Course: Foundations of Higher Math

This is in the chapter titled, "More on Direct Proof and Proof by Contrapositive"

Let $\displaystyle a,b\in Z$, where $\displaystyle a\neq 0$ and $\displaystyle b\neq 0$. Prove that if $\displaystyle a|b$ and $\displaystyle b|a$ , then $\displaystyle a=b$ or $\displaystyle a=-b$

Proof by contrapositive seems too difficult, so I'm trying a direct proof.

Assume that $\displaystyle a|b$ and $\displaystyle b|a$, i.e. $\displaystyle b=ax$ and $\displaystyle a=by$, for some $\displaystyle x,y\in Z$

Then,

$\displaystyle a=b \Rightarrow by=ax \Rightarrow (ax)y=(by)x \Rightarrow a(xy)=b(yx)$. Then divide both sides by (xy), so $\displaystyle a=b$

Is this sufficient?