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.

From what you wrote, $a=akm$, so $a(1-km)=0$. Also $b=bmk$, so $b(1-km)=0$. Thus either $a=b=0$ (and hence $|a|=|b|$), or $mk=1$. The units in $\mathbb Z$ are of course only $+1$ and $-1$.
