If we look at this as geometric series it would look like:
7, 49, 343, 2401, 16807, 117,649.....................
The units digit is the digit on the right of the number. The units digit is repeating like this: 7,9,3,1,7,9. Each time we multiply by 7 the units digit will change and will follow the aforementioned sequence.
Because there are 4 numbers in the sequence, we take 105 and divide it by 4 and we get 26 with a remainder of 1. If we go 4 times, the forth in the sequence is one, the eighth is 1, and the 12th is one. 7^104 will have a units digit of 1. Because we have a remainder of 1, we count up 1 and that gives a units digit of 7.
The answer is the units digit of 7^105 is 7