The unit digit in $${\left( {122} \right)^{173}}$$  is ?

In this case:
122 (powers of the last digit 2 )

2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
2⁶ = 64
2⁷ = 128
2⁸ = 256
2⁹ = 512
……

As you can see, the pattern repeats every 4^th power (2, 4, 8, 6).

We have to find 122¹⁷³
Hence, remainder when 173 is divided by 4 = 1
(which corresponds to the 1^st number of the pattern)
Thus, unit digit of 122¹⁷³ = 2