Lowest Common Multiple Of 24 And 42

Finding the Lowest Common Multiple (LCM) of 24 and 42

This article will guide you through the process of finding the lowest common multiple (LCM) of 24 and 42. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cyclical events. We'll explore different methods to calculate the LCM, making the concept accessible and easy to understand. This comprehensive guide will equip you with the knowledge to confidently tackle LCM problems involving larger numbers.

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. In simpler terms, it's the smallest number that all the given numbers can divide into evenly. For instance, finding the LCM is useful when trying to determine when two events that occur at different intervals will happen simultaneously.

Methods for Calculating the LCM of 24 and 42

There are several methods to calculate the LCM, each with its own advantages. Let's explore two common approaches:

1. Listing Multiples Method

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 24: 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336...
• Multiples of 42: 42, 84, 126, 168, 210, 252, 294, 336...

By comparing the lists, we can see that the smallest common multiple is 168.

2. Prime Factorization Method

This method is generally more efficient, especially for larger numbers. It involves finding the prime factorization of each number and then constructing the LCM using the highest powers of all prime factors present.

• Prime factorization of 24: 2³ x 3¹
• Prime factorization of 42: 2¹ x 3¹ x 7¹

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

• Highest power of 2: 2³ = 8
• Highest power of 3: 3¹ = 3
• Highest power of 7: 7¹ = 7

Now, multiply these highest powers together: 8 x 3 x 7 = 168

Therefore, the LCM of 24 and 42 is 168.

Which Method is Best?

The prime factorization method is generally preferred for its efficiency, particularly when dealing with larger numbers or multiple numbers. The listing multiples method is easier to understand initially but becomes impractical for larger numbers.

Applications of LCM

Understanding the LCM has practical applications in many areas, including:

• Fraction addition and subtraction: Finding a common denominator for fractions involves finding the LCM of the denominators.
• Scheduling: Determining when events with different periodicities coincide. For example, if one event happens every 24 hours and another every 42 hours, the LCM helps find when they'll occur simultaneously again.
• Measurement conversions: Converting units of measurement often involves using LCM to find a common multiple.

This article provided a comprehensive overview of finding the lowest common multiple, focusing on the efficient prime factorization method and comparing it to the listing multiples approach. By mastering these techniques, you'll be well-equipped to solve a wide range of mathematical problems involving LCM.