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2 (a) Find the Highest Common Factor (HCF) of 60 and 84 - Edexcel - GCSE Maths - Question 3 - 2021 - Paper 2

Question 3

2 (a) Find the Highest Common Factor (HCF) of 60 and 84. (b) Find the Lowest Common Multiple (LCM) of 24 and 40.

Worked Solution & Example Answer:2 (a) Find the Highest Common Factor (HCF) of 60 and 84 - Edexcel - GCSE Maths - Question 3 - 2021 - Paper 2

Step 1

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Answer

To find the HCF of 60 and 84, we first find the prime factorization of both numbers:

60:
- Start with 2: 60 ÷ 2 = 30
- 30 ÷ 2 = 15
- 15 ÷ 3 = 5
- Therefore, the prime factors are: 2² × 3¹ × 5¹
84:
- Start with 2: 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- Therefore, the prime factors are: 2² × 3¹ × 7¹

Now, we find the common factors: 2² and 3¹. The HCF is the product of the lowest powers of all common prime factors:

$HCF = 2^2 × 3^1 = 4 × 3 = 12$

Step 2

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Answer

To find the LCM of 24 and 40, we again start with the prime factorization:

24:
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- So, the prime factors are: 2³ × 3¹
40:
- 40 ÷ 2 = 20
- 20 ÷ 2 = 10
- 10 ÷ 2 = 5
- So, the prime factors are: 2³ × 5¹

For the LCM, we take the highest powers of all prime factors:

Thus, the LCM is: $LCM = 2^3 × 3^1 × 5^1 = 8 × 3 × 5 = 120$

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