--- title: "HCF and LCM" subtitle: "Number Foundations · Squares, powers and primes · Lesson 8" --- # HCF and LCM **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** One bus leaves every 12 minutes, another every 18. Both leave at 8:00. When do they next leave together? Primes answer it without listing every time. :::teacher-note **In one sentence:** Students grow two factor trees, sort the primes in a Venn diagram, and find the HCF (shared primes) and LCM (all the primes) of two numbers, then choose between them in context (AC9M7N02, node 3). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict List the factors of each, or think about their primes. :::keep-together **What is the highest common factor of 12 and 18?** ☐ 2 ☐ 3 ☐ 6 ☐ 36 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 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. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Shared and all :::keep-together ### Two factorisations side by side Here is the method for 12 and 18. 1. $12 = 2^2 \times 3, \quad 18 = 2 \times 3^2$ — Write both prime factorisations. 2. $\text{HCF} = 2 \times 3 = 6$ — Shared primes: each prime in both, to its lower power. 3. $\text{LCM} = 2^2 \times 3^2 = 36$ — All the primes, each to its higher power (shared ones once). 4. $6 \times 36 = 216 = 12 \times 18$ — A check: HCF × LCM = the two numbers multiplied. Now try one: the HCF of 24 and 36. **Find the HCF of 24 and 36.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12 24 = 2³ × 3 and 36 = 2² × 3². Shared: 2 × 2 × 3 = 12. ::: :::keep-together ### Find the mistake **Zoe finds the LCM of 6 and 8. Click the line where her working goes wrong.** ``` 1 Find the lowest common multiple of 6 and 8. 2 6 = 2 × 3 and 8 = 2³ 3 Multiply the two numbers: 6 × 8 = 48 4 So the LCM of 6 and 8 is 48. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** 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. ::: :::keep-together ### 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.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $12 = 2^2 \times 3$ 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. ::: :::typical ## One change at a time **A new pair.** Each task changes one thing. **An LCM.** **In context.** **Nothing shared.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## HCF and LCM :::keep-together ### LCM **Find the LCM of 4 and 10.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 20 4 = 2² and 10 = 2 × 5: LCM = 2² × 5 = 20. ::: :::keep-together ### HCF **Find the HCF of 18 and 30.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6 18 = 2 × 3² and 30 = 2 × 3 × 5. Shared: 2 × 3 = 6. ::: :::keep-together ### A common denominator **To add $\tfrac{1}{6} + \tfrac{1}{8}$, both fractions need the same denominator. What is the smallest one that works?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 24 The LCM of 6 and 8 is 24 (the next lessons use this). ::: :::keep-together ### Review: prime factors **Review: what is the prime factorisation of 50?** ☐ $2 \times 25$ ☐ $5 \times 10$ ☐ $2 \times 5^2$ ☐ $2^2 \times 5$ ::: :::teacher-note **Answer:** $2 \times 5^2$ 50 = 2 × 25 = 2 × 5 × 5 = 2 × 5². ::: ## Make: HCF 6, LCM 36 Check: grow trees for your two numbers on paper and compare their primes. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Find the HCF and LCM of 20 and 30.** {{lines-3}} ::: :::teacher-note **Answer:** 20 = 2² × 5 and 30 = 2 × 3 × 5. HCF = 2 × 5 = 10; LCM = 2² × 3 × 5 = 60. :::