Let there be integers \(m\) and \(n\) such that \((m, n)=1\). According to the fundamental theorem of arithmetic, both \(m\) and \(n\) can be written as a product of primes:
\(p_i\) is a prime factor of \(m\) and \(q_j\) is a prime factor of \(n\). Since \((m,n)=1\), then \(p_i \ne q_j\) for all \(i\) and \(j\). For example:
Let \(d\) be a divisor of \(mn\). This means all prime factors of \(d\) are either prime factors of \(m\) or prime factors of \(n\). This means \(d = p_1^{a_1} p_2^{a_2} \ldots p_k^{a_k} q_1^{b_1} q_2^{b_2} \ldots q_h^{b_h}\) where \(a_i \le m_i\) and \(b_i \le n_i\). This means we can represent \(d\) as \((p_1^{a_1} p_2^{a_2}\ldots) (q_1^{b_1} q_3^{b_3}\ldots)=d_1d_2\), where \(d_1\) has all the \(p_i\) factors and \(d_2\) has all the \(q_j\) factors:
Since \(d_1\) has \(p_i\) factors, then \(d_1|m\), and since \(d_2\) has \(q_i\) factors, then \(d_2|n\). Also, since \(p_i \ne q_j\) for all \(i\) and \(j\), then \(d_1\) and \(d_2\) have no common prime factors. Therefore \((d_1, d_2)=1\).
For example, let \(m=4116\) and let \(n=196625\). The prime factors are:
Let \(d=2*5^2*7^2*11=26950\). We know \(d|mn\) as all it's prime factors exists in either \(m\) or \(n\). This means we can do this:
Since \(d_1\) has factors of \(m\), then \(d_1|m\), and since \(d_2\) has factors of \(n\), then \(d_2|n\). Also, since \(d_1\) and \(d_2\) have no common prime factors, then \((d_1, d_2)=1\).