Lowest Common Multiple Of 15 And 24

Finding the Lowest Common Multiple (LCM) of 15 and 24

This article will guide you through the process of finding the lowest common multiple (LCM) of 15 and 24. We'll explore different methods, making it easy to understand and apply this fundamental concept in mathematics. Understanding LCM is crucial in various mathematical operations, particularly in simplifying fractions and solving problems involving ratios and proportions. This comprehensive guide will equip you with the skills to efficiently calculate the LCM of any two numbers.

What is the Lowest Common Multiple (LCM)?

The lowest 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. Think of it as the smallest number that contains both numbers as factors.

Methods for Calculating the LCM of 15 and 24

There are several ways to calculate the LCM of 15 and 24. Let's explore two common and efficient methods:

1. Listing Multiples Method

This method involves listing the multiples of each number until you find the smallest common multiple.

• Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, 135...
• Multiples of 24: 24, 48, 72, 96, 120, 144...

By comparing the lists, we can see that the smallest common multiple is 120.

This method works well for smaller numbers, but it can become time-consuming for larger numbers.

2. Prime Factorization Method

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

• Prime factorization of 15: 3 x 5
• Prime factorization of 24: 2 x 2 x 2 x 3 (or 2³ x 3)

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 5: 5¹ = 5

Now, multiply these highest powers together: 8 x 3 x 5 = 120

Therefore, the LCM of 15 and 24 is 120. This method is more systematic and avoids the need for extensive listing, especially when dealing with larger numbers or multiple numbers.

Applications of LCM

The LCM has many practical applications in various fields, including:

• Fraction addition and subtraction: Finding a common denominator for fractions.
• Scheduling problems: Determining when events will occur simultaneously. For example, if two events happen every 15 days and 24 days respectively, they will occur simultaneously every 120 days.
• Measurement conversions: Converting between different units of measurement.
• Project management: Coordinating tasks with different completion times.

Conclusion

Finding the LCM of 15 and 24, whether using the listing multiples method or the prime factorization method, results in the answer 120. Understanding the LCM is essential for various mathematical applications. The prime factorization method is generally preferred for its efficiency, especially when dealing with larger numbers. Mastering these methods will greatly enhance your mathematical problem-solving skills.