Least Common Multiple Of 10 And 9

Finding the Least Common Multiple (LCM) of 10 and 9

This article will guide you through calculating the least common multiple (LCM) of 10 and 9. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cycles and patterns. This guide will cover different methods for finding the LCM, making it easy to understand regardless of your mathematical background. We'll explore the prime factorization method and the listing multiples method, providing a clear and concise explanation of each.

What is the Least Common Multiple (LCM)?

The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by all of the integers. For instance, the LCM of 2 and 3 is 6, because 6 is the smallest number that is both a multiple of 2 and a multiple of 3. Finding the LCM is a fundamental concept in number theory and has practical applications in various fields.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors. Prime factors are numbers that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, etc.).

1. Find the prime factorization of 10: 10 = 2 x 5
2. Find the prime factorization of 9: 9 = 3 x 3 = 3²
3. Identify the highest power of each prime factor: We have 2, 3, and 5 as prime factors. The highest power of 2 is 2¹, the highest power of 3 is 3², and the highest power of 5 is 5¹.
4. Multiply the highest powers together: 2¹ x 3² x 5¹ = 2 x 9 x 5 = 90

Therefore, the LCM of 10 and 9 is 90.

Method 2: Listing Multiples

This method involves listing out the multiples of each number until you find the smallest common multiple.

1. List the multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100...
2. List the multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99...
3. Identify the smallest common multiple: The smallest number that appears in both lists is 90.

Therefore, the LCM of 10 and 9 is 90.

Conclusion:

Both methods effectively determine the least common multiple of 10 and 9, resulting in the answer 90. The prime factorization method is generally more efficient for larger numbers, while the listing multiples method is easier to visualize for smaller numbers. Understanding both methods provides a versatile approach to solving LCM problems. Remember that the LCM is a crucial concept with practical applications in various mathematical contexts.