What Is The Lcm Of 9 12 15

Finding the Least Common Multiple (LCM) of 9, 12, and 15

This article will guide you through the process of calculating the Least Common Multiple (LCM) of 9, 12, and 15. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cyclical events. This guide will explain the concept clearly and demonstrate different methods to find the LCM, making it easily understandable for everyone.

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 example, the LCM of 2 and 3 is 6, because 6 is the smallest number that is divisible by both 2 and 3.

Methods for Finding the LCM of 9, 12, and 15

There are several ways to calculate the LCM. We'll explore two common methods: the prime factorization method and the listing multiples method.

1. Prime Factorization Method

This method is generally considered the most efficient, especially when dealing with larger numbers. It involves breaking down each number into its prime factors.

• Step 1: Find the prime factorization of each number:

  • 9 = 3 x 3 = 3²
  • 12 = 2 x 2 x 3 = 2² x 3
  • 15 = 3 x 5

• Step 2: Identify the highest power of each prime factor:

  The prime factors present are 2, 3, and 5. The highest power of 2 is 2², the highest power of 3 is 3², and the highest power of 5 is 5.

• Step 3: Multiply the highest powers together:

  LCM(9, 12, 15) = 2² x 3² x 5 = 4 x 9 x 5 = 180

Therefore, the least common multiple of 9, 12, and 15 is 180.

2. Listing Multiples Method

This method is straightforward but can be time-consuming for larger numbers. It involves listing the multiples of each number until you find the smallest common multiple.

• Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, ...
• Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180,...
• Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 180, ...

As you can see, the smallest multiple common to all three numbers is 180.

Conclusion

Both methods will lead to the same answer: the LCM of 9, 12, and 15 is 180. The prime factorization method is generally more efficient for larger numbers, while the listing multiples method is easier to understand for beginners. Understanding LCM is a fundamental skill in mathematics and has practical applications in various fields. Now you can confidently tackle similar problems involving finding the least common multiple of different sets of numbers!