What Is The Lcm Of 24 And 14

Finding the Least Common Multiple (LCM) of 24 and 14

This article will guide you through the process of finding the least common multiple (LCM) of 24 and 14. Understanding LCMs is crucial in various mathematical applications, from simplifying fractions to solving problems in algebra and beyond. This guide will explain the concept and show you different methods to calculate the LCM, ensuring you grasp the fundamentals.

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 the 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 to Find the LCM of 24 and 14

There are several methods to determine the LCM, and we'll explore two common approaches: the prime factorization method and the listing multiples method.

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, 11, etc.).

1. Find the prime factorization of 24: 24 = 2 x 2 x 2 x 3 = 2³ x 3

2. Find the prime factorization of 14: 14 = 2 x 7

3. Identify the highest power of each prime factor present in either factorization: The prime factors are 2, 3, and 7. The highest power of 2 is 2³, the highest power of 3 is 3¹, and the highest power of 7 is 7¹.

4. Multiply the highest powers together: LCM(24, 14) = 2³ x 3 x 7 = 8 x 3 x 7 = 168

Therefore, the least common multiple of 24 and 14 is 168.

Method 2: Listing Multiples

This method is simpler for smaller numbers but becomes less efficient for larger ones.

1. List the multiples of 24: 24, 48, 72, 96, 120, 144, 168, 192...

2. List the multiples of 14: 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, ...

3. Find the smallest common multiple: The smallest number that appears in both lists is 168.

Therefore, the least common multiple of 24 and 14 is 168.

Conclusion:

Both methods demonstrate that the least common multiple of 24 and 14 is 168. The prime factorization method is generally more efficient, especially when dealing with larger numbers. Understanding how to calculate LCMs is a fundamental skill in mathematics and has practical applications across various fields. Remember to practice both methods to solidify your understanding and choose the most efficient method based on the numbers involved.