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.

Hint: 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,
\[\begin{align}
  & \Rightarrow P(3)={{3}^{2}}-3+41 \\
 & \Rightarrow P(3)=9-3+41 \\
 & \Rightarrow P(3)=47.....(2) \\
\end{align}\]
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,
\[\begin{align}
  & \Rightarrow P(5)={{5}^{2}}-5+41 \\
 & \Rightarrow P(5)=25-5+41 \\
 & \Rightarrow P(5)=61.....(2) \\
\end{align}\]
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.