Least Common Multiple Of 6 And 24

Finding the Least Common Multiple (LCM) of 6 and 24: A Step-by-Step Guide

This article will guide you through the process of calculating the least common multiple (LCM) of 6 and 24. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cycles and patterns. We'll explore different methods to find the LCM, ensuring you grasp the concept thoroughly.

What is the Least Common Multiple (LCM)?

The least common multiple (LCM) is the smallest positive integer that is divisible by both of two or more numbers. Think of it as the smallest number that contains both input numbers as factors. For instance, the LCM of 2 and 3 is 6 because 6 is the smallest number divisible by both 2 and 3.

Methods for Finding the LCM of 6 and 24

We'll explore two common methods to determine the LCM of 6 and 24: the listing method and the prime factorization method.

Method 1: Listing Multiples

This method involves listing the multiples of each number until you find the smallest multiple common to both.

• Multiples of 6: 6, 12, 18, 24, 30, 36...
• Multiples of 24: 24, 48, 72...

As you can see, the smallest number that appears in both lists is 24. Therefore, the LCM of 6 and 24 is 24.

Method 2: Prime Factorization

This method uses the prime factorization of each number to find the LCM. Prime factorization breaks a number down into its prime number components.

1. Find the prime factorization of each number:

   • 6 = 2 x 3
   • 24 = 2 x 2 x 2 x 3 = 2³ x 3

2. Identify the highest power of each prime factor present in either factorization:

   • The highest power of 2 is 2³ = 8
   • The highest power of 3 is 3¹ = 3

3. Multiply the highest powers together:

   • LCM(6, 24) = 2³ x 3 = 8 x 3 = 24

Therefore, using prime factorization, we again find that the LCM of 6 and 24 is 24.

Understanding the Relationship Between LCM and GCD

The greatest common divisor (GCD), also known as the highest common factor (HCF), is the largest number that divides both numbers without leaving a remainder. The GCD of 6 and 24 is 6. There's a useful relationship between the LCM and GCD:

LCM(a, b) x GCD(a, b) = a x b

Let's verify this with our numbers:

LCM(6, 24) x GCD(6, 24) = 24 x 6 = 144

6 x 24 = 144

The equation holds true, demonstrating the connection between LCM and GCD.

Conclusion:

We've successfully determined that the least common multiple of 6 and 24 is 24 using two distinct methods. Understanding these methods provides a strong foundation for tackling more complex LCM problems involving larger numbers or multiple numbers. Remember to choose the method you find most efficient and comfortable to use. Mastering LCM is essential for a deeper understanding of number theory and its various applications.