Whole numbers can look unrelated on the surface while sharing the same deeper building blocks. Prime factorisation exposes those building blocks. It shows how a number is assembled from primes and gives a structural route into factors, multiples, highest common factor, lowest common multiple, square roots and index laws.
The goal is not merely to produce a factor tree. It is to understand why prime decomposition is useful and why the final prime factorisation is essentially unique apart from order.
Prime numbers are multiplicative atoms
A prime number has exactly two positive factors: 1 and itself. Composite numbers can be broken into smaller factors. Repeating that process eventually reaches primes.
For example:
84 = 2 × 42
= 2 × 2 × 21
= 2 × 2 × 3 × 7
= 2² × 3 × 7The prime factorisation of 84 is therefore 2² × 3 × 7.
Different factor trees, same prime structure
You could also begin 84 = 12 × 7, then 12 = 3 × 4, then 4 = 2 × 2. The tree looks different, but the prime factors are still 2, 2, 3 and 7.
This is the important invariant: the route can change, but the prime decomposition does not.
Why prime factorisation is useful
- It reveals every factor relationship inside a number.
- It makes HCF and LCM calculations systematic.
- It helps simplify surds and square roots.
- It connects naturally to indices.
- It helps test divisibility and perfect powers.
Worked example 1: factorise 360
360 = 36 × 10
= (2² × 3²) × (2 × 5)
= 2³ × 3² × 5The index notation compresses repeated prime factors.
Worked example 2: identify a perfect square
A positive integer is a perfect square when every prime exponent in its prime factorisation is even.
900 = 2² × 3² × 5²All exponents are even, so 900 is a perfect square. In fact, √900 = 2 × 3 × 5 = 30.
Worked example 3: simplify a square root
√72 = √(2³ × 3²)
= √(2² × 3² × 2)
= 2 × 3 × √2
= 6√2Prime structure makes the repeated pairs visible.
Divisibility becomes structural
If a number contains at least one factor 2 and one factor 3, then it is divisible by 6. If it contains at least two factors of 2 and one factor of 5, then it is divisible by 20.
Prime factorisation therefore turns divisibility from a memorised list of rules into a multiplicative relationship.
Common errors
Stopping at composite factors. A prime factorisation must end with primes only.
Forgetting repeated factors. 72 is not 2 × 3; it is 2³ × 3².
Treating 1 as prime. 1 has only one positive factor and is not prime.
Assuming different factor trees give different answers. The prime factors are unique apart from order.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| Ends with 6 or 15 | Prime definition weak | Test whether every leaf is prime |
| Misses repeated primes | Factor tracking | Write every branch before compressing with indices |
| Cannot use factorisation later | Procedure without structure | Ask what each exponent says about divisibility and powers |
Practice
- Write 180 as a product of prime factors.
- Write 504 as a product of prime factors.
- Is 1,764 a perfect square? Use prime factors to justify.
- Simplify √200 using prime factors.
Answers
1. 2² × 3² × 5. 2. 2³ × 3² × 7. 3. Yes. 1764 = 2² × 3² × 7², so every prime exponent is even and √1764 = 42. 4. 10√2.
Connected routes
Continue to Highest Common Factor and Lowest Common Multiple Through Structure and Indices: Why the Laws Work. Return to the Mathematics Learning Hub for the wider Secondary Mathematics estate.