Let \(n\) be an integer. Show that if \(a, b\) are relatively prime integers, each of which divides n, then \(ab\) divides n.

Since both \(a, b\) divides n, there exist integers \(u, v\) such that

\[ \begin{aligned} au = n \text{ and } bv &= n \end{aligned} \]

Given \(a, b\) are relatively prime integers, there exist integers \(s, t\) such that

\[ \begin{aligned} as + bt &= 1 \\ as \cdot n + bt \cdot n &= n \\ as \cdot bv + bt \cdot au &= n \\ ab \cdot sv + ab \cdot tu &= n \\ \end{aligned} \]

Since \(ab\) divides the left hand side, \(ab \mid n\). \(\Box\)

# posted by rot13(Unafba Pune) @ 11:11 AM