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The unit digit of 3127173 is the unit digit of 7173. The other digits of 3127 are multiples of 10 and so they cannot contribute to the unit digit.
Now the unit digits of the powers of 7 are
Power -- Unit digit
0 -- 1
1 -- 7
2 -- 9
3 -- 3
4 -- 1
and you are back into the loop (of 1-7-9-3).
So, you only need consider 7 to the power 173 modulo 4. That is, the remainder when 173 is divided by 4.
173 = 1 mod 4
So the unit digit of 3127173 is the same as the unit digit of 7173 which is the unit digit of 71 which is 7.
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