What Is The Lcm Of 72 And 120

What is the LCM of 72 and 120? A Comprehensive Guide

Finding the least common multiple (LCM) of two numbers is a fundamental concept in mathematics with applications across various fields. This article will guide you through the process of calculating the LCM of 72 and 120, explaining different methods and providing a clear understanding of the underlying principles. Understanding LCM is crucial for solving problems involving fractions, ratios, and scheduling tasks.

Understanding Least Common Multiple (LCM)

The least common multiple (LCM) 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 prime factors of the given numbers. For instance, the LCM of 4 and 6 is 12, because 12 is the smallest number that is divisible by both 4 and 6. This concept is extremely useful in simplifying fractions and solving problems involving rhythmic cycles or repeating patterns.

Method 1: Prime Factorization

This method is often considered the most reliable for finding the LCM of larger numbers. It involves breaking down each number into its prime factors.

1. Prime Factorization of 72: 72 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²

2. Prime Factorization of 120: 120 = 2 x 2 x 2 x 3 x 5 = 2³ x 3 x 5

3. Identify the highest power of each prime factor: The prime factors present are 2, 3, and 5. The highest power of 2 is 2³, the highest power of 3 is 3², and the highest power of 5 is 5¹.

4. Multiply the highest powers together: LCM(72, 120) = 2³ x 3² x 5 = 8 x 9 x 5 = 360

Therefore, the least common multiple of 72 and 120 is $\boxed{360}$.

Method 2: Listing Multiples

This method is suitable for smaller numbers. It involves listing the multiples of each number until you find the smallest common multiple.

• Multiples of 72: 72, 144, 216, 288, 360, 432...
• Multiples of 120: 120, 240, 360, 480...

The smallest multiple that appears in both lists is 360. Therefore, the LCM of 72 and 120 is 360. This method becomes less efficient with larger numbers.

Method 3: Using the Greatest Common Divisor (GCD)

The LCM and GCD (Greatest Common Divisor) of two numbers are related through the following formula:

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

First, find the GCD of 72 and 120 using the Euclidean algorithm or prime factorization. The GCD of 72 and 120 is 24.

Then, using the formula:

LCM(72, 120) = (72 x 120) / GCD(72, 120) = (8640) / 24 = 360

This method is efficient, especially when dealing with larger numbers where finding the prime factorization might be more time-consuming.

Conclusion

We have demonstrated three different methods to calculate the LCM of 72 and 120, all resulting in the same answer: 360. Choosing the best method depends on the numbers involved and your preference. Understanding the concept of LCM and its different calculation methods is essential for various mathematical applications. Remember to choose the method that best suits the complexity of the numbers involved for efficient calculation.