The number of ordered pairs (m,n) where $m,n\in \{1,2,3,\dots .50\}$ such that $6^{\mathrm{m}}+9^{\mathrm{n}}$ is a multiple of 5 is

$6^m+9^n$

Unit digit of 6m is 6
Unit digit of 9n will be = 9 or 1
For multiple of 5 unit digit of 9n must be = 9
It occur when n = odd

m can be selected in 50 ways and n can be selected in 25 ways
Total number of ordered pair = 50×25 = 1250