Proper Divisor — Definition, Formula & Examples

A proper divisor of a whole number is any factor of that number other than the number itself. For example, the proper divisors of 12 are 1, 2, 3, 4, and 6.

For a positive integer $n$ , a proper divisor of $n$ is any positive integer $d$ such that $d \mid n$ and $d < n$ .

Worked Example

Problem: Find all proper divisors of 18 and their sum.

List all factors: The factors of 18 are the numbers that divide 18 evenly.

1, 2, 3, 6, 9, 18

Remove the number itself: Drop 18 from the list, since a proper divisor must be less than the number.

1, 2, 3, 6, 9

Sum the proper divisors: Add them up.

1 + 2 + 3 + 6 + 9 = 21

Answer: The proper divisors of 18 are 1, 2, 3, 6, and 9, and their sum is 21.

Why It Matters

Proper divisors are the key ingredient in defining perfect numbers. A perfect number like 6 equals the sum of its proper divisors (

1 + 2 + 3 = 6

). This idea appears throughout number theory and in math competitions where problems ask you to classify numbers by how their proper divisor sums compare to the number itself.

Common Mistakes

Mistake: Including the number itself as a proper divisor.

Correction: A proper divisor must be strictly less than the number. The number 12 is a factor of 12, but it is not a proper divisor of 12.