Finding the Highest Common Factor (HCF): A Step-by-Step Guide

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides evenly into two or more numbers. Let’s explore how to find the HCF of 60 and 84.

Method 1: Listing Factors

1. List the Factors of Each Number:
   • Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
   • Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
2. Identify the Common Factors: Look for the factors that are present in both lists:Common Factors: 1, 2, 3, 4, 6, 12
3. Identify the Highest Common Factor: The largest number in the list of common factors is the HCF.Therefore, the HCF of 60 and 84 is 12.

Method 2: Prime Factorisation

1. Find the Prime Factorisation of Each Number:
   • Prime Factorisation of 60: 2^2 \times 3 \times 7
   • Prime Factorisation of 84: 2^2 \times 3 \times 7
2. Identify Common Prime Factors: Identify the prime factors that are common to both numbers. In this case, both 60 and 84 have 2^2 and 3 as prime factors.
3. Multiply the Common Prime Factors: Multiply the common prime factors together, using the lowest power of each common prime factor: HCF = 2^2 \times 3 = 4 \times 3 = 12

Therefore, the HCF of 60 and 84 is 12.

Why This Works

The HCF is the largest number that divides into both 60 and 84. By finding the prime factorization, we break down each number into its fundamental building blocks. The HCF is then constructed by taking the common building blocks (prime factors) with their lowest powers.

Key Skills Used:

• Factorization: Finding the factors of a number.
• Prime Factorisation: Expressing a number as a product of its prime factors.
• Identifying Common Factors: Recognising factors that are shared between two or more numbers.

Need Extra Help with HCF and LCM?

If you’re finding HCF, LCM (Lowest Common Multiple), or other number theory topics challenging, don’t hesitate to seek help. A private GCSE Maths Tutor Online can provide personalised support to help you master these skills. A tutor can work with you one-on-one to explain the concepts, provide practice questions, and help you build your confidence in maths. They can help you develop a systematic approach to finding the HCF and LCM and improve your overall understanding of number theory.