Input a number to check if it's a perfect number, or input a range to generate all perfect numbers within that range.
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Result
What Is a Perfect Number?
A perfect number, also called a complete or ideal number, is a positive integer that equals the sum of its proper divisors (excluding itself). For example:
- 6 is a perfect number because its divisors are 1, 2, and 3, and 1 + 2 + 3 = 6.
- 28 is also a perfect number because its divisors are 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28.
How to Determine if a Number Is a Perfect Number
Find the Divisors: Identify all proper divisors of the number (excluding itself). For example, the divisors of 28 are 1, 2, 4, 7, and 14.
Sum the Divisors: Add up all the divisors. For 28, the sum is 1 + 2 + 4 + 7 + 14 = 28.
Check the Result: If the sum equals the original number, it's a perfect number. Otherwise, it's not a perfect number.
Examples
Example 1: Is 12 a perfect number?
Solution:
1. Find the divisors:
The proper divisors of 12 are 1, 2, 3, 4, and 6.
2. Sum the divisors:
1 + 2 + 3 + 4 + 6 = 16
Thus, 12 is not a perfect number since the sum is greater than 12.
Example 2: Is 496 a perfect number?
Solution:
1. Find the divisors:
The proper divisors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, and 248.
2. Sum the divisors:
1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496
Conclusion: 496 is a perfect number since the sum equals 496.
Commonly Known Perfect Numbers
o date, 51 perfect numbers have been discovered. The first 10 are: