Prime factors: factor trees, HCF and LCM

What does prime factorisation mean?

Prime factorisation means breaking a number down until every piece is a prime number, then writing it as those primes multiplied together. It needs three words children meet at primary school:

• A factor divides into a number exactly. The factors of 12 are 1, 2, 3, 4, 6 and 12.
• A prime number has exactly two factors: 1 and itself. The primes below 20 are 2, 3, 5, 7, 11, 13, 17 and 19. 1 is not prime, because it has only one factor.
• A composite number has more than two factors, so it can be split up further. 12 is composite.

Prime factors are the factors that are also prime. The prime factors of 12 are 2 and 3, and its prime factorisation is 12 = 2 × 2 × 3.

Every whole number above 1 has only one prime factorisation, whichever way you find it (apart from the order the primes are written in). The key stage 3 curriculum calls this the unique factorisation property. It's what makes prime factors so useful: they're like a number's fingerprint.

How to find prime factors with a factor tree

A factor tree is the usual way to find prime factors. Write the number at the top, split it into any factor pair, and keep splitting each branch until it ends in a prime. Circle the primes as you go: they are the ends of the branches.

Worked example: the prime factors of 60

1. Start with 60 and split it into a factor pair you know: 6 × 10 = 60.
2. 6 isn't prime, so split it: 2 × 3 = 6. Both are prime, so circle them.
3. 10 isn't prime either: 2 × 5 = 10. Circle both.
4. Collect the circled primes, smallest first: 2, 2, 3 and 5.

So 60 = 2 × 2 × 3 × 5. Check by multiplying back: 2 × 2 = 4, then 4 × 3 = 12, then 12 × 5 = 60.

Worked example: the prime factors of 84

1. Split 84: 4 × 21 = 84.
2. Split 4: 2 × 2 = 4.
3. Split 21: 3 × 7 = 21.
4. The primes at the ends are 2, 2, 3 and 7.

So 84 = 2 × 2 × 3 × 7. A different start gives the same answer. Starting with 2 × 42 = 84, then 6 × 7 = 42, then 2 × 3 = 6, you still end with 2, 2, 3 and 7. That's the unique factorisation property at work, so your child can start with whichever pair they spot first.

Good times tables make factor trees much quicker, because every split is a times table fact read backwards. The times table grid is a handy way to spot factor pairs.

Writing prime factorisation in index form

Index form uses powers to write repeated primes more neatly. The small raised number says how many times the prime is multiplied: 2² means 2 × 2, and 3³ means 3 × 3 × 3.

So 60 = 2² × 3 × 5, and 84 = 2² × 3 × 7.

For a bigger number, 360 splits as 36 × 10 = 360. Then 36 is 6 × 6, and each 6 is 2 × 3, while 10 is 2 × 5. That gives three 2s, two 3s and one 5, so 360 = 2³ × 3² × 5. Check: 2³ is 8 and 3² is 9, and 8 × 9 = 72, then 72 × 5 = 360. The laws of indices explain powers in more detail.

How to find the HCF using prime factorisation

The highest common factor (HCF) is the biggest number that divides into both numbers. To find it, write both prime factorisations and multiply the primes they share.

Worked example: the HCF of 60 and 84

• 60 = 2 × 2 × 3 × 5
• 84 = 2 × 2 × 3 × 7

Both have 2, 2 and 3. The 5 and the 7 aren't shared. So the HCF is 2 × 2 × 3, which is 12. Check: 60 ÷ 12 = 5 and 84 ÷ 12 = 7.

In index form, take each prime that appears in both numbers, with the smaller power. 60 and 84 both have 2² and 3, so the HCF is 2² × 3 = 12.

The HCF is how you simplify fractions in one step: dividing the top and bottom of ⁶⁰⁄₈₄ by 12 gives ⁵⁄₇.

How to find the LCM using prime factorisation

The lowest common multiple (LCM) is the smallest number that both numbers divide into. To find it, take every prime that appears in either number, each with its bigger power, and multiply them.

Worked example: the LCM of 60 and 84

60 = 2² × 3 × 5 and 84 = 2² × 3 × 7. The primes involved are 2, 3, 5 and 7. The biggest power of 2 is 2², and 3, 5 and 7 each appear once. So the LCM is 2² × 3 × 5 × 7, which is 420. Check: 420 ÷ 60 = 7 and 420 ÷ 84 = 5.

Worked example: buses every 18 and 24 minutes

Two buses leave together, one every 18 minutes and one every 24 minutes. When do they next leave together? That's the LCM.

• 18 = 2 × 3 × 3, which is 2 × 3²
• 24 = 2 × 2 × 2 × 3, which is 2³ × 3

Take the bigger powers: 2³ × 3², which is 8 × 9 = 72. They leave together again after 72 minutes. Their HCF is 2 × 3 = 6.

Common multiples also give the common denominator when adding and subtracting fractions.

When do children learn prime factors?

Children learn the words prime number, prime factor and composite number in Year 5 maths, and recall the primes up to 19. In Year 6 maths they find common factors and common multiples, and link common factors to equivalent fractions. Writing numbers as a product of prime factors, and using it for the HCF and LCM, is in the key stage 3 programme of study, and it comes up again at GCSE.

Key stage 3 covers Years 7 to 9 as one stage, and each school decides the order it teaches topics in. If you want to know when your child will meet prime factorisation, ask the school for its scheme of work or curriculum map. The KS3 maths guide covers the rest of the key stage.

Common mistakes with prime factors

• Counting 1 as prime. 1 never goes in a factor tree's answer.
• Stopping too soon. A branch ending in 4, 9 or 15 isn't finished. Every end must be prime.
• Losing a prime. Writing 60 = 2 × 3 × 5 forgets one of the 2s. Multiplying back catches this: that only makes 30.
• Misreading powers. 2³ means 2 × 2 × 2, which is 8, not 2 × 3.
• Mixing up HCF and LCM. The HCF can't be bigger than the smaller number. The LCM can't be smaller than the bigger number.

How to help at home

• Keep times tables quick. Every step of a factor tree is a times table fact backwards. The times tables pages have practice for each one.
• Play "prime or not?" Call out numbers under 50 and ask whether they're prime. If not, ask for a factor pair.
• Use real timings. Two lights flashing every 6 and 8 seconds, or two clubs meeting every 2 and 3 weeks, are LCM questions.

Prime factorisation questions to try

1. Write 36 as a product of its prime factors, in index form.
2. Write 90 as a product of its prime factors.
3. Is 91 a prime number?
4. Find the HCF of 36 and 90.
5. Find the LCM of 36 and 90.
6. Write 100 as a product of its prime factors, in index form.

Answers

1. 2² × 3² (36 is 2 × 2 × 3 × 3)
2. 2 × 3 × 3 × 5, or 2 × 3² × 5
3. No: 7 × 13 = 91
4. 18 (both share 2 and 3², and 2 × 9 = 18)
5. 180 (2² × 3² × 5, and 36 × 5 = 180)
6. 2² × 5² (100 is 2 × 2 × 5 × 5)

Prime factors lead on to ratio, where the HCF is used to simplify, and to factorising in algebra, which also means finding common factors, this time of expressions.