Question

Which of the following is a perfect number?

a) 6

b) 28

c) Both

d) None

Answer 1

Hint: Here, in this question, we are going to use the fact that a perfect number is such a positive integer which is equal to the sum of all its proper divisors, excluding the number itself. So, we will have to find all the factors of the given numbers and check whether their sum is equal to the given numbers.

Complete step by step solution:

In this given question, we are asked to find out which out of 6 and 28 is a perfect number, both of them are or none of them is.

A perfect number is such a positive integer which is equal to the sum of all its proper divisors, excluding the number itself. Simply, if the sum of all factors of a number is equal to that number then it is called a perfect number.

Now, we know that the factors of 6 are 1, 2, and 3. The sum of all the factors is 6 which is the given number. Hence, 6 is a perfect number.

We know that the factors of 28 are 1, 2, 4, 7 and 14. The sum of all the factors is which is the given number. Hence, 28 is a perfect number.

So, we conclude that both 6 and 28 are perfect numbers.

Therefore, option (c) both is the correct option to this question.

Note: Here, we may see that the proper divisors or factors are all the divisors of a number except the number itself. If their sum is equal to the number then it is a perfect number. Like, the proper divisors of 6 are 1, 2, and 3 whose sum is equal to 6 hence, it is a perfect number.