EDEXCEL GCSE HIGHER MATHS - What are LCM and HCF

EDEXCEL GCSE HIGHER MATHS - What are LCM and HCF?

Lowest Common Multiple (LCM)

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

Finding the LCM:

1. Prime Factorization: Write each number as a product of its prime factors.
2. Identify Common and Uncommon Factors: Identify the common prime factors and their highest powers that appear in any of the numbers.
3. Multiply: Multiply the common and uncommon prime factors together to get the LCM.

Example: Find the LCM of 12 and 18.

• Prime Factorization:
  • 12 = 2 x 2 x 3
  • 18 = 2 x 3 x 3
• Common and Uncommon Factors:
  • Common factors: 2, 3
  • Highest powers: 2² (from 12) and 3² (from 18)
• Multiply: 2² x 3² = 4 x 9 = 36

Therefore, the LCM of 12 and 18 is 36.

Highest Common Factor (HCF)

The HCF of two or more numbers is the largest number that divides into all the given numbers without leaving a remainder.

Finding the HCF:

1. Prime Factorization: Write each number as a product of its prime factors.
2. Identify Common Factors: Identify the common prime factors.
3. Multiply: Multiply the common prime factors together to get the HCF.

Example: Find the HCF of 24 and 36.

• Prime Factorization:
  • 24 = 2 x 2 x 2 x 3
  • 36 = 2 x 2 x 3 x 3
• Common Factors: 2, 2, 3
• Multiply: 2 x 2 x 3 = 12

Therefore, the HCF of 24 and 36 is 12.

Key Concepts:

• The LCM is always greater than or equal to the largest number.
• The HCF is always less than or equal to the smallest number.
• If the HCF of two numbers is 1, they are called coprime or relatively prime.

Applications:

• Fractions: Finding the LCM is crucial when adding or subtracting fractions with different denominators.
• Word Problems: LCM and HCF are used to solve problems involving dividing quantities into equal groups, finding the shortest time for events to occur simultaneously, and other real-world applications.

Practice Exercises:

1. Find the LCM and HCF of:
   • 16 and 24
   • 15 and 35
   • 48 and 72
2. Two buses leave a station at 9:00 am. One bus returns every 20 minutes, and the other returns every 30 minutes. At what time will they both return to the station together?
3. A rectangular garden is 12 meters long and 18 meters wide. What is the largest square tiles that can be used to cover the entire garden without any cutting?

Remember: Practice is key to mastering LCM and HCF concepts. Work through various exercises and apply your knowledge to real-world situations.