What Is The Lowest Common Multiple Of 30 And 42

What is the Lowest Common Multiple (LCM) of 30 and 42? A Step-by-Step Guide

Finding the lowest common multiple (LCM) is a fundamental concept in mathematics, crucial for various applications, from simplifying fractions to solving complex equations. This article will clearly explain how to calculate the LCM of 30 and 42, using multiple methods, ensuring you understand the process completely. This guide is perfect for students learning about LCM, as well as anyone who needs a refresher on this important mathematical skill.

What is the Lowest Common Multiple (LCM)?

The lowest common multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by all the numbers without leaving a remainder. Understanding LCM is essential for various mathematical operations and real-world applications.

Method 1: Prime Factorization

This is arguably the most efficient and commonly used method for finding the LCM of larger numbers. It involves breaking down each number into its prime factors.

1. Find the prime factorization of each number:

   • 30 = 2 x 3 x 5
   • 42 = 2 x 3 x 7

2. Identify the highest power of each prime factor present in either factorization:

   • The prime factors are 2, 3, 5, and 7.
   • The highest power of 2 is 2¹
   • The highest power of 3 is 3¹
   • The highest power of 5 is 5¹
   • The highest power of 7 is 7¹

3. Multiply these highest powers together:

   LCM(30, 42) = 2 x 3 x 5 x 7 = 210

Therefore, the lowest common multiple of 30 and 42 is 210.

Method 2: Listing Multiples

This method is suitable for smaller numbers. It involves listing the multiples of each number until you find the smallest common multiple.

1. List the multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240...
2. List the multiples of 42: 42, 84, 126, 168, 210, 252...

Notice that the smallest multiple common to both lists is 210. This method becomes less practical with larger numbers.

Method 3: Using the Greatest Common Divisor (GCD)

The LCM and GCD (Greatest Common Divisor) are related. You can find the LCM using the GCD and the following formula:

LCM(a, b) = (|a x b|) / GCD(a, b)

1. Find the GCD of 30 and 42 using the Euclidean algorithm or prime factorization:

   • Prime factorization is quicker: 30 = 2 x 3 x 5; 42 = 2 x 3 x 7. The common factors are 2 and 3, so GCD(30, 42) = 2 x 3 = 6

2. Apply the formula:

   LCM(30, 42) = (30 x 42) / 6 = 1260 / 6 = 210

Conclusion:

Regardless of the method employed, the lowest common multiple of 30 and 42 is definitively 210. Choosing the best method depends on the numbers involved. Prime factorization is generally preferred for larger numbers due to its efficiency. Understanding the LCM is crucial for various mathematical operations and problem-solving scenarios. Mastering this concept will solidify your foundation in number theory.