What Is The Lcm Of 40 And 24

What is the LCM of 40 and 24? A Step-by-Step Guide

Finding the least common multiple (LCM) of two numbers is a fundamental concept in mathematics, with applications ranging from simple fraction addition to complex scheduling problems. This article will guide you through calculating the LCM of 40 and 24 using two common methods: prime factorization and the listing method. Understanding these methods will equip you with the skills to find the LCM of any two numbers.

What is the Least Common Multiple (LCM)?

Before diving into the calculations, let's define the LCM. The least common multiple of two or more numbers is the smallest positive number that is a multiple of all the numbers. In simpler terms, it's the smallest number that both numbers can divide into evenly.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors. Prime factors are numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, etc.).

1. Find the prime factorization of 40: 40 = 2 x 2 x 2 x 5 = 2³ x 5

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 are 2, 3, and 5. The highest power of 2 is 2³ (or 8), the highest power of 3 is 3¹, and the highest power of 5 is 5¹.

4. Multiply the highest powers together: LCM(40, 24) = 2³ x 3 x 5 = 8 x 3 x 5 = 120

Therefore, the least common multiple of 40 and 24 is 120.

Method 2: Listing Multiples

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

1. List the multiples of 40: 40, 80, 120, 160, 200...

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

3. Identify the smallest common multiple: The smallest number that appears in both lists is 120.

Therefore, the least common multiple of 40 and 24 is 120.

Conclusion:

Both methods confirm that the least common multiple of 40 and 24 is 120. The prime factorization method is generally more efficient for larger numbers, while the listing method is easier to visualize for smaller numbers. Understanding both techniques provides a versatile approach to solving LCM problems. Remember, mastering LCM calculations is crucial for various mathematical applications and problem-solving scenarios.