Question

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

• A. $4$
• B. $6$
• C. $8$
• D. $1$

Answer 1

The correct option is C $8$

$Re\left(\frac{(11{)}^{1011}+(1011{)}^{11}}{9}\right)=Re\left(\frac{2^{1011}+3^{11}}{9}\right)$
For $Re\left(\frac{2^{1011}}{9}\right)$
$2^{1011}=(9-1{)}^{337}{=}^{337}{C}_0{9}^{337}(-1{)}^0{+}^{337}{C}_1{9}^{336}(-1{)}^1{+}^{337}{C}_2{9}^{335}(-1{)}^2+...{+}^{337}{C}_{337}{9}^0(-1{)}^{337}$
So, remainder is $8$ and $Re\left(\frac{3^{11}}{9}\right)=0$.
So, remainder is $8$.