Proper Factor — Definition, Formula & Examples

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

For a positive integer $n$ , a proper factor (or proper divisor) is a positive integer $d$ such that $d \mid n$ and $d < n$ .

Worked Example

Problem: Find all the proper factors of 18.

Step 1: List every factor of 18 by finding pairs that multiply to 18.

1 \times 18,\; 2 \times 9,\; 3 \times 6

Step 2: The full set of factors is {1, 2, 3, 6, 9, 18}. Remove 18 itself to get the proper factors.

\{1,\, 2,\, 3,\, 6,\, 9\}

Answer: The proper factors of 18 are 1, 2, 3, 6, and 9.

Why It Matters

Proper factors are central to classifying numbers as perfect, abundant, or deficient. A perfect number like 6 equals the sum of its proper factors (

1 + 2 + 3 = 6

), a concept that appears in number theory courses and math competitions.

Common Mistakes

Mistake: Including the number itself as a proper factor.

Correction: By definition, a proper factor must be strictly less than the number. For instance, 12 is a factor of 12, but it is not a proper factor of 12.