Question

The number $111.........1(91\mathrm{times})$ is a

• A. Even number
• B. Prime number
• C. Not prime
• D. None of these

Answer 1

The correct option is C

Not prime

Identify the type of number

The given series is $111.........1(91\mathrm{times})$

Let $S=1+10+{10}^2+{10}^3+....+{10}^{90}$

Here $a=1,r=10$

Therefore

$$\begin{gathered} S_n=\frac{1\left({10}^{91}-1\right)}{10-1} \\ =\frac{{\left({10}^{13}\right)}^7-1}{{10}^{13}-1}·\frac{{10}^{13}-1}{9} \\ =\left[{\left({10}^{13}\right)}^6+{\left({10}^{13}\right)}^5+....+1\right]\left({10}^{12}+{10}^{11}+....+1\right) \end{gathered}$$

Since it is the product of $2$ integers, it is not prime.

Also, we know that $1$ is not an even number.

So, $111.........1(91\mathrm{times})$ is an odd non-prime number.

Hence, the correct option is $\left(\mathrm{C}\right)$.