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

### Concept & Strategy A "Perfect Number" is a positive integer that is equal to the sum of its proper divisors. Proper divisors are all the positive integers that divide the number evenly, excluding the number itself. The strategy here is straightforward: extract the proper divisors for each option and calculate their sum to see which one equals the original number. ### Step-by-Step Solution We will test each option to find the perfect number: * **Test Option (a) 6:** Divisors of $6$ (excluding $6$): $1, 2, 3$ Sum of divisors = $1 + 2 + 3 = 6$ Since $6 = 6$, this is a perfect number! * **Test Option (b) 9:** Divisors of $9$ (excluding $9$): $1, 3$ Sum of divisors = $1 + 3 = 4$ Since $4 \neq 9$, it is not a perfect number. * **Test Option (c) 15:** Divisors of $15$ (excluding $15$): $1, 3, 5$ Sum of divisors = $1 + 3 + 5 = 9$ Since $9 \neq 15$, it is not a perfect number. * **Test Option (d) 21:** Divisors of $21$ (excluding $21$): $1, 3, 7$ Sum of divisors = $1 + 3 + 7 = 11$ Since $11 \neq 21$, it is not a perfect number. ### Exam Strategy & Shortcut **Memorization Benchmark:** Perfect numbers are rare in early mathematics. The first few perfect numbers are mathematically famous: $6, 28, 496,$ and $8128$. Memorizing at least the first two ($6$ and $28$) is highly recommended for aptitude exams as they frequently appear as trivia or direct questions, allowing you to bypass calculation entirely. ### Common Pitfall A common error is forgetting to exclude the number itself ($n$) when calculating the sum of divisors. If you include the number itself, the sum for a perfect number will always equal $2n$ instead of $n$. ### Final Answer **Therefore, the correct answer is $6$.**