Teaching unit · UK Year 7 (Key Stage 3, ages 11 to 12)

Factors, multiples and prime factorisation

Prime numbers, writing a number as a product of its prime factors, and finding the highest common factor and lowest common multiple of 2 numbers

About two lessons of 45 to 60 minutes

Student view

Start here · hook

Every whole number has 1 true 'recipe' of primes

Simplifying a fraction, finding when 2 repeating events line up again (like 2 flashing lights on different cycles), and packing items into equal-sized boxes with nothing left over all rely on the exact same underlying idea: breaking a number down into its factors.

Prime numbers are the 'atoms' of arithmetic. Every whole number greater than 1 breaks down into a UNIQUE product of prime numbers (the unique factorisation property), and once that breakdown is known, finding the highest common factor (HCF) or lowest common multiple (LCM) of any 2 numbers becomes mechanical rather than a guessing game.

60 = 2 x 2 x 3 x 5every composite number has exactly 1 prime 'recipe', however it is found
Simplifying 18/24 to 3/4dividing top and bottom by their HCF, 6
2 traffic lights, 1 changing every 4 seconds and 1 every 6 secondsthey change together again after the LCM, 12 seconds
Packing 18 pencils and 24 rulers into identical boxes with nothing left overthe biggest possible box size uses the HCF, 6

Learning objective

What students will be able to do

Students will identify prime numbers, write any whole number as a product of its prime factors using index notation, and find the highest common factor (HCF) and lowest common multiple (LCM) of 2 numbers.

Success criteria

I can explain what makes a number prime, and recognise that 1 is neither prime nor composite.
I can write a number as a product of its prime factors using a factor tree or repeated division.
I can write a prime factorisation using index (power) notation, e.g. 12 = $2^{2}$ x 3.
I can find the highest common factor (HCF) of 2 numbers.
I can find the lowest common multiple (LCM) of 2 numbers.

Curriculum anchor

Standards this unit teaches

KS3 Maths: NumberUK National Curriculum (England)
Number
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Number" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property".

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 2 factors, itself and 1 (e.g. 2, 3, 5, 7, 11)
Composite number: a whole number greater than 1 with MORE than 2 factors (i.e. not prime)
Factor: a whole number that divides exactly into another number, with no remainder
Multiple: the result of multiplying a number by a whole number (the multiples of 4 are 4, 8, 12, 16...)
Highest common factor (HCF): the largest whole number that is a factor of 2 (or more) given numbers
Lowest common multiple (LCM): the smallest whole number that is a multiple of 2 (or more) given numbers
Prime factorisation: writing a number as a product of prime numbers only, unique for every whole number greater than 1

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. Prime numbers and prime factorisation

Concrete

A prime number has exactly 2 factors: 1 and itself. 7 is prime (only 1 x 7 makes it). 8 is not prime (1 x 8, but also 2 x 4), so 8 is called composite. 1 is a common trap: it only has 1 factor (itself), not 2, so 1 is neither prime nor composite.

To break a composite number down into its prime factors, divide repeatedly by the smallest prime that fits, writing down each division, until only 1 is left. Repeated prime factors are then written compactly with index (power) notation, so 2 x 2 becomes $2^{2}$ .

12 shown as a 3 x 4 rectangle, 1 way to see 12 = 3 x 4. Broken down all the way into PRIMES, 12 = 2 x 2 x 3 =

2^{2}

x 3.

Worked example

Write 60 as a product of its prime factors, using index notation.

60 ÷ 2 = 30 (2 is the smallest prime that divides 60).
30 ÷ 2 = 15 (still divisible by 2).
15 ÷ 3 = 5 (15 is not divisible by 2, but is by 3).
5 is already prime, so stop.

Answer: 60 = 2 x 2 x 3 x 5 = $2^{2}$ x 3 x 5.

Check for understanding, ask

Why is 1 neither a prime nor a composite number?
Write 18 as a product of its prime factors using index notation.

2. Highest common factor and lowest common multiple

Abstract

The HCF of 2 numbers is the biggest number that divides exactly into both. The LCM is the smallest number that both original numbers divide exactly into. Both can be found either by listing factors/multiples directly, or, faster for bigger numbers, from the prime factorisations.

From prime factorisations: the HCF takes the LOWEST power of each prime that appears in BOTH numbers; the LCM takes the HIGHEST power of every prime that appears in EITHER number. There is also a shortcut linking the two: HCF x LCM always equals the product of the original 2 numbers.

The multiples of 4, up to 24.

The multiples of 6, up to 24. Comparing the 2 number lines: 12 and 24 are common multiples of 4 and 6, and 12 is the SMALLEST one, so the LCM of 4 and 6 is 12.

Worked example

Find the HCF and LCM of 18 and 24.

Prime factorise both: 18 = 2 x $3^{2}$ , and 24 = $2^{3}$ x 3.
HCF: take the lowest power of each shared prime. Both share 2 and 3: lowest power of 2 is $2^{1}$ , lowest power of 3 is $3^{1}$ . HCF = 2 x 3 = 6.
LCM: take the highest power of every prime appearing in either number: highest power of 2 is $2^{3}$ , highest power of 3 is $3^{2}$ . LCM = 8 x 9 = 72.
Check with the shortcut: HCF x LCM = 6 x 72 = 432, and 18 x 24 = 432. They match.

Answer: HCF(18, 24) = 6. LCM(18, 24) = 72.

Check for understanding, ask

List the factors of 12 and of 18, then find their HCF directly from the lists.
Why does HCF x LCM always equal the product of the original 2 numbers?

Watch for

Common misconceptions and how to address them

Misconception1 is a prime number.

Why it happens: 1 only divides evenly by itself, which feels 'prime-like', and 1 is often the first number encountered when listing factors.

How to address it: Count 1's factors precisely: only 1 itself, that is just ONE factor, not the 2 factors (1 and itself, as 2 DIFFERENT numbers) that the definition of prime requires. 1 is a special case, neither prime nor composite.

MisconceptionThe HCF of 2 numbers must be one of the 2 original numbers.

Why it happens: Students expect the 'highest' common thing to be one of the numbers being compared, rather than a separate, smaller number.

How to address it: Work a concrete counterexample: HCF(18, 24) = 6, and 6 is neither 18 nor 24. The HCF is the biggest shared FACTOR, which is always less than or equal to the smaller original number, not equal to either original number (unless 1 number is a factor of the other).

MisconceptionThe LCM of 2 numbers is always found by multiplying them together.

Why it happens: This shortcut genuinely works when the 2 numbers share no common factors (are coprime), so students overgeneralise it to every pair of numbers.

How to address it: Show the counterexample directly: LCM(4, 6) is 12, not 4 x 6 = 24, because 4 and 6 share a common factor of 2. Multiplying always works ONLY when the HCF of the 2 numbers is 1.

Misconception'Factor' and 'multiple' mean the same thing.

Why it happens: Both words describe a relationship between 2 numbers connected by multiplication, so the direction is easy to blur.

How to address it: Anchor the direction with a fixed phrase: factors are numbers that divide INTO a given number (factors of 12 are smaller than or equal to 12); multiples are what a given number divides INTO (multiples of 12 are 12 or bigger).

Do it together

Guided practice (with answers)

1. Is 51 a prime number? Explain.
Answer: No, because 51 = 3 x 17, so it has factors other than 1 and itself.
2. Write 36 as a product of its prime factors, using index notation.
Answer: $2^{2}$ x $3^{2}$ , because 36 = 2 x 18 = 2 x 2 x 9 = 2 x 2 x 3 x 3.
3. Find the HCF of 20 and 30.
Answer: 10, because the factors of 20 are 1, 2, 4, 5, 10, 20, and the factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30; the largest number in both lists is 10.
4. Find the LCM of 6 and 8.
Answer: 24, because the multiples of 6 are 6, 12, 18, 24..., and the multiples of 8 are 8, 16, 24...; the smallest number in both lists is 24.
5. List all the common factors of 16 and 24.
Answer: 1, 2, 4, 8, because those are exactly the numbers appearing in both the factor list of 16 (1, 2, 4, 8, 16) and of 24 (1, 2, 3, 4, 6, 8, 12, 24).
6. Using HCF(15, 25) = 5, find LCM(15, 25) with the shortcut.
Answer: 75, because HCF x LCM = 15 x 25 = 375, so LCM = 375 / 5 = 75.

On their own

Independent practice worksheets

Practise prime factorisation and HCF/LCM with computed, never-wrong answer keys.

Year 7 prime factorisation worksheetthe core skill from section 1 Year 7 HCF & LCM worksheetboth direct listing and the prime-factorisation method from section 2

Reach every student

Differentiation

Support

Keep a printed list of primes up to 50 visible until quick recall builds up.
Use a hundred square to shade out multiples of 2, 3, 5 and 7 (a simple sieve) so the remaining unshaded numbers are visibly the primes.
For HCF/LCM, start with the direct listing method (list every factor or the first several multiples) before introducing the faster prime-factorisation shortcut.
Use only 2-digit numbers with small prime factors (2, 3, 5) until the factor-tree process is fluent.

Extension

Find the HCF and LCM of 3 numbers at once, extending the prime-factorisation method (lowest/highest power across all 3).
Investigate the largest known primes and why they are hard to find, connecting to real-world uses of prime numbers in cryptography.
Prove that there are infinitely many primes is too advanced for this stage, but investigate: can you always find a prime bigger than any prime you are given? Try it for a few examples.
Use prime factorisation to quickly determine whether a fraction like 91/119 simplifies, and by how much, without a calculator.

Check it stuck

Assessment: exit ticket

A three-question exit ticket sampling prime recognition, prime factorisation, and HCF/LCM.

1. Explain why 1 is not a prime number.
Answer: A prime number needs exactly 2 factors (1 and itself); 1 only has 1 factor (itself), so it is neither prime nor composite.
2. Write 84 as a product of its prime factors, using index notation.
Answer: $2^{2}$ x 3 x 7, because 84 = 2 x 42 = 2 x 2 x 21 = 2 x 2 x 3 x 7.
3. Find the HCF and LCM of 12 and 18.
Answer: HCF = 6, LCM = 36, because 12 = $2^{2}$ x 3 and 18 = 2 x $3^{2}$ , so HCF = 2 x 3 = 6 and LCM = $2^{2}$ x $3^{2}$ = 36 (check: 12 x 18 = 216 = 6 x 36).

For the teacher

Teacher notes and timings

Rough timing across 2 lessons: Lesson 1 primes and prime factorisation (section 1), Lesson 2 HCF and LCM, both by direct listing and via prime factorisation (section 2), plus the exit ticket.
Every prime factorisation in this unit and its matching worksheet comes from real trial division, verified by reconstructing the original number from the claimed factors (see tests/uks3math6.test.ts), never a hand-typed answer.
The 2-number-line figure in section 2 deliberately uses small, friendly numbers (4 and 6) distinct from the main worked example (18 and 24), so the visual pattern-spotting of common multiples stays uncluttered while the worked example shows the faster prime-factorisation method needed for bigger numbers.
The unique factorisation property (every whole number greater than 1 has exactly 1 prime factorisation, regardless of which primes are tried first or in what order) is worth stating explicitly on the board: it is why 'the' prime factorisation is a meaningful, well-defined thing to ask for.

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