IGCSE Factors and Multiples: Complete Guide for Cambridge IGCSE Mathematics

IGCSE factors and multiples are fundamental topics in Cambridge IGCSE Mathematics that form the foundation for many algebraic and number theory concepts. Mastering prime factorization, HCF (Highest Common Factor), and LCM (Lowest Common Multiple) is essential for solving problems involving fractions, ratios, and algebraic expressions.

This comprehensive IGCSE factors and multiples guide covers everything you need to know, including finding factors and multiples, prime factorization methods, calculating HCF and LCM, step-by-step worked examples, common exam questions, and expert tips from Tutopiya’s IGCSE maths tutors. We’ll also show you how to avoid the most common mistakes that cost students valuable marks.

🎯 What you’ll learn: By the end of this guide, you’ll know how to find all factors of a number, identify prime numbers, use prime factorization to find HCF and LCM, and apply these skills to solve real-world problems in IGCSE exams.

Already studying with Tutopiya? Practice these skills with our dedicated IGCSE Number practice deck featuring exam-style questions and instant feedback.

---

Why IGCSE Factors and Multiples Matter

IGCSE factors and multiples are essential building blocks for many mathematical concepts. Here’s why they’re so important:

• Foundation topic: Required for understanding fractions, ratios, and algebraic simplification
• High frequency: HCF and LCM questions appear regularly in IGCSE maths papers
• Exam weight: Typically worth 4-6 marks per paper
• Real-world applications: Used in scheduling, engineering, and problem-solving scenarios
• Algebra foundation: Essential for factorizing algebraic expressions and solving equations

Key insight from examiners: Students often confuse factors and multiples, or make errors in prime factorization. This guide will help you master these concepts systematically.

---

Understanding Factors: The Basics

A factor of a number is a whole number that divides exactly into that number (with no remainder).

Key points:

• Every number has at least two factors: 1 and itself
• Factors come in pairs (except for perfect squares)
• To find all factors, test numbers from 1 up to the square root

Example: Factors of 12: 1, 2, 3, 4, 6, 12

• 12 ÷ 1 = 12 ✓
• 12 ÷ 2 = 6 ✓
• 12 ÷ 3 = 4 ✓
• 12 ÷ 4 = 3 ✓
• 12 ÷ 6 = 2 ✓
• 12 ÷ 12 = 1 ✓

Finding All Factors: Systematic Method

1. Start with 1 - 1 is always a factor
2. Test numbers in order - Try 2, 3, 4, 5, …
3. Stop at the square root - If you’ve found factors up to √n, you’ve found them all
4. List in pairs - Each factor pairs with another

Example: Find all factors of 36

Solution:

• Test 1: 36 ÷ 1 = 36 → factors: 1, 36
• Test 2: 36 ÷ 2 = 18 → factors: 2, 18
• Test 3: 36 ÷ 3 = 12 → factors: 3, 12
• Test 4: 36 ÷ 4 = 9 → factors: 4, 9
• Test 5: 36 ÷ 5 = 7.2 ✗ (not a factor)
• Test 6: 36 ÷ 6 = 6 → factor: 6 (perfect square, so 6 appears once)

Answer: Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

---

Understanding Multiples: The Basics

A multiple of a number is the result of multiplying that number by any whole number.

Key points:

• Multiples are infinite (you can always multiply by a larger number)
• The first multiple is the number itself (multiply by 1)
• Multiples get larger as you multiply by larger numbers

Example: Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, …

Finding Multiples: Systematic Method

1. Start with the number itself (multiply by 1)
2. Keep adding the number to get the next multiple
3. Or multiply by 2, 3, 4, 5, …

Example: Find the first 6 multiples of 7

Solution:

• 7 × 1 = 7
• 7 × 2 = 14
• 7 × 3 = 21
• 7 × 4 = 28
• 7 × 5 = 35
• 7 × 6 = 42

Answer: 7, 14, 21, 28, 35, 42

---

Prime Numbers and Prime Factorization

Prime Numbers

A prime number is a number greater than 1 that has exactly two factors: 1 and itself.

Prime numbers up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Key points:

• 2 is the only even prime number
• 1 is not a prime number (it only has one factor)
• All other even numbers are composite (have more than 2 factors)

Prime Factorization

Prime factorization means expressing a number as a product of prime numbers.

Method 1: Factor Tree

1. Write the number at the top
2. Find two factors and write them below
3. Continue until all factors are prime
4. Write as a product of primes

Example: Find the prime factorization of 60

Solution using factor tree:

        60
       /  \
      6    10
     / \   / \
    2   3 2   5

Prime factors: 2, 2, 3, 5

Answer: 60 = 2² × 3 × 5

Method 2: Division Method

1. Divide by the smallest prime (2)
2. Continue dividing by primes until you get 1
3. Write all the prime divisors as a product

Example: Find the prime factorization of 84

Solution:

84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1

Answer: 84 = 2² × 3 × 7

---

Highest Common Factor (HCF)

The HCF (also called GCD - Greatest Common Divisor) is the largest number that divides exactly into two or more numbers.

Method 1: Listing Factors

1. List all factors of each number
2. Find the common factors
3. The largest common factor is the HCF

Example: Find the HCF of 24 and 36

Solution:

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Common factors: 1, 2, 3, 4, 6, 12
• Largest: 12

Answer: HCF = 12

Method 2: Prime Factorization (Recommended)

1. Find the prime factorization of each number
2. Identify common prime factors
3. Multiply the lowest power of each common prime

Example: Find the HCF of 60 and 84

Solution:

• 60 = 2² × 3 × 5
• 84 = 2² × 3 × 7
• Common primes: 2² and 3
• HCF = 2² × 3 = 4 × 3 = 12

Answer: HCF = 12

Method 3: Euclidean Algorithm (For Large Numbers)

This method is efficient for large numbers:

1. Divide the larger number by the smaller
2. Replace the larger with the smaller, and the smaller with the remainder
3. Repeat until remainder is 0
4. The last non-zero remainder is the HCF

Example: Find the HCF of 48 and 18

Solution:

48 ÷ 18 = 2 remainder 12
18 ÷ 12 = 1 remainder 6
12 ÷ 6 = 2 remainder 0

Answer: HCF = 6

---

Lowest Common Multiple (LCM)

The LCM is the smallest number that is a multiple of two or more numbers.

Method 1: Listing Multiples

1. List multiples of each number
2. Find the common multiples
3. The smallest common multiple is the LCM

Example: Find the LCM of 4 and 6

Solution:

• Multiples of 4: 4, 8, 12, 16, 20, 24, …
• Multiples of 6: 6, 12, 18, 24, 30, …
• Common multiples: 12, 24, …
• Smallest: 12

Answer: LCM = 12

Method 2: Prime Factorization (Recommended)

1. Find the prime factorization of each number
2. Take the highest power of each prime that appears
3. Multiply them together

Example: Find the LCM of 60 and 84

Solution:

• 60 = 2² × 3 × 5
• 84 = 2² × 3 × 7
• Highest powers: 2², 3, 5, 7
• LCM = 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420

Answer: LCM = 420

Method 3: Formula Method

Formula: HCF × LCM = Product of the two numbers

Therefore: LCM = (Number 1 × Number 2) ÷ HCF

Example: Find the LCM of 24 and 36 (we know HCF = 12)

Solution: LCM = (24 × 36) ÷ 12 = 864 ÷ 12 = 72

Answer: LCM = 72

---

Step-by-Step Method for HCF and LCM Problems

1. Identify what you need - HCF or LCM?
2. Choose your method - Prime factorization is usually most reliable
3. Find prime factorizations - Use factor trees or division method
4. Apply the rules:
   • HCF: Multiply lowest powers of common primes
   • LCM: Multiply highest powers of all primes
5. Check your answer - Verify it makes sense

---

Worked Examples

Example 1: Finding HCF Using Prime Factorization

Find the HCF of 72 and 108

Solution:

1. Prime factorizations:
   • 72 = 2³ × 3²
   • 108 = 2² × 3³
2. Common primes: 2² and 3²
3. HCF = 2² × 3² = 4 × 9 = 36

Answer: HCF = 36

Example 2: Finding LCM Using Prime Factorization

Find the LCM of 18, 24, and 30

Solution:

1. Prime factorizations:
   • 18 = 2 × 3²
   • 24 = 2³ × 3
   • 30 = 2 × 3 × 5
2. Highest powers: 2³, 3², 5
3. LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360

Answer: LCM = 360

Example 3: Word Problem

Two bells ring at intervals of 12 minutes and 18 minutes respectively. If they ring together at 9:00 AM, when will they next ring together?

Solution: This is an LCM problem - we need to find when both bells ring at the same time.

1. Find LCM of 12 and 18:
   • 12 = 2² × 3
   • 18 = 2 × 3²
   • LCM = 2² × 3² = 4 × 9 = 36 minutes
2. They will ring together again 36 minutes after 9:00 AM

Answer: 9:36 AM

Example 4: HCF Word Problem

A rectangular floor measures 240 cm by 180 cm. What is the largest square tile that can be used to cover the floor completely?

Solution: This is an HCF problem - we need the largest square that divides both dimensions.

1. Find HCF of 240 and 180:
   • 240 = 2⁴ × 3 × 5
   • 180 = 2² × 3² × 5
   • HCF = 2² × 3 × 5 = 4 × 3 × 5 = 60 cm
2. The largest square tile is 60 cm × 60 cm

Answer: 60 cm × 60 cm

---

Common Examiner Traps (and How to Dodge Them)

• Confusing factors and multiples - Factors divide into a number; multiples are the number multiplied
• Incomplete prime factorization - Always check you’ve broken down to prime numbers only
• Wrong powers in HCF/LCM - HCF uses lowest powers; LCM uses highest powers
• Forgetting 1 as a factor - 1 is always a factor of every number
• Not simplifying fractions - Use HCF to simplify fractions to lowest terms
• Missing common factors - Double-check your factor lists are complete

---

IGCSE Factors and Multiples Practice Questions

Question 1: Prime Factorization

Express 360 as a product of its prime factors.

Solution:

360 ÷ 2 = 180
180 ÷ 2 = 90
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1

Answer: 360 = 2³ × 3² × 5

Question 2: HCF

Find the HCF of 54 and 72.

Solution:

• 54 = 2 × 3³
• 72 = 2³ × 3²
• HCF = 2 × 3² = 2 × 9 = 18

Answer: HCF = 18

Question 3: LCM

Find the LCM of 15, 20, and 25.

Solution:

• 15 = 3 × 5
• 20 = 2² × 5
• 25 = 5²
• LCM = 2² × 3 × 5² = 4 × 3 × 25 = 300

Answer: LCM = 300

Question 4: Word Problem

Three buses leave a station at intervals of 15 minutes, 20 minutes, and 25 minutes. If they all leave together at 8:00 AM, when will they next leave together?

Solution: Find LCM of 15, 20, and 25:

• 15 = 3 × 5
• 20 = 2² × 5
• 25 = 5²
• LCM = 2² × 3 × 5² = 4 × 3 × 25 = 300 minutes = 5 hours

Answer: 1:00 PM (5 hours after 8:00 AM)

---

Tutopiya Advantage: Personalised IGCSE Factors and Multiples Coaching

• Live whiteboard walkthroughs of prime factorization and HCF/LCM problems
• Exam-docket homework packs mirroring CAIE specimen papers
• Analytics dashboard so parents see accuracy by topic
• Flexible slots with ex-Cambridge markers for last-mile polishing

📞 Ready to turn shaky number skills into exam-ready confidence? Book a free IGCSE maths trial and accelerate your revision plan.

---

Frequently Asked Questions About IGCSE Factors and Multiples

What’s the difference between factors and multiples?

• Factors: Numbers that divide exactly into a given number (e.g., factors of 12 are 1, 2, 3, 4, 6, 12)
• Multiples: Numbers that the given number divides into (e.g., multiples of 5 are 5, 10, 15, 20, …)

How do I find all factors of a number?

Test numbers from 1 up to the square root. Each factor pairs with another. For example, factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.

What is prime factorization?

Prime factorization means expressing a number as a product of prime numbers. For example, 60 = 2² × 3 × 5.

How do I find the HCF?

Method 1: List all factors and find the largest common one. Method 2 (Recommended): Use prime factorization - multiply the lowest power of each common prime.

How do I find the LCM?

Method 1: List multiples and find the smallest common one. Method 2 (Recommended): Use prime factorization - multiply the highest power of each prime that appears.

Can I use a formula for LCM?

Yes! HCF × LCM = Product of the two numbers, so LCM = (Number 1 × Number 2) ÷ HCF

What is a prime number?

A prime number is a number greater than 1 with exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, …

---

Related IGCSE Maths Resources

Strengthen your IGCSE Mathematics preparation with these comprehensive guides:

• IGCSE Set Language and Absolute Value: Complete Guide - Master set notation and absolute value equations
• IGCSE Exponents and Surds: Complete Guide - Master index laws and surd operations
• IGCSE Maths Revision Notes, Syllabus and Preparation Tips - Complete syllabus overview, topic breakdown, and revision strategies
• IGCSE Past Papers Guide - Access free IGCSE past papers and exam resources

---

Next Steps: Master IGCSE Factors and Multiples with Tutopiya

Ready to excel in IGCSE factors and multiples? Our expert IGCSE maths tutors provide:

• Personalized 1-on-1 tutoring tailored to your learning pace
• Exam-focused practice with real Cambridge IGCSE past papers
• Interactive whiteboard sessions for visual learning
• Progress tracking to identify and strengthen weak areas
• Flexible scheduling to fit your revision timetable

Book a free IGCSE maths trial lesson and get personalized support to master factors, multiples, HCF, LCM, and achieve your target grade.

---