Question 202: Let $a, b \in N^{+}$, find all positive integers that cannot be expressed in the form $\frac{a}{b}+\frac{a+1}{b+1}$.
Question 202, Solution: First, we find all positive integers that can be expressed in the form $\frac{a}{b}+\frac{a+1}{b+1}$.
Let the positive integer $n=\frac{a}{b}+\frac{a+1}{b+1}=\frac{2 a b+a+b}{b(b+1)}$, then we know $b|2 a b+a+b \Rightarrow b| a$. Let $a=p b$, then $\frac{a}{b}=p$, so $\frac{a+1}{b+1}=n-p \Rightarrow \frac{p b+1}{b+1}=n-p \Rightarrow \frac{p-1}{b+1}=2 p-n$, thus we know $b+1 \mid p-1$. Let $p-1=q(b+1)$, then $p=q(b+1)+1$, substituting back $\frac{p-1}{b+1}=2 p-n$ we get: $n=2 p-\frac{p-1}{b+1}=2[q(b+1)+1]-q=(2 b+1) q+2$. Therefore, we know $n \geq 2$, and $n-2$ is a multiple of some odd number greater than 1.

On the other hand, for any positive integer $n$, if $n \geq 2, n-2$ has an odd factor greater than 1, let's assume $n=(2 b+1) q+2$. We take $a=q b(b+1)+b$, then $a \in N^{+}$, and $\frac{a}{b}+\frac{a+1}{b+1}=(2 b+1) q+2=n$. This means that if a positive integer $n \geq 2, n-2$ has an odd factor greater than 1, then $n$ can be expressed as $\frac{a}{b}+\frac{a+1}{b+1}$.
In summary, all positive integers that cannot be expressed in the form $\frac{a}{b}+\frac{a+1}{b+1}$ are 1 and numbers of the form $2^{m}+2(m \in N)$.