Distinct integers

To minimize the value of $57^m+45^n$ we need to minimize both $m,n$ . The minimum value occurs when $m=n=1$ , but since $m,n$ have to be distinct we go to the next one that is $m=1,n=2$ which is better than $m=2,n=1$ . Therefore the minimum possible value of $57^m+45^n$ is $57^1+45^2=\boxed{2082}$ .