what is the least common multiple of 3 and 6

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14-10-2024

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Unveiling the Least Common Multiple of 3 and 6

The Least Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both. This concept is fundamental in mathematics and finds applications in various fields, such as scheduling and time calculations. Today, we'll explore the LCM of 3 and 6.

What is the LCM of 3 and 6?

The answer is 6. Here's why:

• Multiples of 3: 3, 6, 9, 12, 15...
• Multiples of 6: 6, 12, 18, 24...

As you can see, 6 is the smallest number that appears in both lists.

Finding the LCM: Methods and Explanations

While we visually identified the LCM above, there are systematic methods to calculate it:

1. Prime Factorization Method:

• Factorize each number into its prime factors:
  • 3 = 3
  • 6 = 2 x 3
• Identify the highest power of each prime factor:
  • 2 appears once in the factorization of 6.
  • 3 appears once in the factorization of both 3 and 6.
• Multiply the highest powers together: 2 x 3 = 6.

2. Listing Multiples Method:

This is the method we used initially. Simply list the multiples of both numbers until you find the smallest common one.

3. Using the Formula:

LCM (a, b) = (a * b) / GCD (a, b)

Where GCD is the Greatest Common Divisor.

• GCD (3, 6) = 3
• LCM (3, 6) = (3 * 6) / 3 = 6

Practical Application:

Imagine you have two machines that run cycles: one completes a cycle every 3 minutes, the other every 6 minutes. The LCM, 6 minutes, represents the time it takes for both machines to complete a cycle at the same time.

Conclusion:

Understanding the concept of LCM is crucial for various mathematical and practical scenarios. We've explored how to calculate the LCM of 3 and 6 using different methods, showcasing its applications in everyday life.