GCSE Maths Practice: factors-and-multiples

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor (GCD), also known as the highest common factor (HCF), is the largest number that divides two or more numbers exactly. In higher-tier GCSE Maths, questions often use numbers with high powers or multiple terms. Recognising how powers of primes behave is crucial for finding the GCD quickly and accurately.

Prime Factorisation and Exponents

Every whole number greater than one can be expressed as a product of prime numbers. When a number includes powers, such as 128 = 2⁷, these exponents make the process easier: simply compare the powers of each common prime factor across the numbers. The lowest power represents how many of that prime all numbers share.

Worked Example (Different Numbers)

Find the GCD of 40, 120, and 160:

40 = 2^3 × 5
120 = 2^3 × 3 × 5
160 = 2^5 × 5
Common primes: 2^3 and 5 → GCD = 2^3 × 5 = 40

This example demonstrates that even when numbers share extra factors (like 3 in 120 or higher powers of 2 in 160), only the smallest power of each shared prime is used for the GCD.

Alternative Method – Euclidean Algorithm for Three Numbers

The Euclidean algorithm works for any number of integers. Start by finding the GCD of the first two numbers, then use that result with the third.

Example: Find the GCD of 72, 108, and 180.
GCD(72,108) = 36
Then GCD(36,180) = 36
Therefore, overall GCD = 36

Common Pitfalls

• Using the highest powers instead of the lowest in prime factorisation.
• Forgetting to include all common primes before multiplying.
• Assuming a number divides another without checking (for example, 128 does not divide 96 exactly).
• Confusing GCD with LCM, which uses highest powers.

Real-Life Applications

The GCD is not just theoretical; it appears in ratio simplification, measurement conversions, and computer algorithms. For example, in digital imaging or data compression, simplifying ratios like 96:128:32 helps reduce file sizes or maintain consistent proportions. Similarly, when cutting cables of different lengths (32 m, 96 m, 128 m), the largest equal sections that leave no remainder are 16 m long—the GCD of those lengths.

FAQ

Q: Why does the GCD of powers of 2 always equal another power of 2?
A: Because powers of 2 have only one prime base. The lowest exponent among them determines the common factor.

Q: How can I check my answer without factorising fully?
A: Divide all given numbers by your GCD. If every result is a whole number, the GCD is correct.

Q: What if all numbers are multiples of a single prime?
A: Then their GCD is that prime raised to the smallest power appearing in any number.

Study Tip

When revising, practise spotting common powers of 2, 3, and 5 by memory. This skill makes prime factorisation faster during timed questions. Remember: for powers, the key phrase is “take the smallest power of each common prime.”

Summary

Finding the GCD of several numbers requires precision and logical reasoning. Whether you use prime factorisation or the Euclidean algorithm, always aim to identify shared primes and take their lowest powers. This technique strengthens number fluency and supports higher-level GCSE topics such as simplifying fractions, algebraic factorisation, and ratio problems.