Example 1 In the donation activity of a school to the Hope Project, the total donation amount of $m$ boys and 11 girls in Class A is equal to the total donation amount of 9 boys and $n$ girls in Class B, which is $(m n+9 m+11 n+145)$ yuan. It is known that the donation amount per person is the same, and it is an integer number of yuan. Find the donation amount per person.
(1996, National Junior High School Mathematics League)
Solution: Let the donation amount per person be $x$ yuan. Since the total donation amounts of the two classes are equal and the donation amount per person is the same, the number of people in the two classes is also equal, denoted as $y$. Then,
$$
y=m+11=n+9 \text{, }
$$

and $\square D \square D \square D \square$
$$
\begin{array}{l}
m n+9 m+11 n+145 \\
=(m+11)(n+9)+46=y^{2}+46 .
\end{array}
$$

Therefore, $y^{2}+46=x y$. This gives $(x-y) y=46$.
Since $x$ and $y$ are integers and $y \geqslant 11$, then
$$
(x-y) y=46=1 \times 46=2 \times 23 \text{. }
$$

Thus, we have $\left\{\begin{array}{l}x-y=1, \\ y=46\end{array}\right.$, or $\left\{\begin{array}{l}x-y=2, \\ y=23 .\end{array}\right.$
That is, $\left\{\begin{array}{l}x=47, \\ y=46\end{array}\right.$ or $\left\{\begin{array}{l}x=25, \\ y=23 .\end{array}\right.$
Therefore, when the number of people in each class is 46, the donation amount per person is 47 yuan; when the number of people in each class is 23, the donation amount per person is 25 yuan.