you can continue  $$1-\frac {z}{x}=k$$ $k$ is integer therefore $x|z$
