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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?

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    Hint: IF $x$ is an integer and $x\neq 0$ then $|x|\geq 1$.2012-10-19

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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.

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    @Draconar: Absolutely. In this case, though, I can just finish the argument, and you can compare that with what you have. Multiplying $|n|\ge 1$ by $|a|$, we get $|b|=|n||a|\ge 1\cdot|a|=|a|$, which is exactly what we wanted to prove.2012-10-19