The remainder when (11) ^ 1011 + (1011) ^ 11 is divided by 9 is...

${11}^{1011} = {\left({11}^{3}\right)}^{327} = {\left(9k + 8\right)}^{327}$

when divided by $9$ is equivalent to the division of

${8}^{327} = {\left(9 - 1\right)}^{327}$

i.e. remainder$= - 1 + 9 = 8$

Now, ${\left(1011\right)}^{11} = {\left(9m + 3\right)}^{11}$

when divided by $9$ is equivalent to the division of

${3}^{11} = {\left(9\right)}^{5} \times 3$

i.e. remainder$= 0$

$∴ {11}^{1011} + {1011}^{11}$ has same remainder as $8$.