GCSE Maths Practice: factors-and-multiples

Question 3 of 10

This GCSE Maths question tests your understanding of the Highest Common Factor (HCF) — the largest number that divides both numbers exactly, a core concept for simplifying fractions and ratios.

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\( \begin{array}{l}\text{What is the highest common factor (HCF) of 36 and 60?}\end{array} \)

Choose one option:

To find the HCF efficiently, break both numbers into prime factors and multiply the primes they share. This method is quick and accurate for exams.

Understanding the Highest Common Factor (HCF)

The Highest Common Factor (HCF) is the largest number that divides two or more numbers exactly. It’s also called the Greatest Common Divisor (GCD). In GCSE Maths, finding the HCF helps simplify fractions, solve ratio problems, and identify number patterns efficiently.

Step-by-Step Method

  1. List all factors of each number. A factor divides a number with no remainder.
  2. Identify common factors that appear in both lists.
  3. Pick the highest number common to both lists — that is the HCF.

Worked Examples (Different Values)

  • Example 1: HCF of 18 and 45.
    18 → 1, 2, 3, 6, 9, 18.
    45 → 1, 3, 5, 9, 15, 45.
    Common: 1, 3, 9 → HCF = 9.
  • Example 2: HCF of 20 and 28.
    20 → 1, 2, 4, 5, 10, 20.
    28 → 1, 2, 4, 7, 14, 28.
    Common: 1, 2, 4 → HCF = 4.
  • Example 3: HCF of 63 and 84.
    63 = 3 × 3 × 7, 84 = 2 × 2 × 3 × 7. Shared = 3 × 7 = 21 → HCF = 21.

Alternative Method: Prime Factorisation

Prime factorisation is a faster, more reliable method for large numbers.

  1. Break both numbers into their prime factors.
  2. Identify the prime numbers they share.
  3. Multiply those shared primes to find the HCF.

Example: 36 = 2² × 3², 60 = 2² × 3 × 5 → shared = 2² × 3 = 12.

Alternative Method: Euclidean Algorithm

This method works well when numbers are large:

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

Example: 84 ÷ 36 = 2 remainder 12 → 36 ÷ 12 = 3 remainder 0 → HCF = 12.

Common Mistakes

  • Confusing HCF and LCM: HCF is the largest shared factor; LCM is the smallest shared multiple.
  • Missing factors: Students often stop listing too early and miss the largest one.
  • Skipping 1: 1 is always a factor, even if it’s not the HCF.

Real-Life Applications

HCF calculations are used when dividing or grouping things equally. For example:

  • Packaging: If you have 36 pencils and 60 pens, the HCF (12) tells you the largest number of sets you can make with no leftovers.
  • Cooking: Scaling recipes evenly uses the HCF to maintain correct proportions.
  • Fractions: Simplifying 36⁄60 uses the HCF = 12 → 3⁄5 in simplest form.

Frequently Asked Questions

Q1: What if two numbers have no common factors other than 1?
A: They are called co-prime numbers.

Q2: Can I use multiplication tables to find HCF quickly?

A: Yes, for smaller numbers. For larger ones, prime factorisation is faster.

Q3: Why is HCF useful in ratios?

A: Dividing both parts of a ratio by the HCF simplifies it to its lowest terms.

GCSE Study Tip

Always check your HCF by dividing it back into both numbers — if there’s no remainder, you’re correct. Prime factor trees make this process clearer and reduce errors in timed exams.

Summary

The Highest Common Factor (HCF) is the largest number that divides both values exactly. You can find it by listing factors, prime factorisation, or the Euclidean method. In this question, the HCF of 36 and 60 is 12. Mastering this method is essential for simplifying fractions, dividing quantities, and solving ratio problems confidently in GCSE Maths.