If \(a|b\) and \(a|c\), then there are integers \(d\) and \(e\), such that \(ad = b\) and \(ae = c\). That means:
\[mb + nc = m(ad) + n(ae) = a(md+ne)\]
where \(m, n \in ℤ\). In other words:
\[mb + nc = ak\]
where \(k \in ℤ\). This shows that \(a|(mb + nc)\). As a corollary, we can also show that \(a|b+c\) and \(a|b-c\).