The acronym LCM stands for Least Common Multiple, also known as the Lowest Common Multiple. In mathematics, the least common multiple of two or more numbers is the smallest positive integer that is divisible by each of the given numbers. Understanding how to find the LCM is a fundamental skill in arithmetic and has practical applications in various real-life situations. There are several effective methods to calculate the LCM of numbers. This article will guide you through different techniques on how to find the LCM, ensuring you grasp each method clearly and can apply them effectively.
What is the Least Common Multiple (LCM)?
The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the given numbers. Think of multiples as the numbers you get when you skip count. For example, multiples of 2 are 2, 4, 6, 8, 10, and so on. Multiples of 5 are 5, 10, 15, 20, 25, and so on.
Let’s visualize this on a number line to understand it better. Consider finding the LCM of 2 and 5. We list out the multiples of each number and identify the common ones.
As you can see, the common multiples of 2 and 5 are 10, 20, 30, and so forth. The smallest number among these common multiples is 10. Therefore, the least common multiple of 2 and 5 is 10. We can write this as LCM(2, 5) = 10.
How to Find the LCM: Different Methods Explained
There are primarily three methods to find the least common multiple of numbers. Each method offers a different approach and can be more suitable depending on the numbers you are working with. Below, we will explore each method in detail, providing step-by-step instructions and examples to illustrate how to find the LCM effectively.
- Listing Multiples Method
- Prime Factorization Method
- Division Method
1. LCM by Listing Multiples Method
The listing multiples method is a straightforward way to find the LCM, especially for smaller numbers. It involves listing out the multiples of each number until you find a common multiple. The smallest of these common multiples is the LCM. Here’s how to find the LCM using the listing method:
Steps:
- Step 1: List Multiples: Write down the first few multiples of each number you want to find the LCM for. Start with the number itself and continue by adding the number to the previous multiple.
- Step 2: Identify Common Multiples: Look at the lists of multiples you’ve created and identify the multiples that appear in all lists. These are the common multiples.
- Step 3: Determine the Least Common Multiple: From the common multiples you’ve identified, select the smallest one. This smallest common multiple is the LCM of the given numbers.
Example: Let’s find the least common multiple (LCM) of 4 and 6 using the listing multiples method.
Solution:
- 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, …
By examining the lists, we can see that the common multiples of 4 and 6 include 24, 48, and so on. The smallest among these is 24.
Therefore, the least common multiple (LCM) of 4 and 6 is 24.
2. LCM by Prime Factorization Method
The prime factorization method is a more systematic approach, particularly useful for larger numbers. This method relies on breaking down each number into its prime factors. Understanding prime factorization is key to understanding how to find the LCM using this method.
Steps:
- Step 1: Prime Factorization: Find the prime factorization of each number. This involves expressing each number as a product of its prime factors. You can use a factor tree or repeated division to find the prime factors.
- Step 2: Identify Highest Powers: For each prime factor that appears in any of the factorizations, identify the highest power to which it is raised.
- Step 3: Calculate LCM: Multiply together each of these prime factors raised to their highest powers. The resulting product is the LCM of the given numbers.
Example: Find the least common multiple (LCM) of 30 and 45 using prime factorization.
Solution:
-
Step 1: Prime Factorization:
- Prime factorization of 30: 30 = 2 × 3 × 5
- Prime factorization of 45: 45 = 3 × 3 × 5 = 3² × 5
-
Step 2: Identify Highest Powers:
- Prime factors involved are 2, 3, and 5.
- Highest power of 2: 2¹ (from 30)
- Highest power of 3: 3² (from 45)
- Highest power of 5: 5¹ (common to both)
-
Step 3: Calculate LCM:
- LCM(30, 45) = 2¹ × 3² × 5¹ = 2 × 9 × 5 = 90
Therefore, the LCM of 30 and 45 is 90.
3. LCM by Division Method
The division method is often considered the most efficient method for finding the LCM, especially when dealing with more than two numbers. It involves dividing the numbers by prime numbers in a systematic way. This method is a practical way to learn how to find the LCM quickly.
Steps:
- Step 1: Set up Division: Write the numbers you want to find the LCM for in a row, separated by commas. Draw a vertical line to the left of the numbers and a horizontal line above them.
- Step 2: Divide by Prime Numbers: Start dividing the numbers by the smallest prime number, 2. If any of the numbers is divisible by 2, divide them and write the quotients below. If a number is not divisible, simply bring it down to the next row.
- Step 3: Repeat Division: Continue this process with the next prime numbers (3, 5, 7, and so on), dividing at least one of the numbers in each row until all the quotients are 1.
- Step 4: Calculate LCM: Multiply all the prime divisors on the left side of the vertical line. This product is the LCM of the original numbers.
Example: Find the least common multiple (LCM) of 12 and 18 using the division method.
Solution:
| 2 | 12, 18 |
|---|---|
| 2 | 6, 9 |
| 3 | 3, 9 |
| 3 | 1, 3 |
| 1, 1 |
- Divide 12 and 18 by 2. We get 6 and 9.
- Divide 6 by 2. 9 is not divisible by 2, so we bring it down. We get 3 and 9.
- Divide 3 and 9 by 3. We get 1 and 3.
- Divide 3 by 3. We get 1.
Now, multiply the prime divisors: 2 × 2 × 3 × 3 = 36.
Therefore, the LCM of 12 and 18 is 36.
The division method is often preferred for its efficiency and applicability to more than two numbers. You can also use online LCM calculators to verify your answers and practice how to find the LCM.
LCM Formula: Using HCF to Find LCM
There’s a useful formula that connects the LCM and the Highest Common Factor (HCF) of two numbers. If you know the HCF of two numbers, you can easily calculate their LCM using this formula. This formula provides another way to how to find the LCM.
For two numbers, ‘a’ and ‘b’, the LCM formula is:
LCM (a, b) = (a × b) / HCF (a, b)
This formula states that the LCM of two numbers is equal to the product of the numbers divided by their HCF.
Example: Find the LCM of 24 and 36, given that their HCF is 12.
Solution:
Using the LCM formula:
LCM (24, 36) = (24 × 36) / HCF (24, 36)
LCM (24, 36) = (24 × 36) / 12
LCM (24, 36) = 864 / 12
LCM (24, 36) = 72
Therefore, the LCM of 24 and 36 is 72.
Relationship Between LCM and HCF
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), and the Least Common Multiple (LCM) are related concepts in number theory. The HCF is the largest number that divides two or more numbers without leaving a remainder, while the LCM is the smallest number that is a multiple of those numbers.
For any two positive integers ‘a’ and ‘b’, the relationship between their LCM and HCF is given by:
LCM (a, b) × HCF (a, b) = a × b
This relationship highlights that the product of two numbers is always equal to the product of their LCM and HCF. This is a fundamental concept in understanding how to find the LCM and HCF and their interrelation.
Difference Between LCM and HCF
While both LCM and HCF are important in number theory, they represent different concepts. Here’s a table summarizing the key differences between LCM and HCF:
| Feature | LCM (Least Common Multiple) | HCF (Highest Common Factor) |
|---|---|---|
| Definition | Smallest common multiple of given numbers | Largest common factor of given numbers |
| Size relative to numbers | Greater than or equal to the largest number in the set | Less than or equal to the smallest number in the set |
| For prime numbers | Product of the prime numbers | Always 1 |
| Use case | Finding when events will recur simultaneously, fractions with common denominators | Simplifying fractions, dividing things into equal sections |
LCM of Three or More Numbers
The methods we discussed for finding the LCM of two numbers can also be extended to find the LCM of three or more numbers. For example, you can use the listing method, prime factorization method, or division method to find the LCM of multiple numbers.
Example: Find the LCM of 10, 15, and 20 using the prime factorization method.
Solution:
- Prime factorization of 10: 2 × 5
- Prime factorization of 15: 3 × 5
- Prime factorization of 20: 2² × 5
Highest powers of prime factors: 2², 3¹, 5¹
LCM (10, 15, 20) = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60
Thus, the LCM of 10, 15, and 20 is 60.
You can similarly apply the listing method or division method to find the LCM of three or more numbers.
LCM Examples with Answers
Let’s work through some examples to solidify your understanding of how to find the LCM.
Example 1: Find the LCM of 8 and 10 using the prime factorization method.
Solution:
- Prime factorization of 8: 2 × 2 × 2 = 2³
- Prime factorization of 10: 2 × 5
LCM (8, 10) = 2³ × 5¹ = 8 × 5 = 40
Example 2: What is the LCM of 9 and 15 using the division method?
Solution:
| 3 | 9, 15 |
|---|---|
| 3 | 3, 5 |
| 5 | 1, 5 |
| 1, 1 |
LCM (9, 15) = 3 × 3 × 5 = 45
Example 3: What is the LCM of 5, 10, and 15 using the listing multiples method?
Solution:
- Multiples of 5: 5, 10, 15, 20, 25, 30, …
- Multiples of 10: 10, 20, 30, …
- Multiples of 15: 15, 30, …
LCM (5, 10, 15) = 30
Practice Questions on LCM
- Find the LCM of 16 and 24 using the division method.
- What is the LCM of 7, 11, and 14?
Answers:
- Check Answer (Answer: 48)
- Check Answer (Answer: 154)
FAQs about Least Common Multiple (LCM)
### What is LCM in Math?
In mathematics, the least common multiple (LCM) is the smallest positive integer that is a multiple of two or more given numbers. It is a fundamental concept in number theory and arithmetic.
### How to Find the LCM of Two Numbers?
You can find the LCM of two numbers using three primary methods: listing multiples, prime factorization, and division method. These methods are detailed in the sections above with step-by-step instructions and examples on how to find the LCM.
### What is the Fastest Way to Find the LCM?
The division method is generally considered the fastest and most efficient way to find the LCM, especially for larger numbers or when dealing with more than two numbers.
### What is the LCM of 15 and 20?
Using the division method:
| 2 | 15, 20 |
|---|---|
| 2 | 15, 10 |
| 3 | 15, 5 |
| 5 | 5, 5 |
| 1, 1 |
LCM (15, 20) = 2 × 2 × 3 × 5 = 60. Thus, the LCM of 15 and 20 is 60.
### What is the Difference Between LCM and HCF?
The LCM is the least common multiple, the smallest number that is a multiple of given numbers. HCF is the highest common factor, the largest number that divides given numbers. Refer to the table above for a detailed comparison.
### What is the Relationship Between HCF and LCM of Two Numbers?
For any two numbers ‘a’ and ‘b’, the relationship is: LCM (a, b) × HCF (a, b) = a × b.
### What is the Least Common Multiple of 6 and 9?
Using the listing method:
- Multiples of 6: 6, 12, 18, …
- Multiples of 9: 9, 18, …
LCM (6, 9) = 18. The LCM of 6 and 9 is 18.
### How To Find Lcm of 3 Numbers?
The LCM of three or more numbers can be found using the same methods as for two numbers: listing multiples, prime factorization, or division method. The prime factorization and division methods are generally more efficient for three or more numbers.
### How to Find LCM using Prime Factorization?
To find the LCM using prime factorization, you need to:
- Find the prime factorization of each number.
- Identify the highest power of each prime factor that appears in any of the factorizations.
- Multiply these highest powers together to get the LCM.
### What is the LCM of Two Coprime Numbers?
The LCM of two coprime numbers (numbers with no common factors other than 1) is simply their product. For example, the LCM of 5 and 7 (coprime numbers) is 5 × 7 = 35.
### How is LCM used in Real Life?
LCM has many practical applications, such as:
- Scheduling events: Finding out when events that occur at regular intervals will happen at the same time again.
- Fractions: Finding a common denominator when adding or subtracting fractions.
- Problem-solving: Solving problems involving repeating patterns or cycles.
### What does Lowest Common Multiple Mean?
Lowest Common Multiple is another term for Least Common Multiple (LCM). It refers to the smallest positive number that is a common multiple of a set of numbers.
### How to Find the Lowest Common Multiple of 12, 15 and 30?
Using the division method:
| 2 | 12, 15, 30 |
|---|---|
| 2 | 6, 15, 15 |
| 3 | 3, 15, 15 |
| 5 | 1, 5, 5 |
| 1, 1, 1 |
LCM (12, 15, 30) = 2 × 2 × 3 × 5 = 60. The LCM of 12, 15, and 30 is 60.
