HongKong 2022-23 IMO Selection Tests

Answer: 276 Since $20^{22}$ is a square number, $m^{2}n$, and hence $n$, must also be a square number. Write $n=k^2$. Then the equation becomes $m^2{k}^2=20^{22}$, or $mk=20^{11}$. Each positive factor of $20^{11}$ corresponds to a choice of $m$, which in turn corresponds to a solution $(m,n)$ to the original equation. The answer is thus equal to the number of positive factors of $20^{11}=2^{22}{5}^{11}$, which is $(22+1)(11+1)=276$.