Every whole number bigger than 1 is built by multiplying prime numbers together — and there is only one way to do it. That unique list of prime building blocks is the number's prime factorisation.

For example, 24 is made from three 2s and one 3:

No other set of primes multiplies to 24. This "one and only one way" fact is so important it has a grand name: the fundamental theorem of arithmetic. Primes really are the atoms of the whole numbers.

Build it with a factor tree

The easiest way to find the prime factorisation is a factor tree. Split the number into any two factors, then split those, and keep going until every number at the bottom — every leaf — is prime. The leaves are the answer.

Watch it grow for 24. Step through the splits; the highlighted leaves at the end are its prime factors. (It doesn't matter which split you pick first — every choice lands on the same primes.)

Writing the answer

Once the leaves are found, collect repeats into powers. Three 2s and one 3 become:

By convention we write the primes in increasing order. Because the factorisation is unique, two people who split 24 differently — say 2 \times 12 versus 4 \times 6 — still end up with exactly 2^3 \times 3.

See it explained

Sal Khan builds prime factorisations with factor trees from scratch.