@neatlobster Good question! Since 8 = -1 mod 9, and the powers of -1 go -1, 1, -1, 1, etc, all (-1)^{even} = 1 mod 9 and all (-1)^{odd} = -1 mod 9. Since we are talking about modulos, and 8 is the same as -1 mod 9, this means (8)^{even} = 1 mod 9, and (8)^{odd} = -1 mod 9.