We group the sum as follow:
[(1)^99 + (11)^99 + ... + (91)^99] + [(2)^99 + (22)^99 + ... (92)^99] + ...... + [(9)^99 + (19)^99 + ... + (99)^99] + [(10)^99 + (20)^99 + (30)^99 + ... (90)^99]
All the terms in a single group have the same last digit (i.e. last digits of 199 + 1199 + ... + 9199 are same, is 1, & similarly for the other groups).
Also, there are 10 terms in each group except for the last one. Therefore the last digit of the sum of terms in first 9 groups is 0. (as whatever be the last digit, we have to multiply it by 10) And the last digit of the sum of the terms in the group 10 is obviously 0.
Hence, the last digit of the series is 0.

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