Prime Factorisation

The building blocks of numbers

Every whole number bigger than 1 is built by multiplying prime numbers together. Just as molecules are built from atoms, numbers are built from primes. Prime factorisation is the process of breaking a number down into the exact primes that multiply to make it.

This is powerful because of one remarkable fact: every number has only one prime factorisation. This rule is so important it has a grand name — the Fundamental Theorem of Arithmetic. If you need to review what makes a number prime, see prime numbers.

The factor tree method

The easiest way to find prime factors is a factor tree. You split a number into any two factors, then keep splitting until every branch ends in a prime.

Example 1 — Factorise 36.

1. Start with 36. Split it: 36 = 6 × 6.
2. Split each 6: 6 = 2 × 3. The 2s and 3s are prime, so those branches stop.
3. Collect the primes at the ends of the branches: 2, 3, 2, 3.
4. Write them in order: 36 = 2 × 2 × 3 × 3.

Writing the answer with powers

When a prime repeats, we tidy the answer using powers (index form). Instead of writing 2 × 2 × 2, we write 2³.

So from Example 1:

36 = 2 × 2 × 3 × 3 = 2² × 3²

Example 2 — Factorise 24.

1. 24 = 4 × 6.
2. 4 = 2 × 2, and 6 = 2 × 3.
3. Primes collected: 2, 2, 2, 3.
4. In index form: 24 = 2³ × 3.

Number	Factor tree path	Product of primes
8	2 × 4 → 2 × 2 × 2	2³
30	5 × 6 → 5 × 2 × 3	2 × 3 × 5
45	5 × 9 → 5 × 3 × 3	3² × 5
60	6 × 10 → 2 × 3 × 2 × 5	2² × 3 × 5

Notice that even though you could split 60 differently, the final primes are always the same.

Why it doesn't matter where you start

You might begin 36 as 4 × 9 instead of 6 × 6. Try it: 4 = 2 × 2 and 9 = 3 × 3, giving 2 × 2 × 3 × 3 again. Whatever path you take, the primes at the bottom are identical. That guaranteed uniqueness is exactly why prime factorisation is so trusted.

Using prime factors for HCF and LCM

The biggest payoff is a fast method for the highest common factor and lowest common multiple. Write both numbers in prime form:

24 = 2³ × 3 36 = 2² × 3²

• HCF: take the shared primes at their lowest power: 2² × 3 = 12.
• LCM: take every prime at its highest power: 2³ × 3² = 72.

This beats listing factors once numbers get large. There is a full lesson on this in lowest common multiple and highest common factor.

A practice activity

Build a "factor tree forest":

1. Choose five numbers between 20 and 100.
2. Draw a factor tree for each, splitting until every branch ends in a prime.
3. Write each answer in index form (using powers).
4. Swap with a partner and check their trees give the same primes even if they split differently.
5. Challenge: pick two of your numbers and use their prime factors to find their HCF and LCM.

Where this leads

Prime factorisation underpins simplifying fractions, finding common denominators, and the HCF/LCM shortcuts above. Master the factor tree and index form now, and a whole family of number problems will become quick and reliable.