Official solution
\[{\mathop{\rm Re}\nolimits} \left( {{{{{(11)}^{1011}} + {{(1011)}^{11}}} \over 9}} \right) = {\mathop{\rm Re}\nolimits} \left( {{{{2^{1011}} + {3^{11}}} \over 9}} \right)\]
For \[{\mathop{\rm Re}\nolimits} \left( {{{{2^{1011}}} \over 9}} \right)\]
\[{2^{1011}} = {(9 - 1)^{337}} = {}^{337}{C_0}{9^{337}}{( - 1)^0} + {}^{337}{C_1}{9^{336}}{( - 1)^1} + {}^{337}{C_2}{9^{335}}{( - 1)^2} + \,\,.....\,\, + \,\,{}^{337}{C_{337}}{9^0}{( - 1)^{337}}\]
So, remainder is 8
and \[{\mathop{\rm Re}\nolimits} \left( {{{{3^{11}}} \over 9}} \right) = 0\]
So, remainder is 8