6.5. Solutions Proof

Solutions to Section 3.2

Solution to Exercise (5†)

The claim is: \((r \mid s) \wedge (s \mid t) \to r \mid t\).

Proof. We need to show something is true for all integers \(r, s, t\), so take arbitrary integers \(k, m, n\) such that: \(k \mid m\), \(m \mid n\).

Now we need to prove that \(k \mid n\) holds.

Since \(k \mid m\), we know that \(m = ak\) for some integer \(a\). Similarly \(n = bm\) for some integer \(b\).

Thus \(n = bm = bak = ck\) with integer \(c= ba\).

Thus \(k \mid n\).

Since \(k,m,n\) were arbitrary, this holds for all integers.

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