Least Common Multiple Of 42 And 24

Finding the Least Common Multiple (LCM) of 42 and 24

This article will guide you through the process of calculating the least common multiple (LCM) of 42 and 24. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cycles and repetitions. We'll explore several methods to determine the LCM, making it easy to understand regardless of your mathematical background.

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. In simpler terms, it's the smallest number that contains all the numbers as factors. For example, the LCM of 2 and 3 is 6 because 6 is the smallest number divisible by both 2 and 3.

Methods for Calculating the LCM of 42 and 24

There are several effective methods to find the LCM of 42 and 24. We'll explore two common approaches: the prime factorization method and the listing multiples method.

1. Prime Factorization Method

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.).

• Prime factorization of 42: 2 x 3 x 7
• Prime factorization of 24: 2 x 2 x 2 x 3 (or 2³ x 3)

Now, to find the LCM, we take the highest power of each prime factor present in either factorization and multiply them together:

LCM(42, 24) = 2³ x 3 x 7 = 8 x 3 x 7 = 168

Therefore, the least common multiple of 42 and 24 is 168.

2. Listing Multiples Method

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

• Multiples of 42: 42, 84, 126, 168, 210, ...
• Multiples of 24: 24, 48, 72, 96, 120, 144, 168, ...

By comparing the lists, we see that the smallest common multiple is 168. This method is simple for smaller numbers but can become cumbersome with larger numbers.

Choosing the Best Method

The prime factorization method is generally more efficient, especially when dealing with larger numbers. The listing multiples method is easier to visualize but less practical for larger numbers. Understanding both methods provides flexibility in approaching LCM problems.

Applications of LCM

The LCM has practical applications in various fields, including:

• Fraction addition and subtraction: Finding a common denominator.
• Scheduling problems: Determining when events will occur simultaneously (e.g., buses arriving at a stop).
• Cyclic patterns: Identifying when repeating patterns align.

Conclusion

Calculating the least common multiple is a fundamental mathematical skill with many practical applications. By understanding and applying the prime factorization method or the listing multiples method, you can efficiently determine the LCM of any two or more integers. Remember to choose the method that best suits the numbers involved for optimal efficiency. The LCM of 42 and 24, as demonstrated above, is definitively 168.