Least Common Multiple Of 6 7 9

Finding the Least Common Multiple (LCM) of 6, 7, and 9

This article will guide you through calculating the least common multiple (LCM) of 6, 7, and 9. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cycles and rhythms. We'll explore different methods to arrive at the solution, making it easy to understand, regardless of your mathematical background.

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 integers. In simpler terms, it's the smallest number that all the given numbers can divide into evenly. For example, the LCM of 2 and 3 is 6, because 6 is the smallest number divisible by both 2 and 3.

Methods for Finding the LCM of 6, 7, and 9

There are several ways to find the LCM of 6, 7, and 9. We'll explore two common methods: the prime factorization method and the listing multiples method.

Method 1: Prime Factorization

This method is generally preferred for larger numbers or when dealing with more than two numbers. It involves finding the prime factors of each number and then constructing the LCM using the highest powers of each prime factor.

1. Find the prime factorization of each number:

   • 6 = 2 x 3
   • 7 = 7 (7 is a prime number)
   • 9 = 3 x 3 = 3²

2. Identify the highest power of each prime factor:

   • The prime factors are 2, 3, and 7.
   • The highest power of 2 is 2¹ = 2.
   • The highest power of 3 is 3² = 9.
   • The highest power of 7 is 7¹ = 7.

3. Multiply the highest powers together:

   • LCM(6, 7, 9) = 2 x 3² x 7 = 2 x 9 x 7 = 126

Therefore, the least common multiple of 6, 7, and 9 is 126.

Method 2: Listing Multiples

This method is simpler for smaller numbers but can become cumbersome with larger numbers. It involves listing the multiples of each number until you find the smallest common multiple.

1. List the multiples of each number:

   • Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126...
   • Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126...
   • Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126...

2. Find the smallest common multiple:

   • By comparing the lists, we can see that the smallest number appearing in all three lists is 126.

Therefore, the least common multiple of 6, 7, and 9 is 126.

Conclusion:

Both methods lead to the same answer: the LCM of 6, 7, and 9 is 126. The prime factorization method is generally more efficient for larger numbers, while the listing multiples method is easier to visualize for smaller numbers. Understanding the concept of LCM and mastering these methods will significantly improve your ability to solve various mathematical problems. Remember to always check your work and choose the method best suited to the numbers involved.