GCSE Maths Practice: factors-and-multiples

Understanding the Highest Common Factor (HCF)

The Highest Common Factor (HCF), also called the Greatest Common Divisor (GCD), is the largest number that divides two or more numbers exactly, leaving no remainder. It represents the greatest shared factor between the given numbers. Finding the HCF is a key GCSE Maths skill used in simplifying fractions, ratio problems, and algebraic expressions.

Step-by-Step Method

1. List all factors of each number. Factors are numbers that divide exactly with no remainder.
2. Identify the common factors that appear in both lists.
3. Choose the largest common one — that is the HCF.

Worked Examples (Different Values)

• Example 1: Find the HCF of 18 and 45.
  Factors of 18: 1, 2, 3, 6, 9, 18.
  Factors of 45: 1, 3, 5, 9, 15, 45.
  Common factors: 1, 3, 9. → HCF = 9.
• Example 2: Find the HCF of 32 and 56.
  Factors of 32: 1, 2, 4, 8, 16, 32.
  Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56.
  Common factors: 1, 2, 4, 8 → HCF = 8.
• Example 3: Find the HCF of 81 and 108.
  81 = 3 × 3 × 3 × 3, 108 = 2 × 2 × 3 × 3 × 3. → Common 3 × 3 × 3 = 27.

Alternative Method 1: Prime Factorisation

1. Break each number into its prime factors.
2. Find the primes they share and multiply them together.

Example: HCF of 24 and 60 → 24 = 2³ × 3, 60 = 2² × 3 × 5. Shared primes = 2² × 3 = 12.

Alternative Method 2: Division (Euclidean) Method

This method works well for larger numbers:

1. Divide the larger number by the smaller.
2. Use the remainder to divide the previous divisor.
3. Continue until the remainder is 0 — the last divisor is the HCF.

Example: Find HCF of 84 and 126 → 126 ÷ 84 = 1 remainder 42 → 84 ÷ 42 = 2 remainder 0 → HCF = 42.

Common Mistakes

• Stopping before listing all factors and missing the largest one.
• Confusing HCF with LCM (Lowest Common Multiple) — HCF finds the biggest shared factor, LCM finds the smallest shared multiple.
• Forgetting to include 1 as a factor of every number.

Real-Life Applications

The HCF is used in real-world maths whenever you need to divide things into equal parts without leftovers. For example:

• Packaging: Grouping items evenly into boxes without any left over.
• Recipes: Reducing ingredient quantities to make smaller but identical batches.
• Fractions: Simplifying fractions like 24⁄60 by dividing top and bottom by 12 (the HCF).

Frequently Asked Questions

Q1: What is the difference between HCF and GCD?
A: They mean exactly the same thing — two different names for the largest common factor.

Q2: Can the HCF ever be 1?
A: Yes. If two numbers share no common factors except 1, they are called co-prime.

Q3: How can I check if my answer is right?
A: The HCF should divide both numbers with no remainder. Quick test: 24 ÷ 12 = 2 and 60 ÷ 12 = 5 → works perfectly.

GCSE Study Tip

When working with larger numbers, use prime factorisation to save time and reduce mistakes. Practise by writing factor trees for several pairs of numbers and circle the common primes.

Summary

The Highest Common Factor (HCF) is the largest number that divides two or more numbers exactly. You can find it by listing factors, prime factorisation, or using division. In this question, the HCF of 24 and 60 is 12. Understanding this method is vital for simplifying fractions, solving ratio problems, and improving overall number fluency in GCSE Maths.