17. How many pairs $(m, n)$ of non-negative integers are there such that $m \neq n$ and $\frac{50688}{m+n}$ is an odd positive power of 2 ?
(2 marks)
有多少對非負整數 $(m, n)$ 滿足 $m \neq n$ 且 $\frac{50688}{m+n}$ 是 2 的正奇數暮?
（２分）
17. 33760
17. Since $50688=2^{9} \times 99$, we must have $m+n=2^{k} \times 99$ where $k$ is one of $0,2,4,6,8$. Forgetting about $m \neq n$ for the moment, there are $2^{k} \times 99+1$ choices of $m$ for each $k$, as $m$ can range from 0 to $2^{k} \times 99$. This leads to a total of $\left(2^{0}+2^{2}+2^{4}+2^{6}+2^{8}\right) \times 99+5=33764$ pairs of $(m, n)$. Among these, 4 pairs violate the condition $m \neq n$, as $m=n$ is possible only when $k$ is $2,4,6$ or 8 . Hence the answer is $33764-4=33760$.