Factor Calculator

Factoring a number means writing it as a product of smaller whole numbers. This calculator finds every factor by testing divisors up to the square root of the number, pairing each divisor it finds with its partner, and then breaks the number into primes.

What is a factor?

A factor, or divisor, of a whole number is a whole number that divides into it exactly, leaving no remainder. 6 is a factor of 24 because 24 ÷ 6 = 4 with nothing left over, while 5 is not, because 24 ÷ 5 leaves 4. Every whole number greater than 1 has at least two factors, 1 and itself.

Factors come in pairs. Whenever d divides n, so does n ÷ d, and the two multiply back to n. The factor pairs of 120 are 1 × 120, 2 × 60, 3 × 40, 4 × 30, 5 × 24, 6 × 20, 8 × 15 and 10 × 12, giving sixteen factors in all. The smaller member of each pair is never more than the square root of the number, so you only need to test divisors up to √n to find them all. For a perfect square such as 36, one pair repeats a number (6 × 6), which is why perfect squares are the only numbers with an odd count of factors.

A number whose only factors are 1 and itself is prime: 2, 3, 5, 7, 11, 13 and so on. A number with more factors is composite. The number 1 is neither. The fundamental theorem of arithmetic says every whole number above 1 can be written as a product of primes in exactly one way, apart from order. For 120 that is 2 × 2 × 2 × 3 × 5, or 2³ × 3 × 5.

The prime factorization reveals how many factors a number has. Add one to each exponent and multiply: 120 = 2³ × 3¹ × 5¹ has (3 + 1)(1 + 1)(1 + 1) = 16 factors. Highly composite numbers like 12, 60, 360 and 5,040 have more factors than any smaller number, which is why they were favored for measuring time and angles.

Negative numbers have factors too. Since (−4) × (−30) = 120 as well, the negatives of every positive factor also divide 120. By convention, factor lists usually show only the positive factors, as this calculator does.

Factors are useful well beyond arithmetic practice. They let you split items into equal groups, simplify fractions, find common denominators, and factor polynomials in algebra. Large prime factorizations are also central to cryptography: multiplying two big primes is easy, but recovering them from the product is extremely hard, and that one-way property protects much of the internet's traffic.

Frequently asked questions