Prime Factorisation Calculator

What is the Prime Factorisation Calculator?

A prime factorisation calculator breaks a whole number down into the set of prime numbers that multiply together to produce it — every whole number greater than 1 has exactly one such combination, known as its prime factorisation. It works using trial division: starting at 2, it repeatedly divides the number by the smallest possible divisor for as long as that divisor divides evenly, then moves to the next candidate divisor, continuing until only 1 remains. To factorise a number by hand, divide it by 2 repeatedly until it's no longer even, then try dividing by 3, 5, 7 and so on, recording each prime divisor you use — if you reach a divisor whose square exceeds what's left of the number, whatever remains is itself prime.

How it works

Prime factorisation breaks a number down into the set of prime numbers that multiply together to produce it — every whole number greater than 1 has exactly one such set (this is known as the Fundamental Theorem of Arithmetic). The standard method (trial division) repeatedly divides the number by the smallest possible prime until what remains is itself prime.

UK context

Prime factorisation underlies a surprising amount of practical mathematics and computing — it's used to find the greatest common divisor and lowest common multiple of numbers efficiently, and forms the mathematical foundation of several encryption systems, which rely on the fact that factorising very large numbers is computationally difficult.

Tips

• A number with only one prime factor (appearing once) is itself a prime number — the factorisation of a prime number is just the number itself.
• The number 1 has no prime factors at all (it's neither prime nor composite), which is why prime factorisation calculators generally start from 2.

A quick note

This page explains the prime factorisation method used in the calculator above.