IGCSE Extended Mathematics (0580) - Study Report

Chapter: Number

Topic: 1.1 Types of number

Subtopic: Identify and use of: common factors

Prepared for: Grade 9 and 10 Students

Author: Your Expert Teacher and Curriculum Designer

Foreword for the Student

Hello, future mathematicians! Welcome to this deep dive into the world of common factors. You might be thinking, "Factors? I learned about those years ago!" And you're right. But in IGCSE Extended Maths, we take familiar concepts and elevate them. We don't just find factors; we master them, we use them as tools, and we understand their profound importance in solving complex problems.

This study note is designed to be your comprehensive guide. It's more than just a list of rules; it's a conversation. We'll start from the very beginning, ensuring our foundation is rock-solid, and then build up to the advanced techniques and problem-solving strategies you'll need to excel in your exams. My goal is to make you not just a student who can find a common factor, but a mathematician who understands why they matter.

Treat this document as your personal tutor. Read it, highlight it, write notes in the margins of your printout, and most importantly, work through the examples. The journey from understanding to mastery is paved with practice. Let's begin!

1. Introduction to the Subtopic: Common Factors

At its heart, mathematics is the study of patterns and relationships. The concept of "common factors" is one of the first and most fundamental relationships we explore between numbers. It’s about finding what numbers have in common, their shared DNA, so to speak.

This subtopic is the bedrock for many other areas in your IGCSE course. Mastering it will make simplifying fractions a breeze, solving certain types of word problems intuitive, and even give you a glimpse into more advanced topics like algebra and cryptography.

Key Terms and Their Importance

Before we dive deep, let's get our vocabulary straight. Using the correct terminology is crucial in mathematics, as it allows us to communicate complex ideas with precision.

Factor: A factor of a whole number is any whole number that divides into it exactly, leaving no remainder. For example, 4 is a factor of 20 because 20 ÷ 4 = 5. Think of factors as the "building blocks" of a number through multiplication.
Common Factor: A common factor is a number that is a factor of two or more other numbers. It's a "shared" factor. For example, the common factors of 12 and 18 are {1, 2, 3, 6}.
Highest Common Factor (HCF): This is the star of the show. The HCF is the largest number that is a common factor of two or more numbers. From our example, the HCF of 12 and 18 is 6. You will also see this called the Greatest Common Divisor (GCD).

Why is the HCF so important?
The HCF is not just an academic exercise. It is the most efficient tool for a variety of tasks. When we simplify a fraction, we are dividing the top and bottom by their HCF. When we solve problems about arranging items into the largest possible identical groups, we are looking for the HCF. It represents the point of maximum commonality.

2. Detailed Explanation with Examples

Part A: How to Find the Factors of a Single Number

Method 1: The Pairing or "Rainbow" Method

Start with 1 and the number itself.
Try the next whole number (2, 3, 4, etc.) and see if it divides exactly. If it does, write down the number and its pair.
Stop when the numbers you are testing meet in the middle.
List all the factors in ascending order.

Example: The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Example: The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

Part B: Finding Common Factors and the HCF by Listing

Example: Find the HCF of 24 and 40.

Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}
Factors of 40: {1, 2, 4, 5, 8, 10, 20, 40}
Common Factors: {1, 2, 4, 8}
Therefore, the HCF is 8.

Part C: The Professional Method - HCF using Prime Factorization

This is the most efficient method for IGCSE. It relies on breaking numbers down into their prime factors.

Step 1: Find the Prime Factorization of Each Number.

Use a factor tree. For example, for 84:

The prime factorization of 84 is 2 × 2 × 3 × 7. In index form, this is 2² × 3 × 7.

Step 2: The Rule for HCF
To find the HCF, multiply the common prime factors raised to their lowest powers.

Example: Find the HCF of 84 and 180.

Prime factorization of 84 = 2² × 3¹ × 7¹
Prime factorization of 180 = 2² × 3² × 5¹

The common prime factors are 2 and 3.
The lowest power of 2 is 2².
The lowest power of 3 is 3¹.
HCF = 2² × 3¹ = 4 × 3 = 12.

Part D: Visualizing HCF with Venn Diagrams

A Venn diagram is a great visual tool. The HCF is the product of all the numbers in the intersection (the overlapping part).

Example: Find the HCF of 48 and 60.

Prime factors of 48: {2, 2, 2, 2, 3}
Prime factors of 60: {2, 2, 3, 5}

The common factors {2, 2, 3} go in the intersection. The remaining factors {2, 2} for 48 and {5} for 60 go in the outer parts.

HCF = Product of numbers in the intersection = 2 × 2 × 3 = 12.

3. Key Concepts and Principles

Fundamental Theorem of Arithmetic: Every integer greater than 1 is either a prime number or can be written as a unique product of prime numbers. This is why the prime factorization method works.
HCF Rule (Formal): HCF(a, b) = p₁^{min(x₁, y₁)} * p₂^{min(x₂, y₂)} * ... which means "take common primes to their lowest power".
Coprime Numbers: Two numbers are coprime (or relatively prime) if their HCF is 1. They share no common prime factors.

Crucial IGCSE Formula:
For any two positive integers `a` and `b`:
HCF(a, b) × LCM(a, b) = a × b

4. Practical Applications

Simplifying Fractions: To simplify 72/108, you divide the numerator and denominator by their HCF, which is 36. This gives 2/3.
Tiling and Flooring: To tile a 420cm by 540cm room with the largest possible identical square tiles, the tile side length must be the HCF(420, 540), which is 60cm.
Organizing and Grouping: To make the greatest number of identical goody bags from 180 pencils, 240 erasers, and 300 stickers, you find the HCF(180, 240, 300), which is 60. You can make 60 bags.

5. IGCSE Exam-Style Questions

Look for keywords like "greatest", "largest", "maximum", and "identical groups" to identify HCF problems.

Problem 1: Pure Calculation

Question:
(a) Write 504 as a product of its prime factors.
(b) Find the Highest Common Factor (HCF) of 504 and 210.

Solution:
(a) 504 = 2 × 252 = 2 × 2 × 126 = 2 × 2 × 2 × 63 = 2 × 2 × 2 × 3 × 3 × 7.
Answer: 504 = 2³ × 3² × 7

(b) We have 504 = 2³ × 3² × 7. And 210 = 2 × 3 × 5 × 7.
Common primes are 2, 3, and 7. Lowest powers are 2¹, 3¹, and 7¹.
HCF = 2 × 3 × 7 = 42.
Answer: 42

Problem 2: The Abstract Question

Question:
The numbers P and Q are written as products of their prime factors.
P = 2⁴ × 3² × 5
Q = 2² × 3³ × 7
(a) Find the HCF of P and Q.

Solution:
(a) Common primes are 2 and 3. Lowest power of 2 is 2². Lowest power of 3 is 3².
HCF = 2² × 3² = 4 × 9 = 36.
Answer: 36

Problem 3: Using the HCF x LCM Formula

Question:
The HCF of two numbers is 18. Their product is 3888. Find their LCM.

Solution:
Using the formula: HCF × LCM = Product.
18 × LCM = 3888.
LCM = 3888 ÷ 18 = 216.
Answer: 216

6. Summary Points for Quick Revision

HCF: Highest Common Factor.
Pro Method: Use prime factorization.
The Rule: Multiply common prime factors at their lowest powers.
Key Formula: HCF × LCM = a × b.
Word Problem Keywords: "largest", "greatest", "identical groups".

7. Questions and Answers

Q1: What is the HCF of two different prime numbers, like 17 and 23?

A: The HCF is always 1. Prime numbers only have factors of 1 and themselves, so their only common factor is 1.

Q2: Can the HCF of two numbers ever be one of the numbers itself?

A: Yes. This happens when the smaller number is a factor of the larger number. The HCF of 9 and 27 is 9.

Q3: Explain the relationship HCF × LCM = a × b. Why is it useful?

A: It means the product of the HCF and LCM equals the product of the numbers themselves. It's a huge time-saver in exams, allowing you to find the LCM quickly if you already know the HCF (or vice-versa).

Q4: A student says the HCF of 120 and 250 is 20. How can you quickly check this?

A: Check if 20 divides into both numbers. 120 ÷ 20 = 6 (Yes). 250 ÷ 20 = 12.5 (No). Since 20 is not a factor of 250, it cannot be the HCF. The answer is incorrect.

Study Notes

Study Notes

Identify and use of: common factors