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 perfect number is :

Options

A. 9
B. 15
C. 21
D. 6

Answer

To determine if a number \( n \) is a perfect number, we need to find the sum of all its divisors excluding \( n \) itself and check if this sum equals \( n \). Let's go through the options step by step.

### Step-by-Step Solution:

1. **Understanding the Definition of Perfect Number**:
- A number \( n \) is perfect if the sum of its divisors (excluding \( n \)) equals \( n \).

2. **Examining Option A: \( n = 9 \)**:
- **Find Divisors**: The divisors of 9 are 1, 3, and 9.
- **Sum of Divisors Excluding 9**: \( 1 + 3 = 4 \).
- **Check Equality**: \( 4 \neq 9 \). Therefore, 9 is not a perfect number.

3. **Examining Option B: \( n = 15 \)**:
- **Find Divisors**: The divisors of 15 are 1, 3, 5, and 15.
- **Sum of Divisors Excluding 15**: \( 1 + 3 + 5 = 9 \).
- **Check Equality**: \( 9 \neq 15 \). Therefore, 15 is not a perfect number.

4. **Examining Option C: \( n = 21 \)**:
- **Find Divisors**: The divisors of 21 are 1, 3, 7, and 21.
- **Sum of Divisors Excluding 21**: \( 1 + 3 + 7 = 11 \).
- **Check Equality**: \( 11 \neq 21 \). Therefore, 21 is not a perfect number.

5. **Examining Option D: \( n = 6 \)**:
- **Find Divisors**: The divisors of 6 are 1, 2, 3, and 6.
- **Sum of Divisors Excluding 6**: \( 1 + 2 + 3 = 6 \).
- **Check Equality**: \( 6 = 6 \). Therefore, 6 is a perfect number.

### Conclusion:
The perfect number from the given options is **Option D: 6**.