lcm for 15 and 9

Guri Bee

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18-10-2024

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Finding the Least Common Multiple (LCM) of 15 and 9: A Step-by-Step Guide

The Least Common Multiple (LCM) of two numbers is the smallest positive number that is a multiple of both. Understanding LCMs is crucial in various mathematical applications, such as solving problems involving fractions and ratios. In this article, we'll explore how to find the LCM of 15 and 9 using a simple method.

1. Prime Factorization:

The first step is to find the prime factorization of each number.

• 15: 3 x 5
• 9: 3 x 3

2. Identify Common and Unique Factors:

Now, let's identify the common and unique prime factors:

• Common factor: 3
• Unique factors: 3 (for 9) and 5 (for 15)

3. Calculate the LCM:

To calculate the LCM, we multiply the highest powers of all the prime factors involved:

• LCM(15, 9) = 3² x 5 = 9 x 5 = 45

Therefore, the Least Common Multiple of 15 and 9 is 45.

Understanding the Concept:

• Why do we use prime factorization? Breaking down numbers into their prime factors helps us visualize their common factors and ensures we don't miss any factors when calculating the LCM.
• Why do we multiply the highest powers? The highest powers ensure that we include all the factors of both numbers, guaranteeing the resulting number is a multiple of both.

Real-World Application:

Let's say you have two measuring cups: one that holds 15 milliliters and another that holds 9 milliliters. If you want to measure out the same amount of liquid using both cups, you need to find the LCM of 15 and 9. The LCM, 45, tells you the smallest amount you can measure out using both cups without having any leftover liquid.

Additional Insights:

• Another method for finding the LCM: You can also use the "prime factorization method" or the "division method" to find the LCM.
• LCM and GCD: The LCM and Greatest Common Divisor (GCD) are related concepts. The product of the LCM and GCD of two numbers is always equal to the product of the two numbers. In our example, GCD(15, 9) = 3. Therefore, LCM(15, 9) x GCD(15, 9) = 15 x 9 = 135, which is true.

Conclusion:

Finding the LCM of two numbers is a straightforward process. Understanding the concept of prime factorization and applying it to the calculation is crucial for solving problems involving LCMs. Remember that LCMs are not just theoretical concepts; they have practical applications in various fields, making them essential for understanding and solving real-world problems.