**Perfect numbers:**

Perfect numbers are the least known numbers which are the positive integers and they can be expressed as the sum of their proper divisors.

The known perfect numbers are 6, 28, 496 and 8128 only. 6 is the first and smallest perfect number and it has proper divisors as 1, 2, and 3. 6 is the improper divisor of 6.

Hence, we can write 6 as, 6 = 1+ 2+ 3= 6

Euclid find out the basics of perfect numbers over 2,000 years ago. And the well-known perfect numbers are only 6, 28, 496 and 8128 which all are the even numbers only. No one can found the perfect numbers other than these four even numbers till.

The following table shows the perfect numbers, with their proper and improper divisors.

**NOTE:**

Also, perfect numbers can be expressed as the sum of proper and improper divisors also. Thus, perfect number is written as the sum of its all divisors and that sum we will get will be two times that perfect number.

**For example:**

- 6 is the perfect number having divisors 1, 2, 3, and 6.
- So we can write, 6 → 1 + 2 + 3 + 6 = 12 = 2 times the perfect number 6

- Similarly, 28 is the perfect number having divisors 1, 2, 4, 7, 14 and 28.
- So we can write as, 28 → 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 times the perfect number 28