The ones digits of powers of 23 cycle as do the ones digits of powers of 3, and as 3x3=9, 3x9=27, 3x7=21, 3x1=3, we see that the ones digits cycle through the values 3, 9, 7, 1. Thus, to find the ones digit of , we need only find mod 4. Now, 23 mod 4 = 3, and so we see that

mod 4 = 1, and thus mod 4 is 3 when n is odd and 1 when n is even. And we see that since 23 is odd, is odd, and so mod 4 = 3, and thus the ones digit of is the same as that of , which is 7.

--Kevin C.