Factor Calculator

About Factors and Divisors

A factor (or divisor) of an integer n is any integer that divides n exactly with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Factors always come in pairs — for every factor f of n, n/f is also a factor. Pairs are typically listed from smallest to largest: (1, n), (2, n/2), and so on.

The most efficient algorithm for finding all factors is trial division up to √n. For each integer i from 1 to √n, if i divides n, then both i and n/i are factors. This runs in O(√n) time. A number is prime if its only factors are 1 and itself — in other words, it has exactly two distinct factors. All other integers greater than 1 are composite.

The number of divisors function τ(n) counts how many factors n has. If the prime factorization of n is p₁^e₁ × p₂^e₂ × … × pₖ^eₖ, then τ(n) = (e₁ + 1)(e₂ + 1) … (eₖ + 1). For example, 360 = 2³ × 3² × 5¹, so τ(360) = (3+1)(2+1)(1+1) = 24 divisors.