Question

Assertion :(A): Let $P\left(n\right)=111\dots 1(91$ times$)$, then $P\left(n\right)$ is a prime number. Reason: (R): Every prime number has at most and at least two factors.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct explanation of (A).
• C. (A)is true but (R} is false.
• D. (A)is false but (R) is true.

Answer 1

The correct option is D (A)is false but (R) is true.

Given: $P\left(n\right)$ is a prime number

$\mathrm{1111..}\left(91\right)times=1+{10}^1+.....1\times {10}^{90}$

$=\frac{{10}^{91}-1}{10-1}=\frac{{{\left(\right(10)}^{13})}^7-1}{10-1}$

$=\frac{{(10}^{13}-1){{\left[\right(\left(10\right)}^{13})}^6+{{\left(\right(10)}^{13})}^5+....{{\left(\right(10)}^{13})}^1+1]}{10-1}$

Since $\frac{{(10}^{13}-1)}{10-1}$ is an integer greater than 1

and ${{\left[\right(\left(10\right)}^{13})}^6+{{\left(\right(10)}^{13})}^5+....{{\left(\right(10)}^{13})}^1+1]$ is also an integer greater than 1 so it is one of the factor of the given number

Hence $P\left(n\right)$ is not a prime number

So assertion is false and reason is true as prime number has atmost and atleast two factors i.e $1$ and itself