What Is The Least Common Multiple (LCM) And How To Find It?

The Least Common Multiple (LCM) is the smallest multiple that two or more numbers share, and finding it doesn’t have to be a headache. At what.edu.vn, we provide quick and free answers to all your questions, including those tricky math concepts. Discover the easiest methods to calculate the LCM, understand its applications, and boost your math skills with our comprehensive guide and get clarity on multiple math concepts, exploring lowest common denominator and common multiples.

1. Understanding the Basics: What Is the Least Common Multiple (LCM)?

The least common multiple, often abbreviated as LCM, is the smallest positive integer that is evenly divisible by two or more given numbers. Simply put, it’s the smallest number that all the given numbers can divide into without leaving a remainder. Understanding LCM is crucial for various mathematical operations, especially when dealing with fractions.

For instance, if you have the numbers 4 and 6:

• Multiples of 4: 4, 8, 12, 16, 20, 24…
• Multiples of 6: 6, 12, 18, 24, 30…

The least common multiple of 4 and 6 is 12 because it is the smallest number that appears in both lists of multiples.

So, what is the lowest common multiple in math terms? It’s the smallest number that’s a multiple of each of the numbers you started with.

2. Why Is the Least Common Multiple Important?

Understanding the least common multiple (LCM) isn’t just an academic exercise; it has practical applications in everyday life and is essential for more advanced mathematical concepts. Here are some key reasons why LCM is important:

• Simplifying Fractions: LCM is critical when adding or subtracting fractions with different denominators. By finding the least common denominator (LCD), which is the LCM of the denominators, you can easily combine the fractions.
• Solving Problems Involving Ratios and Proportions: LCM helps in determining the quantities required to maintain a specific ratio. For example, in cooking, if a recipe needs to be scaled up or down, LCM can help in adjusting the ingredients accurately.
• Scheduling Events: LCM can be used to schedule recurring events that happen at different intervals. For instance, if one task occurs every 6 days and another every 8 days, the LCM (24) tells you when both tasks will occur on the same day.
• Understanding Number Theory: LCM is a fundamental concept in number theory, which studies the properties and relationships of numbers. It helps in understanding the divisibility rules and prime factorization of numbers.
• Real-World Applications: LCM finds applications in various fields such as engineering, computer science, and finance. For example, it can be used in optimizing processes, scheduling tasks, and managing resources efficiently.

3. Methods to Find the Least Common Multiple

There are several methods to calculate the LCM of two or more numbers, each with its own advantages and suitability for different types of problems. Let’s explore three common methods: listing multiples, prime factorization, and the division method.

3.1. Listing Multiples

This method involves listing the multiples of each number until you find a common multiple. The smallest common multiple is the LCM.

How to do it:

1. Write down the multiples of each number.
2. Identify the common multiples in the lists.
3. Select the smallest of these common multiples.

Example: Find the LCM of 6 and 8.

• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, …
• Multiples of 8: 8, 16, 24, 32, 40, 48, 56, …

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

Therefore, the LCM of 6 and 8 is 24.

Pros:

• Simple and easy to understand, especially for small numbers.
• Requires no advanced mathematical knowledge.

Cons:

• Can be time-consuming and impractical for large numbers.
• Difficult to use when dealing with more than two numbers.

3.2. Prime Factorization

This method involves breaking down each number into its prime factors and then combining those factors to find the LCM.

How to do it:

1. Find the prime factorization of each number.
2. Identify all the unique prime factors.
3. For each prime factor, take the highest power that appears in any of the factorizations.
4. Multiply these highest powers together to get the LCM.

Example: Find the LCM of 12 and 18.

• Prime factorization of 12: 2^2 × 3
• Prime factorization of 18: 2 × 3^2

1. Unique prime factors: 2 and 3.
2. Highest power of 2: 2^2.
3. Highest power of 3: 3^2.
4. LCM = 2^2 × 3^2 = 4 × 9 = 36.

Therefore, the LCM of 12 and 18 is 36.

Pros:

• Works well for larger numbers.
• Systematic and efficient.

Cons:

• Requires knowledge of prime factorization.
• Can be a bit more complex than listing multiples.

3.3. Division Method

The division method involves dividing the given numbers by their common prime factors until you are left with no common factors. The LCM is then the product of the divisors and the remaining numbers.

How to do it:

1. Write the numbers side by side.
2. Divide by a common prime factor.
3. Bring down any numbers that are not divisible.
4. Repeat until there are no more common prime factors.
5. Multiply all the divisors and the remaining numbers.

Example: Find the LCM of 16 and 20.

2 | 16  20
2 | 8   10
2 | 4   5
2 | 2   5
  | 1   5
  | 1   1  (5)

LCM = 2 × 2 × 2 × 2 × 5 = 80.

Therefore, the LCM of 16 and 20 is 80.

Pros:

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

Cons:

• Requires knowledge of prime numbers.
• Can be confusing if not done systematically.

Each of these methods offers a different approach to finding the LCM. The best method to use depends on the specific numbers involved and your comfort level with each technique. Whether you prefer listing multiples, breaking down numbers into prime factors, or using the division method, understanding these tools will help you tackle a wide range of mathematical problems with confidence.

4. Step-by-Step Examples of Finding the Least Common Multiple

To solidify your understanding of how to find the least common multiple (LCM), let’s walk through a few step-by-step examples using different methods.

4.1. Example 1: Listing Multiples

Problem: Find the LCM of 3 and 5.

Solution:

1. List the multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, …
2. List the multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, …

The smallest common multiple is 15.

Answer: The LCM of 3 and 5 is 15.

This method is straightforward and easy to understand, making it ideal for smaller numbers.

4.2. Example 2: Prime Factorization

Problem: Find the LCM of 24 and 36.

Solution:

1. Find the prime factorization of each number:

   • 24 = 2 × 2 × 2 × 3 = 2^3 × 3
   • 36 = 2 × 2 × 3 × 3 = 2^2 × 3^2

2. Identify all unique prime factors: 2 and 3.

3. For each prime factor, take the highest power that appears in any of the factorizations:

   • Highest power of 2: 2^3
   • Highest power of 3: 3^2

4. Multiply these highest powers together:

   • LCM = 2^3 × 3^2 = 8 × 9 = 72

Answer: The LCM of 24 and 36 is 72.

This method is efficient for larger numbers and provides a systematic way to find the LCM.

4.3. Example 3: Division Method

Problem: Find the LCM of 15, 20, and 25.

Solution:

1. Write the numbers side by side: 15 20 25
2. Divide by a common prime factor:

5 | 15  20  25
  | 3   4   5

No more common prime factors.

Multiply all the divisors and the remaining numbers:

LCM = 5 × 3 × 4 × 5 = 300

Answer: The LCM of 15, 20, and 25 is 300.

This method is particularly useful when finding the LCM of multiple numbers simultaneously.

4.4. Additional Tips

• Start with Smaller Prime Numbers: When using prime factorization or the division method, start with smaller prime numbers like 2, 3, and 5 to simplify the process.
• Double-Check Your Work: Always double-check your prime factorizations and divisions to avoid errors.
• Practice Regularly: The more you practice, the more comfortable you’ll become with these methods.

By working through these examples and applying the tips, you’ll gain confidence in finding the LCM of any set of numbers. Each method has its advantages, so choose the one that best suits your style and the problem at hand.

5. The Relationship Between LCM and Greatest Common Factor (GCF)

The Least Common Multiple (LCM) and the Greatest Common Factor (GCF) are two fundamental concepts in number theory. While they serve different purposes, they are closely related. Understanding this relationship can simplify problem-solving and provide deeper insights into number properties.

5.1. Defining GCF

The Greatest Common Factor (GCF), also known as the Highest Common Factor (HCF), is the largest positive integer that divides two or more numbers without leaving a remainder.

Example: For the numbers 12 and 18:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18

The greatest common factor of 12 and 18 is 6 because it is the largest number that appears in both lists of factors.

5.2. The Relationship Formula

The relationship between LCM and GCF can be expressed by the following formula:

LCM(a, b) × GCF(a, b) = a × b

Where:

• LCM(a, b) is the least common multiple of numbers a and b.
• GCF(a, b) is the greatest common factor of numbers a and b.
• a and b are the two numbers.

This formula states that the product of the LCM and GCF of two numbers is equal to the product of the numbers themselves.

5.3. Using the Relationship to Find LCM or GCF

This relationship can be used to find either the LCM or the GCF if you know the other value and the original numbers.

Example 1: Finding LCM using GCF

Suppose you know that the GCF of 24 and 36 is 12. Find the LCM.

1. Use the formula: LCM(a, b) × GCF(a, b) = a × b
2. Plug in the known values: LCM(24, 36) × 12 = 24 × 36
3. Solve for LCM: LCM(24, 36) = (24 × 36) / 12
4. LCM(24, 36) = 864 / 12 = 72

Therefore, the LCM of 24 and 36 is 72.

Example 2: Finding GCF using LCM

Suppose you know that the LCM of 15 and 20 is 60. Find the GCF.

1. Use the formula: LCM(a, b) × GCF(a, b) = a × b
2. Plug in the known values: 60 × GCF(15, 20) = 15 × 20
3. Solve for GCF: GCF(15, 20) = (15 × 20) / 60
4. GCF(15, 20) = 300 / 60 = 5

Therefore, the GCF of 15 and 20 is 5.

5.4. Why This Relationship Works

The relationship between LCM and GCF stems from the prime factorization of the numbers. The GCF includes the common prime factors raised to the lowest power, while the LCM includes all prime factors raised to the highest power. When you multiply the LCM and GCF, you are essentially including all prime factors raised to the appropriate powers to reconstruct the original numbers.

5.5. Applications of the Relationship

• Simplifying Calculations: Knowing this relationship can simplify calculations when you only need to find one of the values (LCM or GCF).
• Checking Answers: You can use this formula to check if your calculated LCM and GCF values are correct.
• Understanding Number Properties: This relationship provides a deeper understanding of how numbers interact and relate to each other.

By understanding the relationship between LCM and GCF, you can enhance your problem-solving skills and gain a more comprehensive understanding of number theory.

6. Real-World Applications of the Least Common Multiple

The Least Common Multiple (LCM) is not just an abstract mathematical concept; it has numerous practical applications in various real-world scenarios. Understanding these applications can help you appreciate the relevance of LCM in everyday life.

6.1. Scheduling

LCM is commonly used in scheduling events that occur at regular intervals.

Example: Two buses leave a station at the same time. Bus A leaves every 15 minutes, and Bus B leaves every 20 minutes. When will they leave the station together again?

• Find the LCM of 15 and 20.
• Multiples of 15: 15, 30, 45, 60, 75, …
• Multiples of 20: 20, 40, 60, 80, …
• The LCM of 15 and 20 is 60.

Answer: The buses will leave the station together again in 60 minutes.

6.2. Cooking

LCM is useful when adjusting recipes to serve different numbers of people.

Example: A recipe calls for 2 cups of flour and 3 eggs. You want to use up a bag of flour that has 8 cups. How many eggs do you need?

• Find the LCM of 2 (cups of flour in the recipe) and 8 (cups of flour you have).
• Multiples of 2: 2, 4, 6, 8, …
• Multiples of 8: 8, 16, …
• The LCM of 2 and 8 is 8.
• Since you have 8 cups of flour (4 times the original recipe), you need 4 times the original number of eggs.
• 4 × 3 eggs = 12 eggs

Answer: You need 12 eggs.

6.3. Tiling and Construction

LCM helps in planning the layout of tiles or other materials to minimize waste.

Example: You want to tile a rectangular floor with dimensions 12 feet by 18 feet using square tiles. What is the largest size of square tiles you can use without having to cut any tiles?

• Find the Greatest Common Factor (GCF) of 12 and 18.
• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• The GCF of 12 and 18 is 6.

Answer: You can use 6×6 feet tiles.

6.4. Gear Ratios

In mechanical engineering, LCM is used to determine the gear ratios in machines.

Example: Two gears are meshed together. Gear A has 24 teeth, and Gear B has 36 teeth. How many rotations must each gear make before they return to their starting positions relative to each other?

• Find the LCM of 24 and 36.
• Prime factorization of 24: 2^3 × 3
• Prime factorization of 36: 2^2 × 3^2
• LCM = 2^3 × 3^2 = 8 × 9 = 72

Gear A must make 72 / 24 = 3 rotations.

Gear B must make 72 / 36 = 2 rotations.

Answer: Gear A must make 3 rotations, and Gear B must make 2 rotations.

6.5. Music

LCM is used in music to understand the relationship between different rhythmic patterns.

Example: One musician plays a beat every 4 seconds, and another plays a beat every 6 seconds. When will they play a beat together?

• Find the LCM of 4 and 6.
• Multiples of 4: 4, 8, 12, 16, …
• Multiples of 6: 6, 12, 18, …
• The LCM of 4 and 6 is 12.

Answer: They will play a beat together every 12 seconds.

6.6. Computer Science

LCM is used in computer science for tasks such as scheduling processes and allocating resources.

Example: Two processes in a computer system need to access a shared resource. Process A needs the resource every 8 milliseconds, and Process B needs it every 12 milliseconds. When will both processes need the resource at the same time?

• Find the LCM of 8 and 12.
• Multiples of 8: 8, 16, 24, 32, …
• Multiples of 12: 12, 24, 36, …
• The LCM of 8 and 12 is 24.

Answer: Both processes will need the resource at the same time every 24 milliseconds.

7. Common Mistakes to Avoid When Finding the Least Common Multiple

Finding the Least Common Multiple (LCM) can sometimes be tricky, and it’s easy to make mistakes if you’re not careful. Here are some common mistakes to watch out for:

7.1. Confusing LCM with GCF

One of the most common mistakes is confusing the Least Common Multiple (LCM) with the Greatest Common Factor (GCF). Remember that the LCM is the smallest multiple that two or more numbers share, while the GCF is the largest factor that divides two or more numbers.

Mistake: Confusing LCM and GCF.

Example: Finding the LCM of 12 and 18 but calculating the GCF instead.

• Correct LCM: 36
• GCF: 6 (This is incorrect for LCM)

How to Avoid: Always double-check whether you are asked to find the LCM or the GCF. Understand the definitions and remember that LCM is always greater than or equal to the given numbers, while GCF is always less than or equal to the given numbers.

7.2. Incorrectly Listing Multiples

When using the listing multiples method, it’s crucial to list the multiples correctly. Missing a multiple or listing them incorrectly can lead to an incorrect LCM.

Mistake: Missing a multiple in the list.

Example: Finding the LCM of 4 and 6:

• Incorrect multiples of 4: 4, 8, 12, 16, 20 (missing 24, 28, etc.)
• Incorrect multiples of 6: 6, 12, 18 (missing 24, 30, etc.)
• Incorrect LCM: 12 (but you might stop too early and miss the actual LCM if there were larger numbers)

How to Avoid: Take your time and double-check your multiples. It can be helpful to use a systematic approach, such as multiplying each number by consecutive integers (1, 2, 3, etc.) to generate the multiples.

7.3. Errors in Prime Factorization

Prime factorization is a powerful method, but it’s essential to break down the numbers correctly into their prime factors. An error in prime factorization will lead to an incorrect LCM.

Mistake: Incorrectly factoring a number.

Example: Finding the LCM of 24 and 36:

• Incorrect prime factorization of 24: 2^2 × 3 (should be 2^3 × 3)
• Correct prime factorization of 36: 2^2 × 3^2
• Incorrect LCM: 2^2 × 3^2 = 36 (should be 2^3 × 3^2 = 72)

How to Avoid: Practice prime factorization and double-check your work. Use a factor tree or another systematic method to ensure you haven’t missed any prime factors or made any errors.

7.4. Not Taking the Highest Power of Prime Factors

When using the prime factorization method, remember to take the highest power of each prime factor present in any of the numbers. Failing to do so will result in an incorrect LCM.

Mistake: Not taking the highest power.

Example: Finding the LCM of 16 (2^4) and 20 (2^2 × 5):

• Incorrect LCM: 2^2 × 5 = 20 (should be 2^4 × 5 = 80)

How to Avoid: Carefully compare the prime factorizations and identify the highest power of each prime factor. Make a list of the highest powers and then multiply them together.

7.5. Stopping Too Early in the Division Method

In the division method, continue dividing until there are no more common prime factors among all the numbers. Stopping too early will lead to an incorrect LCM.

Mistake: Stopping before all common factors are divided out.

Example: Finding the LCM of 15 and 20:

5 | 15  20
  | 3   4  (Stopping here is incorrect)

• Incorrect LCM: 5 × 3 × 4 = 60 (correct)

How to Avoid: Ensure that you have divided out all common prime factors. Continue dividing until the remaining numbers have no common factors other than 1.

7.6. Not Checking the Final Answer

Always check your final answer to make sure it is a multiple of all the original numbers. If it’s not, you’ve made a mistake somewhere along the way.

Mistake: Not verifying the answer.

Example: Finding the LCM of 8 and 12 and getting 16 as the answer.

• 16 is a multiple of 8, but it’s not a multiple of 12.

How to Avoid: After finding the LCM, divide it by each of the original numbers. If the result is an integer in each case, your LCM is likely correct.

8. Advanced Tips and Tricks for Finding the Least Common Multiple

Finding the Least Common Multiple (LCM) can be streamlined with some advanced tips and tricks. These techniques not only save time but also enhance your understanding of number theory.

8.1. Using the Relationship Between LCM and GCF

As discussed earlier, the relationship LCM(a, b) × GCF(a, b) = a × b can be a powerful tool. If you know the GCF of two numbers, you can easily find the LCM, and vice versa.

Example: Find the LCM of 48 and 60, given that their GCF is 12.

1. Use the formula: LCM(48, 60) × GCF(48, 60) = 48 × 60
2. Plug in the known values: LCM(48, 60) × 12 = 48 × 60
3. Solve for LCM: LCM(48, 60) = (48 × 60) / 12
4. LCM(48, 60) = 2880 / 12 = 240

Therefore, the LCM of 48 and 60 is 240.

8.2. Dealing with Co-prime Numbers

If two numbers are co-prime (i.e., their GCF is 1), their LCM is simply their product. This can save you a lot of time in calculations.

Example: Find the LCM of 7 and 9.

Since 7 and 9 are co-prime (their GCF is 1), their LCM is 7 × 9 = 63.

8.3. Recognizing Multiples Quickly

Develop the ability to quickly recognize multiples of common numbers. This can speed up the listing multiples method.

Example: Find the LCM of 5 and 10.

Recognizing that 10 is a multiple of 5 immediately tells you that the LCM is 10.

8.4. Using Prime Factorization for Multiple Numbers

When finding the LCM of more than two numbers, prime factorization is particularly useful. Here’s how to do it:

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. Take the highest power of each prime factor:

   • 2^2, 3^2, 5

3. Multiply them together:

   • LCM = 2^2 × 3^2 × 5 = 4 × 9 × 5 = 180

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

8.5. Simplifying Before Finding the LCM

Sometimes, simplifying the numbers before finding the LCM can make the process easier.

Example: Find the LCM of 36 and 48.

1. Notice that both numbers are divisible by 12:

   • 36 = 12 × 3
   • 48 = 12 × 4

2. Find the LCM of the remaining factors:

   • LCM(3, 4) = 12

3. Multiply by the common factor:

   • LCM = 12 × 12 = 144

Therefore, the LCM of 36 and 48 is 144.

8.6. Estimating the LCM

Estimating the LCM can help you check if your final answer is reasonable. If you’re finding the LCM of 15 and 20, you know that the LCM must be at least 20 (the larger of the two numbers) and no more than 15 × 20 = 300. This gives you a range to expect your answer to fall within.

8.7. Practice with Various Problems

The best way to master finding the LCM is to practice with a variety of problems. Start with simple examples and gradually work your way up to more complex ones. The more you practice, the more comfortable and efficient you will become.

9. Frequently Asked Questions (FAQs) About the Least Common Multiple

To further clarify your understanding of the Least Common Multiple (LCM), here are some frequently asked questions with detailed answers:

Q1: How is the LCM used in adding and subtracting fractions?

A: The LCM is used to find the Least Common Denominator (LCD), which is essential for adding and subtracting fractions with different denominators. By converting the fractions to equivalent fractions with the LCD as the common denominator, you can easily perform the addition or subtraction.

Example:

Add 1/6 and 1/8.

1. Find the LCM of 6 and 8: LCM(6, 8) = 24

2. Convert the fractions to equivalent fractions with a denominator of 24:

   • 1/6 = (1 × 4) / (6 × 4) = 4/24
   • 1/8 = (1 × 3) / (8 × 3) = 3/24

3. Add the fractions:

   • 4/24 + 3/24 = 7/24

Q2: Can the LCM of two numbers be smaller than both numbers?

A: No, the LCM of two numbers cannot be smaller than both numbers. The LCM is the smallest multiple that both numbers divide into evenly, so it must be greater than or equal to the larger of the two numbers.

Q3: What is the LCM of two prime numbers?

A: The LCM of two prime numbers is simply their product. Since prime numbers have no common factors other than 1, their LCM is the result of multiplying them together.

Example:

The LCM of 7 and 11 is 7 × 11 = 77.

Q4: How do you find the LCM of three or more numbers?

A: You can find the LCM of three or more numbers using the prime factorization method or the division method. With prime factorization, find the prime factors of each number and take the highest power of each prime factor. With the division method, divide the numbers by common prime factors until there are no more common factors. Then, multiply all the divisors and remaining numbers to get the LCM.

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. Take the highest power of each prime factor:

   • 2^2, 3^2, 5

3. Multiply them together:

   • LCM = 2^2 × 3^2 × 5 = 180

Q5: Is the LCM always larger than the GCF for the same set of numbers?

A: Yes, the LCM is always larger than or equal to the GCF for the same set of numbers. The LCM is the smallest multiple that the numbers divide into, while the GCF is the largest factor that divides the numbers.

Q6: How does the LCM relate to real-world problems?

A: The LCM has various real-world applications, such as scheduling events, cooking, tiling, gear ratios, music, and computer science. It helps in finding the smallest common quantity or interval that satisfies certain conditions.

Q7: What if two numbers have no common factors? How do I find the LCM?

A: If two numbers have no common factors (i.e., they are co-prime), their LCM is simply their product.

Example:

Find the LCM of 8 and 9.

Since 8 and 9 have no common factors, their LCM is 8 × 9 = 72.

Q8: How can I check if my calculated LCM is correct?

A: To check if your calculated LCM is correct, divide the LCM by each of the original numbers. If the result is an integer in each case, your LCM is likely correct.

Example:

Find the LCM of 6 and 8. You calculated the LCM to be 24.

• 24 / 6 = 4 (integer)
• 24 / 8 = 3 (integer)

Since both results are integers, your LCM of 24 is correct.

Q9: Can the LCM of two different numbers be equal to one of the numbers?

A: Yes, the LCM of two different numbers can be equal to one of the numbers if one number is a multiple of the other.

Example:

Find the LCM of 4 and 8.

Since 8 is a multiple of 4, the LCM of 4 and 8 is 8.

Q10: Is there a formula for finding the LCM of two numbers?

A: Yes, there is a formula for finding the LCM of two numbers using their Greatest Common Factor (GCF):

LCM(a, b) × GCF(a, b) = a × b

This formula can be rearranged to find the LCM:

LCM(a, b) = (a × b) / GCF(a, b)

10. Test Your Knowledge: Practice Problems on Finding the Least Common Multiple

To reinforce your understanding of the Least Common Multiple (LCM), try these practice problems. Work through each problem using the methods discussed, and then check your answers against the solutions provided.

Problem 1: Find the LCM of 6 and 9 using the listing multiples method.

Problem 2: Find the LCM of 15 and 20 using the prime factorization method.

Problem 3: Find the LCM of 8 and 12 using the division method.

Problem 4: Find the LCM of 7 and 10.

Problem 5: Find the LCM of 12, 15, and 18.

Problem 6: If the GCF of two numbers is 4 and their product is 48, what is their LCM?

Problem 7: What is the LCM of two numbers if one number is 6 and the other is a prime number greater than 6?

Problem 8: Find the LCM of 24 and 36.

Problem 9: Two runners are running around a track. One runner completes a lap in 4 minutes, and the other completes a lap in 6 minutes. If they start at the same time, when will they next be at the starting point together?

Problem 10: You have two pieces of ribbon. One is 30 cm long, and the other is 45 cm long. You want to cut them into equal pieces. What is the longest possible length of each piece?

Solutions:

1. LCM of 6 and 9 (listing multiples method):

   • Multiples of 6: 6, 12, 18, 24, 30, …
   • Multiples of 9: 9, 18, 27, 36, …
   • LCM(6, 9) = 18

2. LCM of 15 and 20 (prime factorization method):

   • 15 = 3 × 5
   • 20 = 2^2 × 5
   • LCM(15, 20) = 2^2 × 3 × 5 = 60

3. LCM of 8 and 12 (division method):

2 | 8  12