GCSE Maths Practice: factors-and-multiples

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor (GCD) or highest common factor (HCF) is the largest positive integer that divides two or more numbers exactly without leaving a remainder. In GCSE Higher Maths, students often use prime factorisation to find the GCD efficiently, especially for larger numbers that would be tedious to factor by listing.

How Prime Factorisation Works

Prime factorisation breaks a number down into its building blocks—prime numbers. Every integer greater than 1 can be expressed as a product of primes. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. To find the GCD of two numbers:

1. Write both numbers as products of prime factors.
2. Identify all prime factors common to both numbers.
3. Use the lowest powers of those shared primes.
4. Multiply them together to get the GCD.

Worked Example (Different Numbers)

Find the GCD of 180 and 168:

180 = 2 × 2 × 3 × 3 × 5 = 2^2 × 3^2 × 5
168 = 2 × 2 × 2 × 3 × 7 = 2^3 × 3 × 7
Common primes: 2^2 × 3 = 12
Therefore, GCD = 12

This method ensures no factors are missed and is ideal for numbers with many divisors.

Alternative Method – Euclidean Algorithm

When numbers are large, repeatedly dividing can be faster than listing or prime factorisation:

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

Example: Find the GCD of 270 and 198.

270 ÷ 198 = 1 remainder 72
198 ÷ 72 = 2 remainder 54
72 ÷ 54 = 1 remainder 18
54 ÷ 18 = 3 remainder 0
GCD = 18

Common Mistakes

• Multiplying all common primes without taking the smallest powers.
• Forgetting that 1 divides every number and is never the greatest common divisor unless no other factors exist.
• Mixing up GCD (factors) with LCM (multiples).

Applications in Real Life

The GCD is useful in reducing fractions, comparing ratios, and solving problems involving equal groupings. For example, if you have ribbons of 120 cm and 300 cm, and want to cut them into the largest equal pieces, the GCD (60 cm) tells you the maximum length each piece can be without any waste.

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 next number.

Q: Why is the smallest power of each prime used?
A: Because higher powers would not divide both numbers exactly.

Q: What happens if the numbers are coprime?
A: If they share no common prime factors, their GCD is 1.

Study Tip

Practise finding GCDs using both methods: prime factorisation for understanding and the Euclidean algorithm for speed. In exams, always show prime powers clearly to avoid errors.

Summary

Finding the GCD by prime factorisation is a key skill in GCSE Higher Maths. It strengthens understanding of number structure and supports more advanced skills like simplifying algebraic fractions, working with ratios, and identifying patterns in sequences. Once mastered, this topic makes later areas of mathematics much easier to handle.