Prime factor decomposition (also called prime factorisation) means writing any whole number greater than 1 as a product of its prime factors. Every integer has exactly one such representation. The result is usually written in index notation: for example, 360 = 2³ × 3² × 5.
What is a prime factor?
A prime number has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 ...
A factor of a number divides into it exactly. A prime factor is a factor that is also a prime number.
Every whole number greater than 1 is either prime itself or can be broken down into prime factors. This is guaranteed by the Fundamental Theorem of Arithmetic — and the breakdown is always unique.
How do you use a factor tree?
A factor tree breaks a number into two factors, then continues breaking each non-prime factor until all branches end in primes. Circle the primes as you go.
Worked example — decompose 60:
- 60 = 2 × 30
- 30 = 2 × 15
- 15 = 3 × 5 (both prime — stop)
Reading off all the circled primes: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
You can also start with different pairs — e.g. 60 = 4 × 15 = (2 × 2) × (3 × 5). The final answer is the same regardless of how you split the number.
Worked example — decompose 360:
- 360 = 2 × 180
- 180 = 2 × 90
- 90 = 2 × 45
- 45 = 3 × 15
- 15 = 3 × 5 (both prime)
All primes: 2, 2, 2, 3, 3, 5 → 360 = 2³ × 3² × 5
How do you use repeated division?
Repeated division (the "ladder method") divides by the smallest prime that goes in, then continues dividing the result.
Worked example — decompose 504:
| Step | Number | Divide by |
|---|---|---|
| 1 | 504 | 2 → 252 |
| 2 | 252 | 2 → 126 |
| 3 | 126 | 2 → 63 |
| 4 | 63 | 3 → 21 |
| 5 | 21 | 3 → 7 |
| 6 | 7 | 7 → 1 (prime) |
Primes used: 2, 2, 2, 3, 3, 7 → 504 = 2³ × 3² × 7
Both methods give the same answer. Use whichever you find clearer.
How do you use prime factors to find the HCF?
The Highest Common Factor (HCF) is found by identifying the prime factors that both numbers share, then multiplying them together using the lower power of each shared prime.
Worked example — find HCF of 60 and 360:
- 60 = 2² × 3 × 5
- 360 = 2³ × 3² × 5
Shared primes: 2 (lower power = 2²), 3 (lower power = 3¹), 5 (lower power = 5¹).
HCF = 2² × 3 × 5 = 4 × 3 × 5 = 60
How do you use prime factors to find the LCM?
The Lowest Common Multiple (LCM) uses the higher power of every prime that appears in either number.
Worked example — find LCM of 60 and 504:
- 60 = 2² × 3 × 5
- 504 = 2³ × 3² × 7
All primes: 2, 3, 5, 7. Take the highest power of each:
LCM = 2³ × 3² × 5 × 7 = 8 × 9 × 5 × 7 = 2520
What mistakes should you avoid?
- Including 1 as a prime factor. The number 1 is not prime. Prime factors start from 2.
- Stopping the factor tree too early. Keep splitting until every branch ends in a prime number — e.g. 4 is not prime (4 = 2²), so it must be split further.
- Forgetting index notation. Writing 60 = 2 × 2 × 3 × 5 is correct, but 2² × 3 × 5 is the expected form in assessments.
Frequently asked questions
What does "product of prime factors" mean?
"Product" means multiplication. "Product of prime factors" means write the number as a series of primes multiplied together. So 60 written as a product of prime factors is 2 × 2 × 3 × 5, or more neatly 2² × 3 × 5 using index (power) notation.
Does it matter which pair I start with in a factor tree?
No — the Fundamental Theorem of Arithmetic guarantees that every integer has a unique prime factorisation. Starting with 60 = 4 × 15 or 60 = 6 × 10 or 60 = 2 × 30 all lead to the same final answer: 2² × 3 × 5.
Can prime factor decomposition be used for numbers larger than 1000?
Yes. The method is the same — factor tree or repeated division. For large numbers, start with the smallest prime (2) and work upward. For very large numbers, checking divisibility rules helps: a number is divisible by 2 if it ends in an even digit, by 3 if its digit sum is divisible by 3, by 5 if it ends in 0 or 5.
Is prime factor decomposition tested at KS3 or GCSE?
Both. At KS3 the focus is on the method and writing the answer correctly. At GCSE, prime factors are used to find HCF and LCM (often presented in Venn diagrams), and the topic connects to indices and algebraic work. It is a foundational topic that rewards mastery early.
For Socratic KS3 number practice including prime factors, see aitutors.me.