Question

Find Unit digit in 32^32^32

Answer 1

32 = 2^5

32^32 = (2^5)^32 = 2^(5x32) = 2^160

(32^32)^32 = (2^160)^32 = 2^(160x32) = 2^5120

We observe that

2^1 ends in 2

2^2 ends in 4

2^3 ends in 8

2^4 ends in 6

2^5 ends in 2

2^6 ends in 4

2^7 ends in 8

2^8 ends in 6

and so on.

We may conclude (for a non-negative integer k) that

2^(4k+1) ends in 2

2^(4k+2) ends in 4

2^(4k+3) ends in 8

2^(4k) ends in 6

5120 = 4(1280) = 4k, where k = 1280.

Thus, (32^32)^32 = 2^5120 ends in 6.