Prime Factorisation (Prime Factorization): Breaking Numbers into Prime Building Blocks

In one line: Prime factorisation means writing a whole number as a multiplication of only prime numbers, such as 12 = 2 × 2 × 3. A prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole number that divides into another with no remainder.

Why it works

Prime factorisation shows exactly what a number is made of, which makes many other jobs easier. It helps you find the highest common factor (the biggest number that divides two numbers) and the lowest common multiple (the smallest number both divide into). These are used to simplify fractions and add fractions with different denominators. It also helps with square roots and spotting patterns in numbers. Every whole number greater than 1 has exactly one prime factorisation, if you ignore the order, so it acts like a number's unique fingerprint.

Think of it like: Think of prime numbers as basic building bricks, like single-colour LEGO bricks that cannot be split into smaller bricks. Every whole number above 1 can be built by snapping prime bricks together with multiplication. For example, 30 is built from one 2-brick, one 3-brick and one 5-brick (2 × 3 × 5 = 30). Prime factorisation is like taking a model apart until you are left with only the basic bricks, then listing them. (Where this breaks: Real bricks can be stacked into many different shapes, but with numbers the order does not matter: 2 × 3 × 5 and 5 × 3 × 2 are the same factorisation. Also, the bricks only join by multiplying, never by adding. And the number 1 is not a brick at all, because it is not prime and multiplying by 1 changes nothing.)

Worked example

Find the prime factorisation of 360 using a factor tree, and write it using index notation (powers).

Split 360 into two factors, starting with the smallest prime that divides it: 360 = 2 × 180. Circle the 2 because it is prime.
Keep splitting any number that is not prime: 180 = 2 × 90, then 90 = 2 × 45. Circle each 2.
45 is odd, so try the next prime, 3: 45 = 3 × 15, then 15 = 3 × 5. Circle 3, 3 and 5, which are all prime.
Collect every circled prime: 360 = 2 × 2 × 2 × 3 × 3 × 5.
Write repeated primes as powers (index notation, where the small raised number shows how many times a number is multiplied by itself): 360 = 2³ × 3² × 5.
Check by multiplying back: 8 × 9 × 5 = 72 × 5 = 360. Correct.

Answer: 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5

Common mistakes

Including 1 as a prime factor (for example writing 1 × 2 × 3 for 6); remember 1 is not prime, so leave it out.
Stopping at a number that is not prime, such as writing 36 = 4 × 9; keep splitting until every branch ends in a prime (36 = 2 × 2 × 3 × 3).
Forgetting to copy every circled prime from the factor tree into the final answer; check by multiplying your primes back together to see if you get the original number.
Mixing up the powers, such as writing 2 × 3 when you mean 2³; the small raised number counts how many times the prime appears, so 2³ means 2 × 2 × 2.

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Which of these is the prime factorisation of 12?

Questions parents ask

What year do kids learn prime factorisation?

In the UK, children meet prime numbers in Year 5 and usually write numbers as products of prime factors in Year 6 to Year 8. In Australia it is commonly taught in Year 7, often with index notation. In the US, prime factorization usually appears in Grade 6, where it is used to find the greatest common factor and least common multiple.

How can I help my child with prime factorisation at home?

Make sure they know the first primes (2, 3, 5, 7, 11, 13) and their times tables well. Practise drawing factor trees together for numbers like 24, 60 or 100, and always finish by multiplying the primes back to check. Pointing out that different trees for the same number always end with the same primes builds confidence.

Is 1 a prime number?

No. A prime number must have exactly two different factors, 1 and itself, and 1 has only one factor. Leaving 1 out also keeps every number's prime factorisation unique.

Is it 'factorisation' or 'factorization'?

Both are correct and mean the same thing. 'Factorisation' is the usual British and Australian spelling, while 'factorization' is the American spelling, so your child may see either in books and online.