Question

Which of the following numbers is a prime number?
A. 667
B. 861
C. 481
D. 331

Answer 1

Hint: We will take the reference of the definition of prime numbers as the basis to check each of the given options if they satisfy the conditions for prime numbers or not. If the given option has only two factors, that is 1 and the number itself, then it will be considered as a prime number.

Complete step-by-step answer:
It is given in the question that we have to find the prime numbers among the given options of A. 667, B. 861, C. 481 and D. 331. We know that a prime number is a number that has only two factors, which are 1 and the number itself. 2 is considered as the smallest prime number. Some examples of prime numbers are, 2, 3, 5, 7, 11, …. etc. Now, to solve this question, we will check all the given options individually for the conditions of prime numbers. If they have only two factors, then we will consider them as prime numbers. So, let us consider each option one by one.
Option A. 667
We can write 667 as, $1\times 667$ and also as, $23\times 29$. So, we get the four factors of 667 as, 1, 667, 23 and 29. Therefore this number will not be considered as a prime number as it has more than two factors.
Option B. 861.
We can write 861 as, $1\times 861,3\times 287,3\times 7\times 41,7\times 123,21\times 41$. So, we get the factors of 861 as, 1, 861, 3, 287, 7, 41, 123 and 21, so 8 factors. Therefore, we cannot take this number as a prime number too.
Option C. 481.
We can write 481 as, $1\times 481,13\times 37$, which means that its factors are 1, 481, 13 and 37. So, we cannot consider this as a prime number.
Option D. 331.
We can write 331 as, $1\times 331$ only. So, its factors are 1 and 331 only. Therefore, it is a prime number.
Hence, the correct option is option D. 331.

Note:Most of the time, the students check the given number by dividing it with numbers between 2 to 9. If the number is divisible by them, then they consider the number as composite numbers and if it is not divisible, then they directly consider it to be prime numbers. It may not be correct to do it that way as the number may be divisible by a number greater than 9, like the option A. 667, which was divisible by 23 and 29 and not with a number between 2 and 9. We have one more interesting trick to find the prime numbers, as each prime number is either of the form 6n-1 or 6n+1, for integer values of n starting from zero, therefore equate all the options to either of these two and check whether n is integer or not. If n is an integer then it is a prime number and if n is not an integer then it is not a prime number.