According to the fundamental theorem of arithmetic, \(m\) can be represented as:
$$ m = {p_1}^{k_1} * {p_2}^{k_2} * \cdots $$
But we are only interested in the \(p\) in question:
$$ m = p^a * \ldots $$
As for \(n\):
$$ n = p^b * \ldots $$
We can represent \(mn\) as:
$$\begin{align} mn &= (p^a * \ldots) * (p^b * \ldots) \\ &= p^a * p^b * (\ldots) \\ &= p^{a+b} * (\ldots) \end{align} $$
Therefore, \(p^{(a+b)}\) exactly divides \(mn\).