Question

Ali thinks that the value of E will be a prime number where $E = {n^2} + n + 5$ for any whole number value of n. Is Ali correct?
Give reasons to support the answer.

Answer 1

Hint: In this question we have to tell whether the given entity will be a prime number or not for any n belonging to the whole number. Whole numbers are an extended set of natural numbers as it starts from 0. Put various values of n belonging to a set of whole numbers and check whether E is coming out to be prime or not.

Complete step-by-step answer:
Given equation
$E = {n^2} + n + 5$
As we know whole numbers are the positive numbers including zero without any decimal or fractional parts.
So the set of whole numbers is $n = \left\{ {0,1,2,3,4,5................} \right\}$.
Now Ali thinks that the value E will be a prime number for any whole number value of n.
So calculate the value of E for different values of n.
As we know prime numbers are the numbers whose factors are 1 and itself.
So for n = 0.
$ \Rightarrow E = 0 + 0 + 5 = 5$ which is a prime number.
So for n = 1.
$ \Rightarrow E = 1 + 1 + 5 = 7$ which is again a prime number.
So for n = 2.
$ \Rightarrow E = {2^2} + 2 + 5 = 4 + 7 = 11$ which is again a prime number.
So for n = 3.
$ \Rightarrow E = {3^2} + 3 + 5 = 9 + 8 = 17$ which is again a prime number.
So for n = 4.
$ \Rightarrow E = {4^2} + 4 + 5 = 16 + 9 = 25$ which is not a prime number.
As we see 25 is not a prime number, the factors of 25 is
$ \Rightarrow 25 = 1 \times 5 \times 5$ ( it is not a multiplication of 1 and itself) so 25 is not a prime number.
Thus Ali's thinking is wrong.
So Ali is incorrect.
So this is the required answer.

Note: Whenever we face such types of problems the key concept is to have the good understanding of the basic definitions of whole numbers and prime numbers. A prime number is one which is divisible by one or itself. The definition of whole numbers is already being explained earlier. The application of these definitions surely helps in solving problems of this kind.