Factors And Multiples Quizzes

Factors and Multiples Quiz – GCSE Maths Foundation

Difficulty: Foundation

Curriculum: GCSE

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Greatest Common Divisor (GCD) & HCF Quiz – Higher GCSE Maths

Difficulty: Higher

Curriculum: GCSE

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Visual overview of Factors And Multiples.

Introduction

Understanding factors and multiples is a core foundation of GCSE Maths. These concepts underpin divisibility, prime numbers, algebraic simplification, and problem-solving in many other areas. A strong grasp of factors and multiples helps students simplify fractions, find common denominators, and solve number-pattern or sequence questions efficiently.

This topic includes identifying factors and multiples, finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM), and using prime factorisation to break numbers into their building blocks. These skills are vital in both Foundation and Higher papers and appear frequently in exam questions.

Core Concepts

What Are Factors?

A factor is a number that divides another number exactly, leaving no remainder.

Examples

Factors of 12 are 1, 2, 3, 4, 6, and 12.
Factors of 20 are 1, 2, 4, 5, 10, and 20.

To find factors, test divisibility by counting up from 1 to the number’s square root.

Tip: Factors always come in pairs — if 2 is a factor of 20, then 10 is also (because \(2 \times 10 = 20\)).

What Are Multiples?

A multiple of a number is what you get when you multiply it by whole numbers.

Examples

Multiples of 4 → 4, 8, 12, 16, 20, 24, …
Multiples of 7 → 7, 14, 21, 28, 35, …

Multiples grow without limit, while factors are finite. Recognising both helps when finding common multiples or simplifying fractions.

Divisibility Rules

Divisibility rules make it easy to spot factors quickly without full division.

Divisible by	Rule
2	Last digit is even (0, 2, 4, 6, 8)
3	Sum of digits divisible by 3
4	Last two digits form a number divisible by 4
5	Ends in 0 or 5
6	Divisible by both 2 and 3
9	Sum of digits divisible by 9
10	Ends in 0

Tip: Use these checks before calculating HCF or LCM — it saves time in long problems.

Prime Numbers and Prime Factorisation

A prime number has exactly two factors: 1 and itself. Prime factorisation means writing a number as a product of its prime factors.

Examples

Prime numbers up to 20: 2, 3, 5, 7, 11, 13, 17, 19
24 = \(2 \times 2 \times 2 \times 3 = 2^3 \times 3\)
60 = \(2^2 \times 3 \times 5\)

Prime factor trees help visualise how a number breaks into primes, useful for HCF and LCM.

Highest Common Factor (HCF)

The HCF is the largest number that divides exactly into two or more numbers.

Example

Find the HCF of 24 and 36.

24 = \(2^3 \times 3\); 36 = \(2^2 \times 3^2\)

Common factors → \(2^2 \times 3 = 12\)

Answer: HCF = 12

Exam Tip: Use the prime factorisation method instead of listing factors for large numbers — it’s faster and clearer.

Lowest Common Multiple (LCM)

The LCM is the smallest number that is a multiple of each given number.

Example

Find the LCM of 8 and 12.

8 = \(2^3\); 12 = \(2^2 \times 3\)

LCM → take the highest powers → \(2^3 \times 3 = 24\)

Answer: LCM = 24

LCM problems often appear in real contexts like timing events or repeating patterns.

Using HCF and LCM in Problems

Example 1 (Foundation): Sharing items equally

Find the greatest number of groups possible if 18 apples and 24 oranges are shared equally.

HCF(18, 24) = 6 → 6 groups can be made.

Example 2 (Higher): Repeating events

Bus A arrives every 8 minutes, Bus B every 12 minutes. They will arrive together every LCM(8, 12) = 24 minutes.

Worked Examples

Example 1: Find HCF using factors

Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 → HCF = 6

Example 2: Find LCM using multiples

Multiples of 6: 6, 12, 18, 24, 30, 36 Multiples of 8: 8, 16, 24, 32, 40 → LCM = 24

Example 3: Prime factorisation and HCF/LCM

Find HCF and LCM of 45 and 60.

45 = \(3^2 \times 5\), 60 = \(2^2 \times 3 \times 5\)

HCF = \(3 \times 5 = 15\)

LCM = \(2^2 \times 3^2 \times 5 = 180\)

Common Mistakes

Mixing up HCF and LCM in word problems.
Forgetting to use the highest or lowest powers when using prime factors.
Not checking divisibility before assuming numbers are prime.
Leaving out 1 or the number itself when listing factors.

How to avoid: Double-check if the question asks for common factor or multiple; always include 1 and the number itself in factor lists; show working clearly for marks.

Applications

Fractions: Simplifying and finding common denominators.
Sequences: Determining repeating cycles or patterns.
Timetables & Schedules: Calculating when events coincide.
Algebra: Factoring terms with common coefficients.

Strategies & Tips

Use prime factorisation trees for accuracy with large numbers.
Write factors in pairs to reduce errors.
For mental checks, use divisibility rules instead of long division.
Remember: HCF uses the smallest powers, LCM uses the largest powers of each prime.

Summary / Call-to-Action

Mastering factors and multiples strengthens number sense and supports algebra, fractions, and problem-solving. Understanding how to find HCF and LCM efficiently can save valuable exam time.

Attempt quizzes on factors, multiples, and prime factorisation.
Practise real-world HCF/LCM word problems.
Use divisibility rules for quick checks.
Challenge yourself with larger numbers and multi-step reasoning.

Build accuracy through practice and connect these skills to broader GCSE Maths problem-solving questions.