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 \(m+n\) as:
$$\begin{align} m+n &= (p^a * \ldots) + (p^b * \ldots) \\ &= p ^{\min \{ a, b \}} * ((\ldots) + ( \ldots)) \end{align} $$
Therefore, \(p^{\min(a, b)}\) exactly divides \(m+n\).