Prove if $a\mid b$ and $b\mid a$, then $|a|=|b|$ , $a, b$ are integers. Form the assumption, we can say $b=ak$ ,$k$ integer, $a=bm$, $m$ integer.
 Intuitively, this conjecture makes sense. But I can't make further step.

Just continue your idea! $a=bm$, and $b=ak$ implies $a=akm$ and thus $1=km$. Which can only hold for $k=m=\pm1$.
Upd: of course, under $a,b\ne 0$, which comes from the setting.
