2. Find all integers $m, n$ such that $m n \mid \left(3^{m}+1\right), m n \mid \left(3^{n}+1\right)$.
(2005, Korean Mathematical Olympiad)
First prove that at least one of $m$ and $n$ is odd, then prove that at least one of them is 1, and find that the only $(m, n)$ satisfying the condition are $(1,1),(1,2),(2,1)$.