Maths
What is prime factorisation?
Quick answer
Prime factorisation means writing a number as a product of prime numbers. For example, 60 = 2 × 2 × 3 × 5, which can be written as 2² × 3 × 5.
In one sentence
Prime factorisation is breaking a number down into the prime numbers that multiply together to make it.
Explained
A prime number has exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13 and so on (see prime numbers). Any other whole number above 1 is composite, which means it can be split into smaller factors.
Every composite number can be written as a product of primes in just one way, apart from the order. Primes are like the building blocks of numbers.
A factor tree helps you find them.
Step by step
- Start with the number.Take 60.
- Split it into any two factors.60 = 6 × 10.
- Keep splitting.6 = 2 × 3 and 10 = 2 × 5.
- Stop at primes.Circle each prime at the end of a branch: 2, 3, 2, 5.
- Write the product.60 = 2 × 2 × 3 × 5.
You could have started with 60 = 4 × 15 or 2 × 30 and you would end with the same primes. That is a good check.
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To write the answer more neatly, use powers (also called indices): 2 × 2 × 3 × 5 = 2² × 3 × 5. The small number says how many times a prime appears. Read more at powers and indices.
Another method is repeated division. Divide by the smallest prime that fits (2, then 3, then 5 …) until you reach 1. For 84: 84 ÷ 2 = 42, 42 ÷ 2 = 21, 21 ÷ 3 = 7, and 7 ÷ 7 = 1. So 84 = 2² × 3 × 7.
Prime factorisation is a handy tool for finding the HCF and LCM. It also helps with simplifying fractions and with square roots, and the idea sits behind the codes that protect information online. See also factors and multiples.
Examples
| Number | Prime factorisation | Index form |
|---|---|---|
| 12 | 2 × 2 × 3 | 2² × 3 |
| 36 | 2 × 2 × 3 × 3 | 2² × 3² |
| 60 | 2 × 2 × 3 × 5 | 2² × 3 × 5 |
| 84 | 2 × 2 × 3 × 7 | 2² × 3 × 7 |
| 100 | 2 × 2 × 5 × 5 | 2² × 5² |
Common mistakes
- Stopping at a composite number. 6 and 10 are not finished; keep going until every number is prime.
- Treating 1 as a prime. It is not.
- Missing a repeated prime. 36 has two 2s and two 3s.
Quick quiz
Prime factorisation
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Question 1 of 4
What is the prime factorisation of 12?
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Related topics
Prime numbers
A prime number is a whole number greater than 1 that has exactly two factors, 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13.
Factors and multiples
Factors are numbers that divide exactly into a number (the factors of 12 are 1, 2, 3, 4, 6 and 12). Multiples are the numbers in its times table (12, 24, 36…).
Highest common factor and lowest common multiple
The HCF is the largest number that divides exactly into two numbers. The LCM is the smallest number that both divide into. For 12 and 18, the HCF is 6 and the LCM is 36.
Powers and indices
A power (also called an index or exponent) is a small raised number showing how many times to multiply a number by itself. So 2⁴ means 2 × 2 × 2 × 2 = 16.
Scientific notation
Scientific notation writes very large or very small numbers as a number from 1 to 10 times a power of 10. For example, 3,500,000 is 3.5 × 10⁶. In the UK it is called standard form.
Square roots
The square root of a number is the value that, multiplied by itself, gives that number. The square root of 25 is 5, because 5 × 5 = 25. The sign is √.