How to Find All the Factors of a Number

To find every factor of a number, test the integers from 1 up to its square root and pair each divisor with its partner. This finds them all in about √n steps instead of n.

Here's the efficient method, plus patterns that reveal primes and perfect squares.

The square-root pairing method

For each number d from 1 to √n that divides n, both d and n ÷ d are factors. For 36: 1×36, 2×18, 3×12, 4×9, 6×6 → factors 1, 2, 3, 4, 6, 9, 12, 18, 36.

Spotting primes and perfect squares

• Exactly two factors → the number is prime.
• An odd number of factors → a perfect square (one pair repeats, like 6×6).

Counting factors with prime powers

If n = 2³ × 3² × 5, the number of factors is (3+1)(2+1)(1+1) = 24. Get the prime form from the Prime Factorization Calculator.

Try it instantly

The Factors Calculator lists every factor, shows the factor pairs as a diagram, and reports the count and sum.