What Is The Lcm Of 6 8 9

Finding the Least Common Multiple (LCM) of 6, 8, and 9

This article will guide you through the process of calculating the least common multiple (LCM) of 6, 8, and 9. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cycles and timing. This step-by-step explanation will make finding the LCM of any set of numbers easy to understand.

What is the Least Common Multiple (LCM)?

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more numbers. In simpler terms, it's the smallest number that all the given numbers can divide into evenly. For instance, the LCM of 2 and 3 is 6, because 6 is the smallest number that is divisible by both 2 and 3.

Method 1: Prime Factorization

This is generally considered the most efficient method for finding the LCM of larger numbers. It involves breaking down each number into its prime factors.

1. Find the prime factorization of each number:

   • 6 = 2 x 3
   • 8 = 2 x 2 x 2 = 2³
   • 9 = 3 x 3 = 3²

2. Identify the highest power of each prime factor: Looking at the prime factorizations above, we see the prime factors are 2 and 3. The highest power of 2 is 2³ (from the factorization of 8), and the highest power of 3 is 3² (from the factorization of 9).

3. Multiply the highest powers together: Multiply 2³ and 3² to find the LCM.

   2³ x 3² = 8 x 9 = 72

Therefore, the LCM of 6, 8, and 9 is 72.

Method 2: Listing Multiples

This method is more suitable for smaller numbers. It involves listing the multiples of each number until you find the smallest common multiple.

1. List multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...

2. List multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, ...

3. List multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, ...

By comparing the lists, we can see that the smallest common multiple is 72. This method becomes less practical as the numbers get larger.

Conclusion

Both methods will give you the same answer: the LCM of 6, 8, and 9 is 72. The prime factorization method is generally preferred for its efficiency, especially when dealing with larger numbers or a greater number of numbers. Understanding LCM is fundamental for various mathematical concepts and problem-solving. Now you can confidently tackle similar LCM problems!