(related to Proposition: Divisibility Laws)

The divisibility laws follow immediately from the definition of divisors for integers:

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\("\Rightarrow"\) * By hypothesis, $m\mid n \wedge m\mid l.$ * From rule 4 above, we have that \(m\mid nx\) and \(m\mid ly\) for any integers $x,y\in\mathbb Z.$ * From rule 5, it follows that \(m\mid (nx+ly)\) for all integers $x,y\in\mathbb Z.$

\("\Leftarrow"\) * By hypothesis, \(m\mid (nx+ly)\) for all integers $x,y\in\mathbb Z.$ * It follows in particular for \(x=0,y=1\) and \(x=1,y=0\) that \(m\mid n \wedge m\mid l\).

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  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927