Question

A number n is said to be ‘perfect’ if the sum of all its divisors (excluding n itself) is equal to n. An example of a perfect number is
$\left( a \right)$ 9
$\left( b \right)$ 15
$\left( c \right)$ 21
$\left( d \right)$ 6

Answer 1

Hint: In this particular question use the concept that to find the factors of any number use the prime factorization method i.e. write the prime factors of the number, so use these concepts to reach the solution of the question.

Complete step by step answer:
According to question
A number n is said to be ‘perfect’ if the sum of all its divisors (excluding n itself) is equal to n.
So for this check the given options one by one and write all the prime factors of the given number in the options using the prime factorization method.
Prime numbers:
Prime numbers are those numbers which are divisible by only 1 or itself.
For example: 1, 2, 3, 5, 7...
Now check all the options one by one.
Option (a)
9
Now write all the prime factors of 9.
$ \Rightarrow 9 = 1 \times 3 \times 3$
So the sum of all its factors are, 1 + 3 + 3 = 7
So the sum is 7 which is not equal to 9, so 9 is not said to be ‘perfect’.
Option (b)
15
Now write all the prime factors of 15.
$ \Rightarrow 15 = 1 \times 3 \times 5$
So the sum of all its factors are, 1 + 3 + 5 = 9
So the sum is 9 which is not equal to 15, so 15 is not said to be ‘perfect’.
Option (c)
21
Now write all the prime factors of 9.
$ \Rightarrow 21 = 1 \times 3 \times 7$
So the sum of all its factors are, 1 + 3 + 7 = 11
So the sum is 11 which is not equal to 21, so 21 is not said to be ‘perfect’.
Option (d)
6
Now write all the prime factors of 6.
$ \Rightarrow 6 = 1 \times 2 \times 3$
So the sum of all its factors are, 1 + 2 + 3 = 6
So the sum is 6 which is equal to 6, so 6 is said to be ‘perfect’.
So it is the required answer.

So, the correct answer is “Option d”.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall that the prime numbers are those numbers which are divisible by only 1 or itself, so according to this factorize all the numbers given in the options as above and then add all the factors of a particular option and check which option satisfies the condition to be ‘perfect’ as above which is the required answer.