Lowest Common Multiple Of 20 And 24

Finding the Lowest Common Multiple (LCM) of 20 and 24: A Step-by-Step Guide

Finding the lowest common multiple (LCM) is a fundamental concept in mathematics, particularly useful in algebra, number theory, and even practical applications involving scheduling and measurement. This article will guide you through calculating the LCM of 20 and 24 using different methods, ensuring you understand the process and can apply it to other number pairs. We'll cover prime factorization, listing multiples, and using the greatest common divisor (GCD).

What is the Lowest Common Multiple (LCM)?

The lowest common multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the numbers. Understanding LCM is crucial for solving problems involving fractions, simplifying expressions, and finding solutions in various mathematical contexts. In simpler terms, it's the smallest number that both of your chosen numbers can divide into evenly.

Method 1: Prime Factorization

This is often considered the most efficient method for finding the LCM, especially for larger numbers. Let's break down 20 and 24 into their prime factors:

• 20: 2 x 2 x 5 = 2² x 5
• 24: 2 x 2 x 2 x 3 = 2³ x 3

To find the LCM, we take the highest power of each prime factor present in either number's factorization:

• Highest power of 2: 2³ = 8
• Highest power of 3: 3¹ = 3
• Highest power of 5: 5¹ = 5

Now, multiply these highest powers together: 8 x 3 x 5 = 120

Therefore, the LCM of 20 and 24 is 120.

Method 2: Listing Multiples

This method is straightforward but can be time-consuming for larger numbers. We list the multiples of each number until we find the smallest common multiple:

• Multiples of 20: 20, 40, 60, 80, 100, 120, 140...
• Multiples of 24: 24, 48, 72, 96, 120, 144...

The smallest number appearing in both lists is 120. Therefore, the LCM of 20 and 24 is 120.

Method 3: Using the Greatest Common Divisor (GCD)

There's a handy formula connecting the LCM and GCD of two numbers:

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

First, we need to find the GCD of 20 and 24. We can use the Euclidean algorithm for this:

1. Divide 24 by 20: 24 = 20 x 1 + 4
2. Divide 20 by the remainder 4: 20 = 4 x 5 + 0

The last non-zero remainder is 4, so the GCD(20, 24) = 4.

Now, apply the formula:

LCM(20, 24) = (20 x 24) / 4 = 480 / 4 = 120

Therefore, the LCM of 20 and 24 is 120.

Conclusion

We've explored three different methods to determine the LCM of 20 and 24, all leading to the same answer: 120. Choosing the best method depends on the numbers involved and your comfort level with each approach. Prime factorization is generally preferred for efficiency, while listing multiples is a more intuitive method for smaller numbers. Understanding the relationship between LCM and GCD provides a powerful alternative for calculations. Mastering these techniques will enhance your understanding of number theory and problem-solving skills.