Year 9 maths lesson · Factors, Multiples, Primes

HCF and LCM

Find the Highest Common Factor and Lowest Common Multiple of two numbers using prime factorisation and Venn diagrams. The overlap of the Venn diagram gives the HCF; all primes across both circles give the LCM.

  • About 7–9 minutes
  • 8 quick questions as you go
  • Then about 6 minutes of practice
Practice questions → Set for Class

Worked examples

HCF and LCM of 12 and 30

  1. 12 = 2 × 2 × 3 — put the 2×2×3 in the Venn diagram
  2. 30 = 2 × 3 × 5 — put the 2×3×5 in the Venn diagram
  3. Shared primes (overlap): 2, 3 → HCF = 2 × 3 = 6
  4. All primes: 2, 2, 3, 5 → LCM = 2 × 2 × 3 × 5 = 60

HCF of 60 and 90

  1. 60 = 2 × 2 × 3 × 5
  2. 90 = 2 × 3 × 3 × 5
  3. Shared primes (overlap): 2, 3, 5
  4. HCF = 2 × 3 × 5 = 30

LCM of 48 and 72

  1. 48 = 2 × 2 × 2 × 2 × 3
  2. 72 = 2 × 2 × 2 × 3 × 3
  3. Shared primes (overlap): 2, 2, 2, 3
  4. All primes: 2, 2, 2, 2, 3, 3 → LCM = 2⁴ × 3² = 144

Before this lesson

It helps to be comfortable with:

Lesson transcript

Everything the lesson says, word for word

Student: Hey Alex! What are we looking at today?

Teacher: HCF and LCM — highest common factor and lowest common multiple.

Student: Oh yeah, I remember this from last year!

Teacher: Good. Let’s revise. HCF first — think about what the words mean. Highest Common Factor.

Student: So… the highest factor that two numbers have in common?

Teacher: Exactly. Try finding the HCF of 12 and 30.

Student: Okay. Factors of 12 are 1, 2, 3, 4, 6, 12. Factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. Common factors are 1, 2, 3, and 6 — so the highest is 6!

Teacher: Nice work. That method’s fine, but it gets slow for bigger numbers.

Teacher: Now LCM. Same thing — think about the words. Lowest Common Multiple.

Student: The lowest multiple that both numbers share?

Teacher: Right. Try it with 12 and 10.

Student: Okay. Multiples of 12: 12, 24, 36, 48, 60… Multiples of 10: 10, 20, 30, 40, 50, 60… The first one they share is 60. So LCM is 60!

Teacher: Good. But there’s a faster way, using prime factorisations and Venn diagrams.

Teacher: Let’s go back to 12 and 30. First, we need the prime factorisation of each number.

Student: 12 is 2 times 2 times 3. And 30 is 2 times 3 times 5.

Teacher: Now we put these into a Venn diagram. The primes that both numbers share go in the middle.

Teacher: 12 and 30 both have a 2 and a 3 in their factorisation. So 2 and 3 go in the overlap.

Teacher: 12 has an extra 2 that 30 doesn’t share — that goes on 12’s side. And 30 has a 5 that 12 doesn’t — that goes on 30’s side.

Teacher: Let’s take a look at the diagram. If we look at the middle, we can see a 2. That means 2 goes into both 30 and 12. Same with 3 — 3 goes into both 30 and 12. Since 2 and 3 go into them both, so does 6. To find the HCF, we multiply just the primes in the middle. 2 times 3 is 6.

Teacher: And for the LCM? To be a multiple of both numbers, we need all the primes from both. So we multiply everything on the diagram — 2 times 2 times 3 times 5 equals 60.

Student: Oh, that’s way faster than listing everything out!

Teacher: Let’s do a couple more. How about 18 and 24?

Teacher: 18 is 2 times 3 times 3. And 24 is 2 times 2 times 2 times 3.

Teacher: They share a 2 and a 3 in common. 18 has an extra 3, and 24 has two extra 2s.

Teacher: HCF — multiply the middle — 2 times 3 is 6. LCM — multiply everything — 2 times 2 times 2 times 3 times 3 is 72.

Student: I like this method — the Venn diagram makes it so clear!

Teacher: Right, try these questions then.

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