What is 2^{-1} + 2^{-2} + 2^{-3} + 2^{-4} + 2^{-5} + 2^{-6} mod{13}? Express your answer as an integer from 0 to 12, inclusive.
As a modular inverse, 2^{-1} = 6.
Then 2^{-2} = 6^2 = 10
2^{-3} = 6*10 = 60 = 8
2^{-4} = 6*8 = 48 = 9
2^{-5} = 6*9 = 54 = 2
2^{-6} = 6*2 = 12
Adding these up, we get 6 + 10 + 8 + 9 + 2 + 12 = 47 = 8 mod 13.
The answer is 8.
