Perfect Numbers
A perfect number is defined as a positive integer whose positive divisors (excluding the number itself) when added equal the number itself.
The first four perfect numbers are:
- 6
- 28
- 496
- 8128
2n−1(2n − 1)
- For n = 2: 2(2−1)(22 − 1) = 2(3) = 6
- For n = 3: 2(3−1)(23 − 1) = 4(7) = 28
- For n = 5: 2(5−1)(25 − 1) = 16(31) = 496
- For n = 7: 2(7−1)(27 − 1) = 64(127) = 8128
Example:
Show 496 is a perfect number.Solution:
Factors of 28 are 1, 2, 4, 7, 14, and 28Excluding the number 28 from the list of factors, we now have 1, 2, 4, 7, and 14.
The sum of these factors are:
1 + 2 + 4 + 7 + 14 = 28
Therefore, the number 28 is a perfect number.