Teaching unit · UK Year 11 (Key Stage 4 / GCSE Foundation, ages 15 to 16)

Prime factorisation, HCF and LCM

Writing a number as a product of prime factors in index notation, then using it to find the HCF and LCM

About two lessons of 45 to 60 minutes

Start here · hook

Every whole number greater than 1 has exactly ONE list of prime factors

60 can be built as 6 x 10, or 4 x 15, or 2 x 30, many different ways of multiplying to get there. But broken all the way down to PRIME factors, there is only ever one answer: 60 = 2 x 2 x 3 x 5, however you started. This fact is called the unique factorisation theorem, and it is why prime factorisation is such a powerful tool: it is the one true 'fingerprint' of a number.

That fingerprint, written compactly with index notation (2² x 3 x 5 rather than 2 x 2 x 3 x 5), is also the fastest way to find the HCF (highest common factor) and LCM (lowest common multiple) of two numbers, especially for numbers too large to comfortably list every factor or multiple of by hand.

60 = 2² x 3 x 5the unique prime 'fingerprint' of 60, however you factorise it, always the same
HCF of 60 and 84 without listing every factor84 = 2² x 3 x 7, so the HCF only needs the SHARED primes: 2² x 3 = 12
Two events that repeat every 12 and 18 daysthe LCM (36 days) tells you when they next coincide, found from prime factors
Large numbers in cryptographyreal encryption relies on prime factorisation being easy to build but hard to reverse for huge numbers

Learning objective

What students will be able to do

Students will write a number as a product of its prime factors using index notation, work backwards from a prime factorisation to the number itself, and use two numbers' prime factorisations to find their HCF and LCM.

Success criteria

I can write a number as a product of its prime factors, using index (power) notation for repeated primes.
I can work out a number from its prime factorisation by multiplying the prime powers together.
I can find the HCF of two numbers by taking the LOWEST power of every prime common to both factorisations.
I can find the LCM of two numbers by taking the HIGHEST power of every prime present in either factorisation.
I know that HCF x LCM always equals the product of the two original numbers.

Curriculum anchor

Standards this unit teaches

GCSE Number, Structure & Calculation #4UK GCSE Mathematics (DfE, England)
Prime factorisation, product notation and the unique factorisation theorem
Subject content statement (Department for Education, "GCSE mathematics: subject content and assessment objectives", published 1 November 2013, reference DFE-00233-2013, "Number" section, "Structure and calculation" sub-heading, item 4, https://www.gov.uk/government/publications/gcse-mathematics-subject-content-and-assessment-objectives): students should "use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem". This is standard-type content, so ALL GCSE students (Foundation and Higher tier) are taught and assessed on it.

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Prime number: a whole number greater than 1 with exactly two factors, 1 and itself (2, 3, 5, 7, 11...)
Prime factorisation: writing a number as a product of prime numbers only, e.g. 60 = 2 x 2 x 3 x 5
Index notation: writing a repeated factor with a small raised power, e.g. 2 x 2 x 2 written as 2 cubed (2³)
Unique factorisation theorem: the fact that every whole number greater than 1 has exactly ONE prime factorisation, regardless of how you find it
Highest common factor (HCF): the largest number that divides exactly into two (or more) given numbers
Lowest common multiple (LCM): the smallest number that both (or all) given numbers divide into exactly

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Writing a number as a product of prime factors

Concrete

Divide repeatedly by the smallest prime that fits, until only 1 is left, then collect the primes used with index notation for any that repeat. This always works and always gives the same answer, by the unique factorisation theorem.

For 60: 60 / 2 = 30. 30 / 2 = 15. 15 is not divisible by 2, try 3: 15 / 3 = 5. 5 is itself prime, so stop. Collected primes: 2, 2, 3, 5, written compactly as 2² x 3 x 5.

Prime powers of 2 marked on a number line alongside 60 itself, a reminder that 60's factorisation (2² x 3 x 5) uses only 2 of its 2s, not a higher power like 2³ = 8.

Worked example

Write 90 as a product of its prime factors, using index notation.

90 / 2 = 45.
45 is not divisible by 2. 45 / 3 = 15.
15 / 3 = 5.
5 is prime, stop.
Collected primes: 2, 3, 3, 5, so 90 = 2 x 3² x 5.

Answer: 2 x 3² x 5

Check for understanding, ask

Why does dividing by the SMALLEST prime first still give the correct, unique answer?
How do you know when to stop dividing?

2. Using prime factorisations to find the HCF and LCM

Abstract

Once both numbers are written as prime factorisations, the HCF takes the LOWEST power of every prime that appears in BOTH lists (primes missing from one list are simply not included in the HCF). The LCM takes the HIGHEST power of every prime that appears in EITHER list.

60 = 2² x 3 x 5. 84 = 2² x 3 x 7. Shared primes: 2 (lowest power 2² in both) and 3 (lowest power 3¹ in both); 5 and 7 are not shared, so they are excluded from the HCF. HCF = 2² x 3 = 12. For the LCM, take the highest power of EVERY prime seen anywhere: 2² (from either), 3¹ (from either), 5¹ (only in 60), 7¹ (only in 84). LCM = 2² x 3 x 5 x 7 = 420.

Worked example

36 = 2² x 3² and 90 = 2 x 3² x 5. Find the HCF and LCM of 36 and 90.

HCF: shared primes are 2 (lowest power: 2¹, since 90 only has 2¹) and 3 (lowest power: 3², present in both). HCF = 2 x 3² = 18.
LCM: highest power of every prime present anywhere: 2² (from 36), 3² (from either), 5¹ (only in 90). LCM = 2² x 3² x 5 = 180.

Answer: HCF = 18. LCM = 180.

Check for understanding, ask

If two numbers share NO prime factors at all, what is their HCF?
Why does the LCM always use the HIGHEST power, while the HCF always uses the LOWEST power?

Watch for

Common misconceptions and how to address them

MisconceptionThe HCF is found by taking the HIGHEST shared power, and the LCM by taking the LOWEST, the two rules swapped.

Why it happens: Students confuse which extreme (lowest/highest) belongs to which quantity (common FACTOR vs common MULTIPLE).

How to address it: A factor divides IN, so it cannot be bigger than what both numbers actually share, hence the LOWEST shared power. A multiple is built OUT to cover every prime either number needs, hence the HIGHEST power seen anywhere. Say it out loud every time: 'HCF is lowest, LCM is highest.'

MisconceptionA prime that appears in only ONE of the two factorisations should still be included in the HCF.

Why it happens: Students forget that the HCF only uses primes common to BOTH numbers, not every prime seen anywhere (that is the LCM's job).

How to address it: Before finding the HCF, first list only the primes that appear in BOTH factorisations. Any prime present in just one number cannot be part of a number that divides into both, so it is excluded entirely, not given a power of zero and included.

MisconceptionIndex notation exponents are added when multiplying two numbers with the same base, so 2² x 3² means the same as (2 x 3)⁴.

Why it happens: Students over-apply the 'add the powers' rule for multiplying LIKE bases to a case with DIFFERENT bases.

How to address it: 2² x 3² means (2 x 2) x (3 x 3) = 4 x 9 = 36, two separate calculations multiplied together, not one combined base to the 4th power. The 'add powers' rule only applies when the BASE is the same on both sides, e.g. 2² x 2³ = 2⁵.

Do it together

Guided practice (with answers)

1. Write 40 as a product of its prime factors, using index notation.
Answer: 2³ x 5, because 40 = 2 x 2 x 2 x 5.
2. A number has the prime factorisation 2 x 5². Work out the number.
Answer: 50, because 2 x 5 x 5 = 50.
3. 48 = 2⁴ x 3 and 18 = 2 x 3². Find the HCF of 48 and 18.
Answer: 6, because the shared primes are 2 (lowest power 2¹) and 3 (lowest power 3¹), so HCF = 2 x 3 = 6.
4. 48 = 2⁴ x 3 and 18 = 2 x 3². Find the LCM of 48 and 18.
Answer: 144, because the highest powers seen are 2⁴ (from 48) and 3² (from 18), so LCM = 16 x 9 = 144.
5. For 48 and 18, HCF = 6 and LCM = 144. Check HCF x LCM = 48 x 18.
Answer: Both equal 864 (6 x 144 = 864, and 48 x 18 = 864), confirming HCF x LCM always equals the product of the two numbers.

On their own

Independent practice worksheets

Practise writing prime factorisations in index notation and using them to find the HCF and LCM, with computed, never-wrong answer keys.

Prime Factorisation: Product & Index Notation (Year 11)section 1's skill: factorising, and working backwards from a factorisation HCF & LCM from Prime Factors (Year 11)section 2's skill: combining two factorisations into an HCF and LCM Year 10 versions of both worksheetsthe same skills, freshly generated, for an earlier class or extra review

Reach every student

Differentiation

Support

Use a factor-tree layout on paper for the first few factorisations, circling primes as they are found, before moving to the compact 'divide repeatedly' written method.
Write both factorisations directly underneath each other, prime by prime (a 2s row, a 3s row, a 5s row...), so the shared vs unique primes are visually obvious before touching the HCF/LCM rule.
Start HCF/LCM practice with factorisations already given (not requiring students to factorise from scratch first), isolating the combining step.
Keep early numbers to two-prime factorisations (e.g. 2² x 3) before introducing three or more distinct primes.

Extension

Ask students to prove, using a small example, why HCF x LCM always equals the product of the two original numbers (every prime power is counted exactly once between the lowest-power and highest-power picks).
Extend to three numbers at once: find the HCF and LCM of three given prime factorisations, not just two.
Investigate what happens when the two numbers share EVERY prime factor (one number is a multiple of the other): what are the HCF and LCM then?
Connect prime factorisation to simplifying fractions: the HCF of the numerator and denominator is exactly what you divide out to simplify.

Check it stuck

Assessment: exit ticket

A three-question exit ticket sampling prime factorisation and the HCF/LCM method.

1. Write 72 as a product of its prime factors, using index notation.
Answer: 2³ x 3², because 72 = 2 x 2 x 2 x 3 x 3.
2. 24 = 2³ x 3 and 40 = 2³ x 5. Find the HCF of 24 and 40.
Answer: 8, because the only shared prime is 2, with the same lowest power 2³ in both, so HCF = 2³ = 8.
3. 24 = 2³ x 3 and 40 = 2³ x 5. Find the LCM of 24 and 40.
Answer: 120, because the highest powers seen are 2³, 3¹ (only in 24) and 5¹ (only in 40), so LCM = 8 x 3 x 5 = 120.

For the teacher

Teacher notes and timings

Rough timing across two lessons: Lesson 1 prime factorisation with index notation (section 1), Lesson 2 the HCF/LCM combining method (section 2) plus the exit ticket.
This unit assumes comfort recognising prime numbers and basic factor pairs (Grade 4-7 Number Theory). If the factorisation PROCESS itself, not the HCF/LCM combining rule, is the sticking point, revisit that worksheet first.
Curriculum note: this unit cites the DfE 'GCSE mathematics: subject content and assessment objectives' (2013), item 4 of the 'Number' section's 'Structure and calculation' sub-heading, standard type, so required for every GCSE student regardless of tier. This is deliberately a DEEPER treatment than the Grade 4-7 Number Theory worksheet's brute-force 'list every factor/multiple' method: the GCSE-specific skill is using product (index) notation and deriving the HCF/LCM directly from prime powers, which scales to much larger numbers than listing ever could.
Language to repeat: 'HCF is lowest, LCM is highest'; a prime missing from one factorisation is excluded from the HCF but still included (at whatever power it has) in the LCM.
Present and print both work: use the Print button for a clean handout, or work through the 60 = 2² x 3 x 5 factorisation and the 36/90 HCF-LCM example live with the class.

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