Question

When expanded the units place of
${(3127)}^{173}$

Answer 1

Cyclicity of 7 is 4
173 = 4 × 43 + 1
so unit digit is
${7}^{1} = 7$

Answer 2

Given, 3127 has unit digit 7

At power 1, it's unit digit is 7
At power 2, it's 9
At power 3, it's 3
At power 4, it's 1
At power 5, it's 7
At power 6, it's 9
At power 7, its 3
At power 8, its 1
At power 9, its 7

We can see a pattern that the unit digit in the results is (9,3,1,7), (9,3,1,7), .......

We need to see powers from 2 to 173

there are 172 powers.
Dividing them by 4 we get, 172/4 = 43

Since it is completely divisible, the unit digit is 7.

HOPE IT HELPS.