Factors Calculator: All Factors and Factor Pairs of a Number

How it works

d is a factor of n ⇔ n ÷ d leaves no remainder · test d = 1 … √n; each hit gives the pair (d, n/d) · if n = p₁^e₁ × p₂^e₂ × …, the number of factors is (e₁ + 1)(e₂ + 1)…

A factor (or divisor) of n is a whole number that divides n exactly. Factors come in pairs that multiply to n, and the smaller member of each pair is never more than √n, so testing up to the square root finds them all. The prime factorization predicts how many factors there are. A number is prime when its only factors are 1 and itself, perfect when its proper factors add up to it (6, 28, 496…), abundant when they add up to more and deficient when they add up to less.

Worked example

48: testing 1 to 6 (√48 ≈ 6.93) gives the pairs 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8, so the factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 — ten of them, matching 48 = 2⁴ × 3 and (4 + 1)(1 + 1) = 10. They sum to 124; the proper factors sum to 76 > 48, so 48 is abundant.

Sources: Wolfram MathWorld — Divisor · Common Core State Standards — 4.OA.4 (factor pairs, prime and composite)

Frequently asked questions