Question

Consider the statement “$P\left( n \right)={{n}^{2}}-n+41$ is prime”. Then which of the following is true?
$A)$ $P\left( 5 \right)$ is false and $P\left( 3 \right)$ is true.
$B)$ Both $P\left( 5 \right)$ and $P\left( 3 \right)$ are false.
$C)$ $P\left( 5 \right)$ is true and $P\left( 3 \right)$ is false.
$D)$ Both $P\left( 5 \right)$ and $P\left( 3 \right)$ are true.

Answer 1

From the question, it was given that $P\left( n \right)={{n}^{2}}-n+41$ is prime. Let us assume this as equation (1). Now let us substitute the value of n is equal to 3 in equation (1). Let us assume this as equation (2). Now let us substitute the value of n is equal to 5 in equation (1). Let us assume this as equation (3). Now we should check whether $P\left( 3 \right)$ and $P\left( 5 \right)$ are prime or not.

Complete step-by-step solution:
From the question, we were given a statement that “$P\left( n \right)={{n}^{2}}-n+41$ is prime”. From the option, it is clear that we should check whether $P\left( 3 \right)$ and $P\left( 5 \right)$.
By comparing $P\left( n \right)$ with $P\left( 3 \right)$, we can say that the value of n is equal to 3.
We know that $P\left( n \right)={{n}^{2}}-n+41$.
Let us assume
$P\left( n \right)={{n}^{2}}-n+41.......(1)$
Let us substitute the value of n is equal to 3 in equation (1). Then we get,

& \Rightarrow P(3)={{3}^{2}}-3+41 \\\ & \Rightarrow P(3)=9-3+41 \\\ & \Rightarrow P(3)=47.....(2) \\\ \end{aligned}$$ From equation (2), it is clear the value of $$P(3)$$ is equal to 47. We already know that 47 is a prime number. So, we can say that $$P(3)$$ is a prime number. Let us substitute the value of n is equal to 5 in equation (1). Then we get, $$\begin{aligned} & \Rightarrow P(5)={{5}^{2}}-5+41 \\\ & \Rightarrow P(5)=25-5+41 \\\ & \Rightarrow P(5)=61.....(2) \\\ \end{aligned}$$ From equation (2), it is clear the value of $$P(5)$$ is equal to 61. We already know that 61 is a prime number. So, we can say that $$P(5)$$ is a prime number. So, we can say that $$P(3)$$ and $$P(5)$$ are prime numbers. **Hence, option D is correct.** **Note:** Students should know the definition of a prime number. A number is a prime number if the number is having only one and itself as factors. So, it is clear that a prime number will have only two factors. Students should be careful while doing the calculation in this problem. If a small mistake is done, we cannot get a correct answer to this problem.