Let \(a\) and \(b\) be positive integers. According to the fundamental theorem of arithmetic:
$$ \begin{align} a &= {p_1}^{x_1} * {p_2}^{x_2} * {p_3}^{x_3}* \cdots * {p_n}^{x_n} \\ b &= {p_1}^{y_1} * {p_2}^{y_2} * {p_3}^{y_3} * \cdots * {p_n}^{y_n} \\ [a,b] &= {p_1}^{\max(x_1,y_1)} * {p_2}^{\max(x_2,y_2)} * {p_3}^{\max(x_3,y_3)}* \cdots *{p_n}^{\max(x_n,y_n)} \end{align} $$
Let's group the prime factors into two groups:
$$ [a,b] = (m)(n) $$
\(m\) contains all the \(p_i\)'s where \(x_i \ge y_i\) and \(n\) contains all the \(p_i\)'s where \(y_i \gt x_i\). This means \(m|a\) and \(n|b\). Also, since \(m\) and \(n\) don't have any common primes, then \((m,n)=1\).