Three. (20 points) There are $m$ regular $n$-sided polygons, and the sum of the interior angles of these $m$ regular polygons can be divided by 8. Find the minimum value of $m+n$.


Three, from the problem, we know that the total sum of the interior angles of these $m$ regular polygons is $m(n-2) \times 180$.
From $81[180 m(n-2)]$
$\Rightarrow 21 \mathrm{~m}(n-2) \Rightarrow 21 \mathrm{mn}$.
Thus, at least one of $m$ and $n$ is even.
Also, $m \geqslant 1, n \geqslant 3$, and both are integers.
To make $m+n$ the smallest, then
$(m, n)=(1,4),(2,3)$.
Therefore, the minimum value of $m+n$ is 5.