The square of a prime number is prime. A.True B.False C.Data insufficient D.None
Hint: We will use the definition of prime numbers to prove this question. Once we state the definition of what prime numbers are, we will generalise it for the square of prime numbers. Then, we will see how many factors the square of prime numbers contains so as to know if we can include it in the category of prime numbers or not .
Complete step-by-step answer: We need to check the given statement if true or not. The statement is: The square of a prime number is prime. Let us recall the definition of prime numbers. According to the definition, a prime number is a number which is only divisible by 1 and the number itself. Prime numbers are not divisible by any other number except those two (1 and itself). Now, considering the square of the prime numbers, let us say a number m is a prime number. Therefore, its square will be $m^2$ . Let us suppose $m^2$ is a prime number. For $m^2$ to be a prime number, according to definition, it must contain only 2 factors i. e., 1 and $m^2$ . But $m^2$ can also be written as mxm which makes m a factor of $m^2$ as well. So, we get 3 factors of $m^2$ in total i.e., 1, m and $m^2$. Hence, our assumption is wrong that the square of a prime number is also a prime number. Therefore, we can say, by contradiction, that $m^2$ is not a prime number.
Option(B) is correct.
Note: In such questions where we are given a statement to justify whether true or false, we should first prove it by any method i.e., either by assumption or direct substitution to check if the given statement holds for every value or not. For example take the number as 3 square will be 9 . Here 3 is a prime number but 9 is not a prime number.
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