Example 7 In a school's donation activity for the Hope Project, the total donation amount from $m$ boys and 11 girls in Class A is equal to the total donation amount from 9 boys and $n$ girls in Class B, both being $(m n+9 m+11 n+145)$ yuan. It is known that each person's donation amount 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 amount of the two classes is 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,
$$
y=m+11=n+9
$$
․ $m n+9 m+11 n+145$
$$
=(m+11)(n+9)+46=y^{2}+46 \text {. }
$$

Since $x, y$ are integers and $y \geqslant 11$, then
$(x-y) y=46=1 \times 46=2 \times 23$.
Thus, we have $\left\{\begin{array}{l}x-y=1, \\ y=46\end{array}\right.$ or $\left\{\begin{array}{l}x-y=2, \\ y=23,\end{array}\right.$
which gives $\left\{\begin{array}{l}x=47, \\ y=46\end{array}\right.$ 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.