Example 16 (Question from the 44th American High School Mathematics Examination) The number of positive integer solutions $(m, n)$ to the equation $\frac{4}{m}+\frac{2}{n}=1$ is ( ).
A. 1
B. 2
C. 3
D. 4
E. More than 4
Solution: Choose D. Reason: Since $m, n$ are positive integers and $\frac{4}{m}<4, n>2$. The equation $\frac{4}{m}+ \frac{2}{n}=1$ is equivalent to $(m-4)(n-2)=8$, so we only need to find all ways to decompose 8 into the product of positive integers, i.e., $1 \cdot 8, 2 \cdot 4, 4 \cdot 2, 8 \cdot 1$. They correspond to 4 sets of solutions $(m, n)=(5,10),(6,6),(8,4),(12,3)$. Therefore, choose D.