Prime Factorisation
Every time you shop online or send a private message, huge numbers are being split into their
prime building blocks — and the fact that this splitting is fiendishly hard for enormous numbers
is exactly what keeps your bank details safe. That splitting has a name: prime factorisation.
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: its very own
prime fingerprint.
For example, 24 is made from three
2s and one 3:
24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 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 — and just like an atom, a prime cannot be
broken into a product of smaller whole numbers.
A baker has 12 cookies. She can pack them as 2
boxes of 6, or 3 boxes of
4, or 4 boxes of
3. But if she keeps breaking each box down into the
smallest equal groups she can, she always finishes with the same
three prime piles: 2, 2 and
3.
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.)
Three worked examples
Build each tree in your head, collect the prime leaves, and write them smallest-first:
-
12 \to 2 \times 6 \to 2 \times 2 \times 3, so
12 = 2 \times 2 \times 3 = 2^2 \times 3.
-
18 \to 2 \times 9 \to 2 \times 3 \times 3, so
18 = 2 \times 3 \times 3 = 2 \times 3^2.
-
60 \to 6 \times 10 \to (2 \times 3)(2 \times 5), so
60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5.
Two traps to avoid when you grow a factor tree:
-
Keep splitting until every end is prime. A leaf like
6 or 9 is not finished —
it still splits (6 = 2 \times 3). You only stop a branch when
its number is prime.
-
The order you split in doesn't change the answer. Start
24 as 4 \times 6 or as
2 \times 12 — you still end with exactly
2 \times 2 \times 2 \times 3.
Try a fresh number
Here a new number grows its own factor tree, one split at a time, all the way
down to its primes. Step through it, then press Refresh for a different
number and do it again. The highlighted leaves at the bottom are always prime — that is where
each branch stops.
Writing the answer
Once the leaves are found, collect repeats into powers. Three
2s and one 3 become:
24 = 2^3 \times 3
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.
A frog splits 24 as 4 \times 6 first;
a monkey starts with 2 \times 12. They argue about who is right —
but when both keep splitting down to the primes, they land on the very same fingerprint
2 \times 2 \times 2 \times 3. There was nothing to argue about!
See it explained
Sal Khan builds prime factorisations with factor trees from scratch.