GCSE Maths Practice: factors-and-multiples

Question 9 of 10

This question tests your ability to find the greatest common divisor (GCD) of three numbers using the prime factorisation method.

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\( \begin{array}{l}\text{Find the greatest common divisor of }100,150,\text{ and }200.\end{array} \)

Choose one option:

For three numbers, find the GCD of the first two, then use that result with the third number.

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor (GCD), also called the highest common factor (HCF), is the largest integer that divides two or more numbers exactly with no remainder. In GCSE Higher Maths, this concept extends beyond small numbers — it can involve three or more values, prime factorisation, or even algebraic terms.

Prime Factorisation Method

To find the GCD using prime factorisation:

  1. Break each number into its prime factors.
  2. Identify the prime factors common to all numbers.
  3. Multiply those shared factors using the smallest powers.

Example (different numbers):
Find the GCD of 84, 90, and 126.

84 = 2 × 2 × 3 × 7 = 2^2 × 3 × 7
90 = 2 × 3 × 3 × 5 = 2 × 3^2 × 5
126 = 2 × 3 × 3 × 7 = 2 × 3^2 × 7
Common primes: 2 × 3 = 6
Therefore, GCD = 6

Using the Euclidean Algorithm

This algorithm uses division and remainders:

Step 1: Divide the larger number by the smaller.
Step 2: Replace the larger number with the remainder.
Step 3: Repeat until the remainder is 0.
The last non-zero remainder is the GCD.

For instance, the GCD of 210 and 84:

210 ÷ 84 = 2 remainder 42
84 ÷ 42 = 2 remainder 0
GCD = 42

Common Mistakes

  • Forgetting to use the smallest powers in prime factorisation.
  • Stopping after finding one common factor, not the greatest one.
  • Confusing GCD (based on factors) with LCM (based on multiples).

Real-Life Applications

GCD appears frequently in ratio simplification, scaling problems, and algebraic simplifications. For example, if three containers hold 100 ml, 150 ml, and 200 ml of liquid, the largest equal portion that can be measured using all containers is their GCD — 50 ml.

FAQ

Q: Can we find the GCD of more than two numbers?
A: Yes. Find the GCD of the first two, then use that result with the third number.

Q: Why use prime factorisation instead of listing factors?
A: It’s faster for large numbers and avoids missing hidden factors.

Q: What if all numbers are even?
A: Then at least 2 will always be part of the GCD.

Study Tip

When revising, practise both listing and prime factorisation methods. Higher-level questions may involve three numbers or algebraic terms, so train yourself to quickly spot patterns in powers and primes.

Summary

The GCD is a fundamental tool for simplifying problems involving multiple quantities. Understanding how to calculate it efficiently builds a strong base for algebraic manipulation, simultaneous equations, and proportion reasoning later in GCSE Maths.