Mathematics - Binomial Theorem Question with Solution | TestHub

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

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$=0$

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