Factor Calculator

The Formula

Factors are whole numbers that divide evenly into another number with no remainder. Every factor pair (a, b) satisfies a x b = n. Finding all factors of a number is a fundamental number theory skill with applications in simplification, division, and factorization problems.

Variable Definitions

How to Use This Calculator

1. 1

   Enter any positive integer in the input field.

2. 2

   View ALL factors listed in ascending order, from smallest to largest.

3. 3

   See the factor pairs table showing each pair multiplied together to produce the original number.

4. 4

   Check the "Prime?" indicator to instantly determine if the number is prime (exactly two factors: 1 and itself).

5. 5

   The sum of all factors is also displayed, which is useful for classifying numbers as deficient, perfect, or abundant.

Factors come in pairs that multiply to the original number. Each pair represents one way to multiply to reach n.

Understanding the Concept

Finding factors is a fundamental number theory skill with applications throughout mathematics. A factor of n is any integer that divides n evenly with no remainder. Every positive integer greater than 1 has at least two factors: 1 and itself. Numbers with exactly two factors are called prime numbers, while numbers with more than two factors are called composite numbers. The number 1 is neither prime nor composite — it has exactly one factor. To find all factors efficiently, you only need to test divisors up to the square root of n: for each divisor d that divides n evenly, both d and n/d are factors. This is why the calculator can find factors of large numbers quickly. For example, for n = 36, the square root is 6, so testing divisors 1 through 6 reveals: 1 x 36, 2 x 18, 3 x 12, 4 x 9, and 6 x 6. The sum of all proper factors (excluding n itself) determines whether a number is deficient (sum < n), perfect (sum = n, like 6 = 1+2+3), or abundant (sum > n).

Frequently Asked Questions

Sources & References