The number of ordered pairs (m, n) where $\mathrm{m},\mathrm{n},\in \{1,2,3,\dots \dots 100\}$ such that 7^m + 7ⁿ is divisible by 5 is

The number m can be selected from  1 to 100 in 100 ways.

For n=1,2,3,....,100. Unit digit of 7ⁿ = 7 or 9 or 3 or 1 (each occur 25 times) $\therefore$(7^m+7ⁿ) is divisible by 5 if sum of the unit digits of 7^m,7ⁿ is 10 (i.e;7+3 or 3+7 or 9+1 or 1+9) $\therefore$  Corresponding to  each selection of 'm' the number  n can be can be selected in 25 ways . No. of pairs (m,n) = 100 × 25 =2500 ways