1. Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

2. Express 360 as the product of prime factors
What happens if you start with a different pair of factors, for example, 30 and 12?
Start with any factor pair of 360 36 10
Continue with factor pairs 12 3 2 5 2 6
Circle the prime factors 2 3
𝟑𝟔𝟎=𝟐×𝟐×𝟐×𝟑×𝟑×𝟓
= 𝟐 𝟑 × 𝟑 𝟐 ×𝟓

3. Draw factor trees for both numbers
Find the HCF of 180 and 630
180
Draw factor trees for both numbers
630 18 10 63 10 6 3 2 5 9 7 2 5 2 3 3 3
18𝟎=𝟐×𝟐×𝟑×𝟑×𝟓
= 𝟐 𝟐 × 𝟑 𝟐 ×𝟓
𝟔𝟑𝟎=𝟐×𝟑×𝟑×𝟓×𝟕
=𝟐× 𝟑 𝟐 ×𝟓×𝟕

4. Find the HCF of 180 and 630 180 630 2 2 7 3 3 5 18𝟎=𝟐×𝟐×𝟑×𝟑×𝟓
= 𝟐 𝟐 × 𝟑 𝟐 ×𝟓
𝟔𝟑𝟎=𝟐×𝟑×𝟑×𝟓×𝟕
=𝟐× 𝟑 𝟐 ×𝟓×𝟕
Organise the factors into a Venn diagram
180
630
Multiply common factors to find the HCF: 2 2 7 3 3
Place common factors in the overlap 5
𝑯𝑪𝑭=𝟐×𝟑×𝟑×𝟓
=𝟗𝟎

5. Multiply all factors to find the LCM:
Find the LCM of 180 and 630
18𝟎=𝟐×𝟐×𝟑×𝟑×𝟓
= 𝟐 𝟐 × 𝟑 𝟐 ×𝟓
𝟔𝟑𝟎=𝟐×𝟑×𝟑×𝟓×𝟕
=𝟐× 𝟑 𝟐 ×𝟓×𝟕
180
630
Multiply all factors to find the LCM: 2 2 7 3 3 5
𝑳𝑪𝑴=𝟐×𝟐×𝟑×𝟑×𝟓×𝟕
=𝟏𝟐𝟔𝟎

6. Problems in context
Katie is having a birthday party. She buys 36 cupcakes and 84 party rings. Katie wants to make sure that everyone at her party gets the same amount of food. What is the greatest number of people Katie can invite?
Solution: the greatest number of people she can invite is 11 (herself and 11 guests makes 12 which is the HCF of 36 and 84).

7. Problems in context
Two security lights are activated at the same time. One flashes every 20 seconds. The other flashes every 45 seconds. After how long will they both flash at the same time?
Solution: after 180 seconds/3 minutes which is the LCM of 20 and 45.

8. Extension
Susan is working out the HCF and LCM of two integers. The HCF of her two numbers is 21. To work out the LCM, she uses the calculation 2× 3 3 ×7 What could the two numbers be?
Solution: 42 and 189, or 21 and 378, or 63 and 126.
This can be seen by considering the possible positioning of each factor in the Venn diagram.