Question

For each $nϵN,{10}^{2n+1}+1$ is divisible by

• A. $11$
• B. $13$
• C. $27$
• D. none of these

Answer 1

The correct option is A $11$

Sol. 1 : Put $n=1$ and check for option

Sol. 2 : Let ${10}^{(2n+1)}+1$ is divisible by $I$ for all $n$.

Hence, it will also be divisible for $n=n+1$

$S={10}^{(2n+1)}+1=I.\left(m\right)$ $m\in I$

Put $(n+1)$ in place of $n$.

$\Rightarrow S^{{}^{\prime }}={10}^{\left[2\right(n+1)+1]}+1$

$={10}^2.{10}^{(2n+1)}+1$

$=100({10}^{(2n+1)}+1)-99$

$=100.I\left(m\right)-11\times 3^2$

$\therefore I$ must be equal to any of the factors of $99$

Answer $=11$.

Hence, the answer is $11.$