HCF and LCM

Classroom resources

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Students will be able to

• I can find the HCF and LCM of two numbers from their prime factorisations.
• I can explain why the method works.
• I can decide whether a problem needs the HCF or the LCM.

Timing

Predict and Run 8 min · Investigate 15 min · Modify 10 min · Practise and Make 12 min

Before class

Do the Run board yourself first. Students need prime factorisation with exponents (previous lesson).

The one idea: shared primes give the HCF; all the primes, shared ones once, give the LCM. The common mistake is taking the product as the LCM (Zoe's mistake), which counts shared primes twice.

Listing factors and multiples is a fine check, but this lesson aims at the prime method, which the next unit uses for common denominators.

Running the lesson

1. Predict: students who chose 2 or 3 found a common factor but not the highest.
2. Run: ask what is in the middle of the diagram, and why it divides both numbers.
3. Investigate: for the buses, ask what would happen at 6 minutes (the HCF).
4. Modify: for the bags, ask how many apples and pears go in each bag.
5. Make: compare methods; some students list multiples of 6 up to 36.

The hard part (on purpose)

Deciding whether a problem needs the HCF or the LCM. Ask whether the answer should be smaller than the numbers (sharing) or bigger (lining up again) before any calculation.

When students get stuck

Answers to the checks

1. the HCF of 12 and 18: What is the highest common factor of 12 and 18?
   6. 6 divides both 12 and 18, and nothing bigger does. 2 and 3 are common factors but not the highest; 36 is a common multiple.
2. two trees: Grow trees for 36 and 48. Which primes are in the middle of the diagram? Then use the same method on 12 and 18: was your prediction right?
   
   36 = 2² × 3² and 48 = 2⁴ × 3. Shared: 2 × 2 × 3 = 12 (the HCF). All: 2⁴ × 3² = 144 (the LCM).
3. Two factorisations side by side: Find the HCF of 24 and 36.
   12
   24 = 2³ × 3 and 36 = 2² × 3². Shared: 2 × 2 × 3 = 12.
4. Find the mistake: Zoe finds the LCM of 6 and 8. Click the line where her working goes wrong.
   line 3
   Line 3: 6 × 8 counts the shared 2 twice. All the primes, the shared one once: 2³ × 3 = 24. 24 is a multiple of both, and lower than 48.
5. HCF or LCM?: Bus A leaves every 12 minutes and bus B every 18 minutes. Both leave at 8:00. After how many minutes do they next leave together? Write each step with its reason.
   $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$ (prime factorisations, from factor trees); They leave together at common multiples of 12 and 18 (each bus leaves at multiples of its own time); LCM $= 2^2 \times 3^2 = 36$ (the lowest common multiple: all the primes, shared ones once)
   36 minutes, at 8:36: bus A has made 3 trips and bus B 2.
6. A new pair: Grow trees for 24 and 60. Write the HCF and the LCM.
   
   24 = 2³ × 3 and 60 = 2² × 3 × 5. HCF = 2² × 3 = 12. LCM = 2³ × 3 × 5 = 120.
7. An LCM: Find the LCM of 4 and 6.
   12
   4 = 2² and 6 = 2 × 3: LCM = 2² × 3 = 12.
8. In context: 24 apples and 36 pears are packed into identical bags, with nothing left over. What is the greatest number of bags?
   12
   The number of bags must divide 24 and 36: the HCF is 12 (each bag has 2 apples and 3 pears).
9. Nothing shared: Edge case: 7 and 9 share no primes. What are their HCF and LCM?
   HCF 1, LCM 63
   With no shared primes the HCF is 1, and the LCM is the product: 63.
10. LCM: Find the LCM of 4 and 10.
    20
    4 = 2² and 10 = 2 × 5: LCM = 2² × 5 = 20.
11. HCF: Find the HCF of 18 and 30.
    6
    18 = 2 × 3² and 30 = 2 × 3 × 5. Shared: 2 × 3 = 6.
12. A common denominator: To add $\tfrac{1}{6} + \tfrac{1}{8}$, both fractions need the same denominator. What is the smallest one that works?
    24
    The LCM of 6 and 8 is 24 (the next lessons use this).
13. Review: prime factors: Review: what is the prime factorisation of 50?
    $2 \times 5^2$
    50 = 2 × 25 = 2 × 5 × 5 = 2 × 5².
14. HCF 6, LCM 36: Find two numbers, not 6 and 36, whose HCF is 6 and whose LCM is 36. Write how you found them. Then type the smaller of your two numbers.
    $6 = 2 \times 3$ and $36 = 2^2 \times 3^2$ (prime factorisations of the HCF and the LCM); Both numbers contain $2 \times 3$; one more 2 and one more 3 are shared out (the HCF is what both share; the LCM has everything); $2^2 \times 3 = 12$ and $2 \times 3^2 = 18$ (giving both extras to one number gives 6 and 36); $12 \times 18 = 216 = 6 \times 36$ (HCF × LCM is the two numbers multiplied)
    12 and 18: 12 = 2² × 3 and 18 = 2 × 3², HCF 6 and LCM 36. Their product is 216 = 6 × 36.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › A new pair

Open task: any program that does what the step asks.

one change at a time › An LCM

Open task: any program that does what the step asks.

one change at a time › In context

Open task: any program that does what the step asks.

one change at a time › Nothing shared

Open task: any program that does what the step asks.

Stretch and support

Stretch: Check that HCF × LCM = the two numbers multiplied for three pairs. Explain why with the Venn diagram.

Support: Use the board's Venn diagram for every pair, and list factors or multiples as a check.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "HCF and LCM".

The idea: LCM and GCD from prime factorisations: I can Students use prime factorisations to find the lowest common multiple and greatest common divisor of pairs of natural numbers.

What understanding looks like:
- Correctly finds LCM and GCD for a given pair using prime factorisations.
- Can explain why the method works (not just apply it).
- Applies LCM or GCD appropriately when presented with a contextual problem.

Mistakes students often make here:
- A student multiplies the two numbers for the LCM (Zoe's mistake).
- A student gives a common factor that is not the highest.
- A student cannot tell whether a problem needs the HCF or the LCM.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Prime factorisation via factor trees), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "HCF and LCM".

Exit ticket

Find the HCF and LCM of 20 and 30.

20 = 2² × 5 and 30 = 2 × 3 × 5. HCF = 2 × 5 = 10; LCM = 2² × 3 × 5 = 60.

What each step is for

• the HCF of 12 and 18: Committing to an HCF shows whether they want the highest common factor or any common factor, and whether they confuse it with a multiple.
• two trees: Finishing both trees sorts the primes into shared and not shared, and the board multiplies them into the HCF and the LCM.
• shared and all: Shared primes give the HCF and all the primes the LCM; Zoe multiplies the numbers, which counts the shared 2 twice. Context decides which one a problem needs.
• one change at a time: One change at a time: a new pair on the board, an LCM, an HCF in context, then numbers that share no primes.
• HCF and LCM: HCF, LCM and a common denominator, mixed; the product as the LCM is the planted slip; the review keeps prime factorisation fresh.
• HCF 6, LCM 36: Working backwards from an HCF and an LCM to the two numbers uses the primes in both directions, and the product check (HCF × LCM) ties them together.

Where it leads

Next: the fractions unit, where the LCM of two denominators is the lowest common denominator (AC9M7N06).

Where it leads: simplifying fractions in one step (dividing by the HCF), and algebraic HCFs when factorising.