Least Common Multiple Of 21 And 24

Finding the Least Common Multiple (LCM) of 21 and 24: A Step-by-Step Guide

Finding the least common multiple (LCM) is a fundamental concept in mathematics, particularly useful in simplifying fractions and solving problems involving cycles or periodic events. This article will guide you through the process of calculating the LCM of 21 and 24, explaining the methods involved and providing a clear understanding of the underlying principles. Understanding LCMs is crucial for various mathematical applications, including algebra, number theory, and even programming.

What is the Least Common Multiple (LCM)?

The least common multiple of two or more integers is the smallest positive integer that is divisible by all the integers. Think of it as the smallest number that contains all the given numbers as factors. For example, the LCM of 2 and 3 is 6 because 6 is the smallest number divisible by both 2 and 3. In this article, we'll focus on finding the LCM of 21 and 24.

Method 1: Prime Factorization

This method is generally considered the most efficient and reliable way to find the LCM of larger numbers. It involves breaking down each number into its prime factors.

1. Find the prime factorization of 21: 21 = 3 x 7

2. Find the prime factorization of 24: 24 = 2 x 2 x 2 x 3 = 2³ x 3

3. Identify the highest power of each prime factor: The prime factors present are 2, 3, and 7. The highest power of 2 is 2³, the highest power of 3 is 3¹, and the highest power of 7 is 7¹.

4. Multiply the highest powers together: LCM(21, 24) = 2³ x 3 x 7 = 8 x 3 x 7 = 168

Therefore, the least common multiple of 21 and 24 is 168.

Method 2: Listing Multiples

This method is simpler for smaller numbers but becomes less efficient as the numbers increase.

1. List the multiples of 21: 21, 42, 63, 84, 105, 126, 147, 168, ...

2. List the multiples of 24: 24, 48, 72, 96, 120, 144, 168, ...

3. Find the smallest common multiple: The smallest number that appears in both lists is 168.

Therefore, the least common multiple of 21 and 24 is 168.

Method 3: Using the Greatest Common Divisor (GCD)

This method utilizes the relationship between the LCM and the greatest common divisor (GCD). The formula is:

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

1. Find the GCD of 21 and 24 using the Euclidean algorithm or prime factorization: The prime factorization method shows that the only common factor of 21 and 24 is 3. Therefore, GCD(21, 24) = 3.

2. Apply the formula: LCM(21, 24) = (21 x 24) / 3 = 504 / 3 = 168

Therefore, the least common multiple of 21 and 24 is 168.

Conclusion

We've explored three different methods to calculate the LCM of 21 and 24, all arriving at the same answer: 168. The prime factorization method is generally preferred for its efficiency, especially when dealing with larger numbers. Understanding the LCM is essential for various mathematical operations and problem-solving scenarios. Now you have a solid grasp of how to find the LCM and can confidently apply these methods to other number combinations.