Proof
(related to Proposition: Sign of Divisors of Integers)
- Let \(a,b\in\mathbb Z\) and let $a\mid b$, i.e. \(a\) be a divisor of \(b\),
- By definition, we have \(b=qa\) for some \(q\) and \(a\neq 0\).
- The same holds for the negative integer \(-a\neq 0\) and the absolute value for integers, we have that \(|a|\neq 0\).
- It follows
\[\begin{array}{rcl}
-b=(-q)a&\Longrightarrow&a\mid-b\\
b=(-q)(-a)&\Longrightarrow&-a\mid b\\
-b=q(-a)&\Longrightarrow&-a\mid-b\\
|b|=|q||a|&\Longrightarrow&|a|~\mid~|b|\\
\end{array}
\]
∎
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References
Bibliography
- Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
