51. If the sum of the interior angles of $m$ regular $n$-sided polygons can be divided by 27, then the minimum value of $m+n$ is
Answer: 6
Solution: The total sum of the interior angles of $m$ regular $n$-sided polygons is
$$
180(n-2) m
$$

Since
$$
180=2^{2} \times 3^{2} \times 5,
$$

and
$$
27=3^{3},
$$

it is necessary that $n-2, m$ contain at least one factor of 3, and $n \geq 3$,

Therefore,
$$
m=3, n=3 \text { or } n-2=3, m=1,
$$

In both cases,
$$
m+n=6
$$