How To Find Least Common Multiple Of 3 Numbers

Finding the Least Common Multiple (LCM) of three numbers is a fundamental skill in mathematics, especially useful in simplifying fractions, solving word problems, and understanding number theory. The LCM is the smallest number that is a multiple of each of the given numbers. Mastering this skill requires understanding prime factorization, the greatest common divisor (GCD), and a systematic approach to calculation. This comprehensive guide provides a detailed explanation of how to find the LCM of three numbers, complete with examples and practical applications.

Understanding the Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by each of those numbers without leaving a remainder. In simpler terms, it's the smallest number that all the given numbers can divide into evenly. The LCM is crucial in various mathematical operations, such as adding or subtracting fractions with different denominators.

Why is LCM Important?

• Simplifying Fractions: When adding or subtracting fractions with different denominators, you need to find a common denominator. The LCM of the denominators is the least common denominator, which simplifies the process.
• Solving Word Problems: Many real-world problems involve finding a common point or cycle. The LCM helps in determining when events will coincide.
• Number Theory: The LCM is a fundamental concept in number theory, providing insights into the relationships between numbers and their multiples.

Methods to Find the LCM of Three Numbers

There are several methods to find the LCM of three numbers. The most common ones include:

1. Listing Multiples
2. Prime Factorization
3. Using the Greatest Common Divisor (GCD)

Each method has its advantages and is suitable for different types of numbers. Let's explore each method in detail.

Method 1: Listing Multiples

The listing multiples method involves writing out the multiples of each number until you find a common multiple. This method is straightforward and easy to understand, making it suitable for small numbers.

Steps to Find the LCM by Listing Multiples:

1. List Multiples: Write down the multiples of each number.
2. Identify Common Multiples: Look for multiples that are common to all three lists.
3. Find the Least Common Multiple: Identify the smallest common multiple among the lists.

Example:

Find the LCM of 4, 6, and 8.

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, ...
• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...
• Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, ...

The common multiples are 24, 48, and so on. The smallest common multiple is 24.

Therefore, the LCM of 4, 6, and 8 is 24.

Advantages:

• Simple and easy to understand.
• Suitable for small numbers.

Disadvantages:

• Time-consuming for larger numbers.
• Not practical for numbers with no obvious common multiples.

Method 2: Prime Factorization

The prime factorization method involves breaking down each number into its prime factors. This method is more systematic and efficient, especially for larger numbers.

Steps to Find the LCM by Prime Factorization:

1. Prime Factorization: Find the prime factorization of each number.
2. Identify Highest Powers: For each prime factor, identify the highest power that appears in any of the factorizations.
3. Multiply Highest Powers: Multiply together the highest powers of all prime factors.

Example:

Find the LCM of 12, 18, and 30.

1. Prime Factorization:
   • 12 = 2^2 * 3
   • 18 = 2 * 3^2
   • 30 = 2 * 3 * 5
2. Identify Highest Powers:
   • Highest power of 2: 2^2
   • Highest power of 3: 3^2
   • Highest power of 5: 5
3. Multiply Highest Powers:
   • LCM = 2^2 * 3^2 * 5 = 4 * 9 * 5 = 180

Therefore, the LCM of 12, 18, and 30 is 180.

Advantages:

• Systematic and efficient.
• Suitable for larger numbers.

Disadvantages:

• Requires knowledge of prime factorization.
• Can be time-consuming to find prime factors for very large numbers.

Method 3: Using the Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) method involves finding the GCD of the numbers and using it to calculate the LCM. This method is particularly useful when you already know the GCD or can easily find it.

Steps to Find the LCM using GCD:

1. Find the GCD: Find the GCD of the three numbers.

2. Use the Formula: Use the formula:

   LCM(a, b, c) = |a * b * c| / GCD(a * b, a * c, b * c)

   However, a more straightforward approach involves finding the LCM of two numbers first, then finding the LCM of that result with the third number.

   • Find LCM(a, b) = |a * b| / GCD(a, b)
   • Then, find LCM(LCM(a, b), c) = |LCM(a, b) * c| / GCD(LCM(a, b), c)

Example:

Find the LCM of 16, 24, and 40.

1. Find the GCD:

   • GCD(16, 24) = 8
   • GCD(8, 40) = 8

2. Find LCM(16, 24):

   • LCM(16, 24) = |16 * 24| / GCD(16, 24) = 384 / 8 = 48

3. Find LCM(48, 40):

   • GCD(48, 40) = 8
   • LCM(48, 40) = |48 * 40| / GCD(48, 40) = 1920 / 8 = 240

Therefore, the LCM of 16, 24, and 40 is 240.

Advantages:

• Efficient if the GCD is known or easy to find.
• Useful for simplifying calculations.

Disadvantages:

• Requires knowledge of GCD calculation.
• Can be complex if the GCD is not readily apparent.

Step-by-Step Examples

Let's go through a few more examples to solidify your understanding.

Example 1: Finding the LCM of 9, 15, and 21

Method: Prime Factorization

1. Prime Factorization:
   • 9 = 3^2
   • 15 = 3 * 5
   • 21 = 3 * 7
2. Identify Highest Powers:
   • Highest power of 3: 3^2
   • Highest power of 5: 5
   • Highest power of 7: 7
3. Multiply Highest Powers:
   • LCM = 3^2 * 5 * 7 = 9 * 5 * 7 = 315

Therefore, the LCM of 9, 15, and 21 is 315.

Example 2: Finding the LCM of 10, 25, and 35

Method: Prime Factorization

1. Prime Factorization:
   • 10 = 2 * 5
   • 25 = 5^2
   • 35 = 5 * 7
2. Identify Highest Powers:
   • Highest power of 2: 2
   • Highest power of 5: 5^2
   • Highest power of 7: 7
3. Multiply Highest Powers:
   • LCM = 2 * 5^2 * 7 = 2 * 25 * 7 = 350

Therefore, the LCM of 10, 25, and 35 is 350.

Example 3: Finding the LCM of 6, 14, and 16

Method: Prime Factorization

1. Prime Factorization:
   • 6 = 2 * 3
   • 14 = 2 * 7
   • 16 = 2^4
2. Identify Highest Powers:
   • Highest power of 2: 2^4
   • Highest power of 3: 3
   • Highest power of 7: 7
3. Multiply Highest Powers:
   • LCM = 2^4 * 3 * 7 = 16 * 3 * 7 = 336

Therefore, the LCM of 6, 14, and 16 is 336.

Tips and Tricks for Finding the LCM

• Start with Prime Factorization: Prime factorization is often the most efficient method, especially for larger numbers.
• Look for Common Factors: Identifying common factors can simplify the process.
• Use GCD When Possible: If you know the GCD, use it to simplify the LCM calculation.
• Check Your Answer: Ensure that the LCM you find is divisible by all the given numbers.
• Practice Regularly: The more you practice, the better you'll become at finding the LCM.

Practical Applications of LCM

The LCM is not just a theoretical concept; it has numerous practical applications in various fields.

1. Scheduling

Problem: Three buses leave a station at 6:00 AM. The first bus returns every 45 minutes, the second every 60 minutes, and the third every 75 minutes. At what time will all three buses be at the station together again?

Solution:

1. Find the LCM of 45, 60, and 75.

   • Prime Factorization:
     • 45 = 3^2 * 5
     • 60 = 2^2 * 3 * 5
     • 75 = 3 * 5^2
   • Highest Powers:
     • 2^2, 3^2, 5^2
   • LCM = 2^2 * 3^2 * 5^2 = 4 * 9 * 25 = 900

   The LCM is 900 minutes, which is 15 hours.

   The buses will be at the station together again 15 hours after 6:00 AM, which is 9:00 PM.

2. Mixing Ingredients

Problem: A recipe calls for ingredients A, B, and C. Ingredient A is added every 8 minutes, ingredient B every 12 minutes, and ingredient C every 15 minutes. If all ingredients are added together at the start, when will they be added together again?

Solution:

1. Find the LCM of 8, 12, and 15.

   • Prime Factorization:
     • 8 = 2^3
     • 12 = 2^2 * 3
     • 15 = 3 * 5
   • Highest Powers:
     • 2^3, 3, 5
   • LCM = 2^3 * 3 * 5 = 8 * 3 * 5 = 120

   The LCM is 120 minutes, which is 2 hours.

   The ingredients will be added together again after 2 hours.

3. Tiling and Patterns

Problem: You want to tile a floor using square tiles of sizes 6 inches, 8 inches, and 10 inches. What is the smallest square area you can tile completely using each size of tile?

Solution:

1. Find the LCM of 6, 8, and 10.

   • Prime Factorization:
     • 6 = 2 * 3
     • 8 = 2^3
     • 10 = 2 * 5
   • Highest Powers:
     • 2^3, 3, 5
   • LCM = 2^3 * 3 * 5 = 8 * 3 * 5 = 120

   The LCM is 120 inches.

   The smallest square area you can tile completely using each size of tile is 120 inches by 120 inches, which is 14,400 square inches.

Common Mistakes to Avoid

• Incorrect Prime Factorization: Ensure that you correctly identify the prime factors of each number.
• Missing Common Factors: Double-check that you have identified all common factors.
• Using GCD Incorrectly: Make sure you are using the GCD formula correctly.
• Not Checking the Answer: Always verify that the LCM you find is divisible by all the given numbers.
• Rushing the Process: Take your time and be systematic to avoid errors.

Conclusion

Finding the Least Common Multiple (LCM) of three numbers is a valuable skill with numerous practical applications. By understanding the methods of listing multiples, prime factorization, and using the Greatest Common Divisor (GCD), you can efficiently calculate the LCM for any set of numbers. Remember to practice regularly, avoid common mistakes, and apply these techniques to real-world problems to strengthen your understanding. With this comprehensive guide, you are well-equipped to tackle any LCM-related challenge with confidence and precision.