What Is The Least Common Multiple Of 6 And 24

What is the Least Common Multiple (LCM) of 6 and 24? A Step-by-Step Guide

Finding the least common multiple (LCM) is a fundamental concept in mathematics, particularly useful in various fields like simplifying fractions, solving problems involving cycles, and even in music theory. This article will clearly explain how to calculate the LCM of 6 and 24, providing you with a simple, step-by-step method that you can apply to any pair of numbers. Understanding this will help you master this crucial mathematical skill.

What is the 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. In simpler terms, it's the smallest number that both numbers divide into evenly. This is different from the greatest common divisor (GCD), which is the largest number that divides both numbers without leaving a remainder.

Methods for Finding the LCM of 6 and 24

There are several ways to find the LCM of 6 and 24. We'll explore two common and effective methods:

1. Listing Multiples Method

This method is straightforward, especially for smaller numbers:

• List the multiples of 6: 6, 12, 18, 24, 30, 36...
• List the multiples of 24: 24, 48, 72...

Notice that the smallest number that appears in both lists is 24. Therefore, the LCM of 6 and 24 is 24.

2. Prime Factorization Method

This method is more efficient for larger numbers and provides a deeper understanding of the concept. It involves finding the prime factorization of each number:

• Prime factorization of 6: 2 x 3
• Prime factorization of 24: 2 x 2 x 2 x 3 (or 2³ x 3)

Next, 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

Finally, multiply these highest powers together: 8 x 3 = 24. This gives us the LCM of 6 and 24.

Understanding the Result: Why 24 is the LCM of 6 and 24

The LCM, 24, is divisible by both 6 (24 ÷ 6 = 4) and 24 (24 ÷ 24 = 1) without leaving any remainder. It's the smallest number that satisfies this condition.

Applying the LCM in Real-World Scenarios

The LCM has practical applications in various real-world situations. For example:

• Scheduling: If event A happens every 6 days and event B happens every 24 days, the LCM helps determine when both events will occur on the same day again (in 24 days).
• Fraction Arithmetic: When adding or subtracting fractions, finding the LCM of the denominators is crucial for finding a common denominator.

This comprehensive guide explains how to easily find the least common multiple of 6 and 24, using both the listing method and the more efficient prime factorization method. Remember that understanding the concept of LCM is important for various mathematical operations and problem-solving scenarios.