What is a Perfect Number?

A Perfect Number is nothing but a whole number, which is identical to the sum of its each and every proper divisor.

Perfect numbers

What are Divisors?
Divisors are same as the factors of a number.  A divisor or factor is a number which divides a number evenly and gives the remainder as zero. Finding factors of a number is simple. For example, consider the number 9.  The divisors or factors of 9 are 1, 3, 9, because only these three numbers can divide 9 without leaving any remainder.

Mathematical problems are solved by finding factors and multiples as we did now. In the above case, 3 is the factor of 9 as it divides 9 evenly. At the same time, 9 is a multiple of 3 because 3 X 3 = 9. This signifies that finding factors and multiples are necessary to find the perfect numbers and to solve many algebraic expressions.

What are Proper Divisors?
Proper divisors are the divisors of the number excluding the number itself. As already discussed, the divisors or factors of 9 are 1, 3 and 9. The proper divisors of number 9 are 1, 3, and 9. Number 9 is not a proper divisor of 9 because it is the same as the original number for which the divisors are identified.

What are Perfect Numbers?
Knowing what proper divisors are, now let us look at the question: What is a perfect numbers? Identify the proper divisors of a number and add them. If the result of addition is same as the actual number for which the divisors are identified, then that number is a perfect number.

For example, we identified 1 and 3 as proper divisors of9. Now add these proper divisors:
1 + 3 = 4

Here the sum is 4. Instead, if the sum was 9 (the actual number for which you found the divisors), then 9 will be called as perfect number. In this case, sum is 4 and not 9. This signifies that 9 is not a perfect number.

Let us take up another example. Are 6 a perfect number? The divisors or factors of 6 are 1, 2, 3, and 6. The proper divisors of number 6 will then be 1, 2 and 3. Now, add these proper divisors.  Addition of 1, 2 and 3 will result in 6. The result 6 is same as the number 6 for which we identified these proper divisors. Thus, 6 is a perfect number.

The other perfect numbers are:
• 28
• 496
• 8128