What is last digit (in a different radix)?

Question

What is last digit of $AD^{AD}$ (hexadecimal) in octal?

How is it found?

Answer 1

Since $(2n+1)^2=8\,\frac{n(n+1)}{2}+1\equiv1\quad\mod8,$ we have $x^y\equiv x\quad\mod8$ for odd $x,y.$ Since the last octal digit of hexadecimal AD is 5, that is your answer.

Answer 2

AD = 10*16 + 13 = 173.

173^173 mod 8 = (173 mod 8)^173 mod 8 = 5^173 mod 8

5^1 mod 8 = 5, 5^2 mod 8 = 1, 5^3 mod 8 = 5 etc., so 5^173 mod 8 = 5.