How To Find The Lowest Common Multiple Of 3 Numbers

Finding the Least Common Multiple (LCM) of three numbers is a fundamental skill in mathematics, especially useful in arithmetic and algebra. The LCM is the smallest number that is a multiple of each of the given numbers. Understanding how to calculate it simplifies many mathematical problems, such as adding fractions with different denominators.

Understanding the Least Common Multiple (LCM)

The Least Common Multiple (LCM) of a set of numbers is the smallest positive integer that is divisible by all the numbers in the set. For example, consider the numbers 4, 6, and 8. Multiples of 4 include 4, 8, 12, 16, 20, 24, 28, ...; multiples of 6 include 6, 12, 18, 24, 30, ...; and multiples of 8 include 8, 16, 24, 32, .... The smallest number that appears in all three lists is 24, so the LCM of 4, 6, and 8 is 24.

Why is LCM Important?

The concept of LCM is crucial in various mathematical applications:

• 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.
• Time and Work Problems: LCM is useful in problems involving cycles or periodic events. For instance, if three machines perform a task at different intervals, the LCM can help determine when they will all perform the task simultaneously.
• Algebra: LCM is used in simplifying algebraic expressions and solving equations involving fractions or rational expressions.

Methods to Find the LCM of Three Numbers

There are several methods to calculate the LCM of three numbers, each with its own advantages. The most common methods are:

1. Listing Multiples
2. Prime Factorization
3. Division Method

Let's explore each of these methods in detail.

1. Listing Multiples

The most straightforward method is to list the multiples of each number until you find a common multiple. This method is best suited for small numbers because it can become cumbersome with larger numbers.

Steps:

1. List Multiples: Write down the multiples of each number.
2. Identify Common Multiples: Look for the multiples that appear in all three lists.
3. Find the Smallest: The smallest common multiple is the LCM.

Example:

Find the LCM of 3, 4, and 6.

• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...
• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, ...
• Multiples of 6: 6, 12, 18, 24, 30, ...

The smallest multiple common to all three numbers is 12. Therefore, the LCM of 3, 4, and 6 is 12.

Advantages:

• Simple to understand and apply.
• Good for small numbers.

Disadvantages:

• Time-consuming for larger numbers.
• Requires careful listing and comparison of multiples.

2. Prime Factorization

The prime factorization method involves breaking down each number into its prime factors and then using these factors to determine the LCM.

Steps:

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 all the highest powers of the 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 reliable.
• Effective for larger numbers.

Disadvantages:

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

3. Division Method

The division method is a systematic approach that involves dividing the numbers by their common prime factors until all the quotients are 1.

Steps:

1. Set Up Division: Write the numbers in a row, separated by commas or spaces.
2. Divide by Prime Factor: Find a prime number that divides at least two of the numbers. Divide the numbers by this prime factor and write the quotients below. If a number is not divisible, simply bring it down to the next row.
3. Repeat: Continue dividing by prime factors until all the numbers have been reduced to 1.
4. Multiply Divisors: Multiply all the prime factors used as divisors to get the LCM.

Example:

Find the LCM of 16, 24, and 36.

2 | 16, 24, 36
2 |  8, 12, 18
2 |  4,  6,  9
2 |  2,  3,  9
3 |  1,  3,  9
3 |  1,  1,  3
  |  1,  1,  1

Multiply all the prime factors: 2 * 2 * 2 * 2 * 3 * 3 = 144

Therefore, the LCM of 16, 24, and 36 is 144.

Advantages:

• Efficient and organized.
• Works well for multiple numbers.

Disadvantages:

• Requires familiarity with prime numbers.
• Can be prone to errors if not performed systematically.

Step-by-Step Examples

To further illustrate these methods, let's work through a few more examples.

Example 1: Finding the LCM of 8, 12, and 15

Method 1: Listing Multiples

• Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, ...
• Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ...
• Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, ...

The LCM of 8, 12, and 15 is 120.

Method 2: Prime Factorization

• 8 = 2^3
• 12 = 2^2 * 3
• 15 = 3 * 5

LCM = 2^3 * 3 * 5 = 8 * 3 * 5 = 120

Method 3: Division Method

2 | 8, 12, 15
2 | 4, 6, 15
2 | 2, 3, 15
3 | 1, 3, 15
5 | 1, 1, 5
  | 1, 1, 1

LCM = 2 * 2 * 2 * 3 * 5 = 120

Example 2: Finding the LCM of 9, 15, and 21

Method 1: Listing Multiples

• Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 261, 270, 279, 288, 297, 306, 315, ...
• Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270, 285, 300, 315, ...
• Multiples of 21: 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231, 252, 273, 294, 315, ...

The LCM of 9, 15, and 21 is 315.

Method 2: Prime Factorization

• 9 = 3^2
• 15 = 3 * 5
• 21 = 3 * 7

LCM = 3^2 * 5 * 7 = 9 * 5 * 7 = 315

Method 3: Division Method

3 | 9, 15, 21
3 | 3, 5, 7
5 | 1, 5, 7
7 | 1, 1, 7
  | 1, 1, 1

LCM = 3 * 3 * 5 * 7 = 315

Example 3: Finding the LCM of 14, 20, and 35

Method 1: Listing Multiples

• Multiples of 14: 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, ...
• Multiples of 20: 20, 40, 60, 80, 100, 120, 140, ...
• Multiples of 35: 35, 70, 105, 140, ...

The LCM of 14, 20, and 35 is 140.

Method 2: Prime Factorization

• 14 = 2 * 7
• 20 = 2^2 * 5
• 35 = 5 * 7

LCM = 2^2 * 5 * 7 = 4 * 5 * 7 = 140

Method 3: Division Method

2 | 14, 20, 35
2 | 7, 10, 35
5 | 7, 5, 35
7 | 7, 1, 7
  | 1, 1, 1

LCM = 2 * 2 * 5 * 7 = 140

Tips and Tricks

• Start with the Largest Number: When listing multiples, starting with the largest number can sometimes speed up the process.
• Use Prime Factorization for Large Numbers: For larger numbers, the prime factorization method is generally more efficient.
• Check Divisibility Rules: Use divisibility rules to quickly identify prime factors (e.g., a number ending in 0 or 5 is divisible by 5, a number is divisible by 3 if the sum of its digits is divisible by 3).
• Double-Check Your Work: Always double-check your calculations to avoid errors.

Common Mistakes to Avoid

• Incorrect Prime Factorization: Ensure you correctly break down each number into its prime factors. A mistake here will lead to an incorrect LCM.
• Missing Common Factors: When using the division method, make sure you divide by all common prime factors until the numbers are reduced to 1.
• Arithmetic Errors: Simple arithmetic errors can lead to incorrect results. Take your time and double-check each step.
• Stopping Too Early: Ensure you list enough multiples to find the smallest common one.

Real-World Applications

The concept of LCM is not just theoretical; it has practical applications in everyday life:

• Scheduling: If you have multiple tasks that occur at different intervals, the LCM can help you determine when they will all occur simultaneously. For example, scheduling maintenance for machines with different maintenance cycles.
• Cooking: In cooking, you might need to adjust ingredient quantities based on different serving sizes. LCM can help you scale recipes correctly.
• Travel: Planning trips involving different modes of transportation (e.g., buses, trains) with varying schedules can be simplified using LCM to coordinate arrival and departure times.
• Construction: When working with materials of different lengths, LCM can help you determine the shortest length you can cut to get equal-sized pieces without waste.

Advanced Topics and Extensions

• LCM of More Than Three Numbers: The same methods can be extended to find the LCM of more than three numbers. Just ensure you include all numbers in the process.
• Greatest Common Divisor (GCD): Understanding GCD (also known as Highest Common Factor or HCF) is closely related to LCM. The product of two numbers is equal to the product of their LCM and GCD.
• LCM in Algebra: LCM is used to simplify algebraic expressions involving rational functions.

Conclusion

Finding the LCM of three numbers is a valuable skill that simplifies many mathematical problems. Whether you choose to list multiples, use prime factorization, or apply the division method, understanding the underlying principles will help you solve LCM problems efficiently and accurately. With practice, you can master this skill and apply it to various real-world scenarios.