how to prove if $a|b$ and $b\neq 0$, then $|a|\leq|b|$

Question

where the conditions are: $a \neq 0$, $b \neq 0$ and $a$ and $b$ are integers.

maybe i'm missing something very basic about the properties of an absolute values.

My approach was to supposed, on the contrary, that |b| >= |a|, but I'm always getting that |b| is indeed >= |a|

could someone help me?

Answer 1

HINT: If $a\mid b$, then there is an integer $n$ such that $b=na$, and therefore $|b|=|n||a|$. Since $b\ne 0$, we know that $n\ne 0$, and therefore $|n|$ is a positive integer. Therefore $|n|\ge 1$. If you now multiply this inequality by the right thing, you’ll get the desired result.