The remainder when \(11^{1011} + 1011^{11}\) is divided by \(9\) is

11 = 2 (mod 9) and 2^6 = 1 (mod 9), 1011 = 3 (mod 6), so 2^1011 = 8 (mod 9). 1011 is divisible by 3, so 1011^11 = 0 (mod 9). Remainder 8.