Prime Factorisation: The Factor Tree Method + Examples

Prime factorisation expresses a number as a product of its prime factors. For example 60 = 2 × 2 × 3 × 5, or 2² × 3 × 5 using indices. It is taught in Year 4 and becomes the tool for LCM, HCF, square roots and simplifying fractions.

First, Prime Numbers

A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Two points that are always trap questions:

• 1 is not prime, because it has only one factor.
• 2 is the only even prime.

See the full discussion in prime numbers.

The Factor Tree Method

The standard primary approach.

Example: prime factorisation of 72

1. Divide by the smallest prime that works: 72 ÷ 2 = 36
2. Continue: 36 ÷ 2 = 18
3. Continue: 18 ÷ 2 = 9
4. 9 is not divisible by 2, so move to the next prime: 9 ÷ 3 = 3
5. 3 is prime → stop

The result is 2 × 2 × 2 × 3 × 3 = 2³ × 3²

Always start with the smallest prime. Starting anywhere gives the same final answer — a number's prime factorisation is unique — but the route is longer and more error-prone.

The Repeated Division Method

Tidier for large numbers. Keep dividing by primes until you reach 1:

Collect the divisors: 2³ × 3².

What It Is Actually For

Finding the HCF

Take the common factors with the lowest indices.

72 = 2³ × 3² and 60 = 2² × 3 × 5 → HCF = 2² × 3 = 12

Finding the LCM

Take every factor with the highest index.

LCM = 2³ × 3² × 5 = 360

A memory hook: HCF takes the lowest, LCM takes the highest. More in how to find the LCM and HCF.

Simplifying Surds

√72 = √(2³ × 3²) = √(2² × 3² × 2) = 6√2

Simplifying Fractions

Factors shared by numerator and denominator cancel directly.

The Mistakes That Actually Cost Marks

1. Stopping before every factor is prime. Writing 72 = 8 × 9 and stopping. Neither 8 nor 9 is prime, so both must be broken down further. This is the number one cause.

2. Including 1 as a prime factor. The number 1 never appears in a prime factorisation.

3. Swapping lowest and highest indices when finding the LCM and HCF. If the HCF comes out larger than one of the original numbers, it is definitely reversed — and a child can catch that without help.

4. Miscounting repeated factors. Three 2s written as 2² instead of 2³. Recounting the branches of the tree fixes it.

5. Assuming large numbers must be hard. 1,000 = 2³ × 5³ and takes four steps. What matters is not the size of the number but whether the child knows the primes below 20.