The "perfect number trick" involves identifying numbers that are equal to the sum of their proper divisors (excluding the number itself).
Understanding Perfect Numbers
A perfect number is a positive integer that is equal to the sum of its proper divisors. This means you find all the numbers that divide evenly into the number, excluding the number itself, and then add those divisors together. If the sum equals the original number, it's a perfect number!
How to Determine if a Number is Perfect:
- List the divisors: Write down all the numbers that divide evenly into the number, excluding the number itself.
- Sum the divisors: Add all the divisors together.
- Compare the sum to the original number: If the sum of the divisors equals the original number, then the number is perfect.
Examples:
Let's look at some examples to illustrate the concept:
- 6: The divisors of 6 (excluding 6) are 1, 2, and 3. 1 + 2 + 3 = 6. Therefore, 6 is a perfect number.
- 28: The divisors of 28 (excluding 28) are 1, 2, 4, 7, and 14. 1 + 2 + 4 + 7 + 14 = 28. Therefore, 28 is a perfect number.
Table of First Few Perfect Numbers
| Perfect Number | Divisors (excluding itself) | Sum of Divisors |
|---|---|---|
| 6 | 1, 2, 3 | 6 |
| 28 | 1, 2, 4, 7, 14 | 28 |
| 496 | 1, 2, 4, 8, 16, 31, 62, 124, 248 | 496 |
