Re: How do I do this problem without a calculator?

Well, 6^10 = 1 mod 11 and it's even, so 6^10 = 12 mod 22. Therefore
6^19 = 12*6^9 mod 22. As 12 = 11+1 and 6^9 is even, 11*6^9 = 0. and
thus 6^19 = 6^9 mod 22.

Now 6*6^9 = 6^10 = 12, meaning 6^9 solves 6x = 12 in Z_22. Since gcd
(6, 22) = 2, the possible values for x are 2 and 2+11 = 13, but the
latter is ruled out because 6^9 is even.

Therefore 6^9 and 6^19 have residue 2 in Z_22.

--OL

.

Loading