How Many Perfect Numbers are there Between 1 and 100?

Between 1 and 100, there are two perfect numbers, namely 6 and 28.

Let’s examine the two perfect numbers between 1 and 100:

Perfect Number 6:

The divisors of 6 are 1, 2, and 3.

The sum of these divisors (excluding 6 itself) is 1+2+3=6.

Therefore, 6 is a perfect number.

Perfect Number 28:

The divisors of 28 are 1, 2, 4, 7, and 14.

The sum of these divisors (excluding 28 itself) is 1+2+4+7+14=28.

Therefore, 28 is also a perfect number.

Perfect numbers are relatively rare, and the discovery of more perfect numbers becomes increasingly challenging. Euler proved that if 2^p−1 is a prime number, then 2^p−1(2^p − 1) is a perfect number, where p is a prime number. The known perfect numbers follow this formula, with 6 corresponding to 2²(2³ − 1) and 28 corresponding to 2³(2⁴ − 1).

In the range from 1 to 100, these are the only two known perfect numbers, and they have been known since ancient times.