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

MCQ

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

A
$1$
B
$4$
C
$6$
✓
$8$

✓

Answer

Correct option: D.

$8$

d
$\operatorname{Re}\left(\frac{(11)^{1011}+(1011)^{11}}{9}\right)=\operatorname{Re}\left(\frac{2^{1011}+3^{11}}{9}\right)$

For $\operatorname{Re}\left(\frac{2^{1011}}{9}\right)$

$2^{1011}=(9-1)^{337}={ }^{337} C_{0} 9^{337}(-1)^{0}$

$+{ }^{337} C_{1} 9^{336}(-1)^{1}$

$+{ }^{337} C_{2} 9^{335}(-1)^{2}+\ldots \ldots$

$+{ }^{337} C_{337} 9^{0}(-1)^{337}$

so, remainder is $8$ and $\operatorname{Re}\left(\frac{3^{11}}{9}\right)=0$

So, remainder is $8$

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