Least Common Multiple Of 4 6 7

Finding the Least Common Multiple (LCM) of 4, 6, and 7

This article will guide you through the process of calculating the least common multiple (LCM) of 4, 6, and 7. The LCM is the smallest positive integer that is divisible by all the given numbers. This concept is fundamental in various mathematical applications, including simplifying fractions and solving problems involving cyclical events. Understanding how to find the LCM is crucial for anyone studying mathematics, particularly at the middle school and high school levels. We'll explore several methods to solve this, ensuring you grasp the underlying principles.

Method 1: Prime Factorization

This is a widely used and efficient method for finding the LCM. It involves breaking down each number into its prime factors. Let's begin:

4: The prime factorization of 4 is 2 x 2, or 2².
6: The prime factorization of 6 is 2 x 3.
7: The prime factorization of 7 is 7 (7 is a prime number).

Now, to find the LCM, we take the highest power of each prime factor present in the factorizations:

The highest power of 2 is 2².
The highest power of 3 is 3.
The highest power of 7 is 7.

Multiply these highest powers together: 2² x 3 x 7 = 4 x 3 x 7 = 84.

Therefore, the least common multiple of 4, 6, and 7 is 84.

Method 2: Listing Multiples

This method is more straightforward for smaller numbers but can become cumbersome for larger ones. We list the multiples of each number until we find the smallest common multiple.

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84...
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84...
Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84...

Notice that 84 is the smallest number that appears in all three lists. Thus, the LCM of 4, 6, and 7 is 84.

Method 3: Using the Greatest Common Divisor (GCD)

The LCM and GCD are related. There's a formula that connects them:

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

This can be extended to more than two numbers, but it's more complex. While it's possible to use this method for 4, 6, and 7, the prime factorization method is generally more efficient for multiple numbers.

To illustrate, let's find the LCM of 4 and 6 first using this method. The GCD of 4 and 6 is 2. Therefore, LCM(4,6) = (4 x 6) / 2 = 12. Then we find the LCM of 12 and 7 using the same method. The GCD of 12 and 7 is 1. Thus LCM(12,7) = (12 x 7) / 1 = 84.

Conclusion

We've explored three different methods to calculate the least common multiple of 4, 6, and 7. The prime factorization method is often the most efficient and conceptually clear, especially when dealing with larger numbers or multiple numbers. Remember to choose the method that best suits your understanding and the complexity of the problem. Regardless of the method, the answer remains consistent: the LCM of 4, 6, and 7 is 84.