Least Common Multiple Of 20 And 24

Finding the Least Common Multiple (LCM) of 20 and 24: A Step-by-Step Guide

Finding the least common multiple (LCM) is a fundamental concept in mathematics with applications ranging from simple fraction addition to more complex scheduling problems. This article will provide a clear, step-by-step explanation of how to calculate the LCM of 20 and 24, along with different methods you can use. Understanding this process will solidify your grasp of LCM and improve your problem-solving skills in various mathematical contexts. We'll explore both the prime factorization method and the least common multiple formula method, ensuring a comprehensive understanding.

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 both numbers can divide into evenly. For instance, finding the LCM is crucial when adding or subtracting fractions with different denominators—you need the LCM to find the least common denominator.

Method 1: Prime Factorization

This is a widely used and generally preferred method for finding the LCM. It involves breaking down each number into its prime factors.

1. Find the prime factorization of each number:

   • 20 = 2 x 2 x 5 = 2² x 5
   • 24 = 2 x 2 x 2 x 3 = 2³ x 3

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

   • The prime factors are 2, 3, and 5.
   • The highest power of 2 is 2³ = 8.
   • The highest power of 3 is 3¹ = 3.
   • The highest power of 5 is 5¹ = 5.

3. Multiply the highest powers together:

   • LCM(20, 24) = 2³ x 3 x 5 = 8 x 3 x 5 = 120

Therefore, the least common multiple of 20 and 24 is 120.

Method 2: Using the Formula (LCM and GCD)

This method utilizes the relationship between the LCM and the greatest common divisor (GCD) of two numbers. The formula is:

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

Where:

• a and b are the two numbers (20 and 24 in this case)
• GCD(a, b) is the greatest common divisor of a and b.

1. Find the GCD of 20 and 24:

   One way to find the GCD is using the Euclidean algorithm. However, for smaller numbers, you can list the factors:

   Factors of 20: 1, 2, 4, 5, 10, 20 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

   The greatest common factor is 4. Therefore, GCD(20, 24) = 4.

2. Apply the formula:

   LCM(20, 24) = (20 x 24) / 4 = 480 / 4 = 120

Again, the least common multiple of 20 and 24 is 120.

Conclusion:

Both methods effectively determine the LCM of 20 and 24, resulting in the same answer: 120. The prime factorization method is often preferred for its simplicity and direct approach, especially when dealing with larger numbers. The formula method, however, highlights the mathematical relationship between LCM and GCD, offering a different perspective on the problem. Choosing the best method depends on your comfort level and the specific numbers involved. Understanding both approaches will enhance your mathematical skills and problem-solving capabilities.