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