EDEXCEL GCSE FOUNDATION MATHS - What is LCM or HCF

EDEXCEL GCSE Foundation Maths: What is LCM and HCF?

Introduction

The Lowest Common Multiple (LCM) and Highest Common Factor (HCF) are two essential concepts in number theory that find applications in various mathematical problems, including fractions, ratios, and problem-solving.

LCM: The smallest common multiple of two or more numbers.

HCF: The largest number that divides two or more numbers exactly.

Finding the LCM

1. Listing Multiples Method:

• List the multiples of each number until you find a common multiple.
• The smallest common multiple is the LCM.

Example: Find the LCM of 4 and 6.

Multiples of 4: 4, 8, 12, 16, 20, 24... Multiples of 6: 6, 12, 18, 24, 30...

Therefore, the LCM of 4 and 6 is 12.

2. Prime Factorization Method:

• Find the prime factorization of each number.
• Identify the highest power of each prime factor present in the numbers.
• Multiply these highest powers together to get the LCM.

Example: Find the LCM of 12 and 18.

12 = 2² x 3 18 = 2 x 3²

LCM = 2² x 3² = 36

Finding the HCF

1. Listing Factors Method:

• List all the factors of each number.
• Identify the common factors.
• The largest common factor is the HCF.

Example: Find the HCF of 12 and 18.

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

Therefore, the HCF of 12 and 18 is 6.

2. Prime Factorization Method:

• Find the prime factorization of each number.
• Identify the common prime factors.
• Multiply these common prime factors, raising each to the lowest power present in the factorizations.

Example: Find the HCF of 12 and 18.

12 = 2² x 3 18 = 2 x 3²

HCF = 2 x 3 = 6

Applications of LCM and HCF

• Fractions: Finding the least common denominator (LCD) for adding or subtracting fractions.
• Ratios: Simplifying ratios to their simplest form.
• Problem Solving: Solving problems involving quantities that repeat at regular intervals.

Example: Two buses leave a station at 9:00 AM. Bus A departs every 20 minutes and Bus B departs every 30 minutes. At what time will they both leave the station again?

To solve this, we need to find the LCM of 20 and 30.

LCM (20, 30) = 60 minutes = 1 hour

Therefore, both buses will leave the station again at 10:00 AM.

Key Points

• LCM and HCF are important concepts in number theory.
• Understanding these concepts can help solve various mathematical problems.
• Multiple methods exist for finding LCM and HCF, including listing multiples/factors and prime factorization.

By practicing these concepts, you will gain a strong understanding of LCM and HCF, which will be helpful in your GCSE Mathematics journey.