# How to Find the Least Common Multiple

- Author: Noreen Niazi
- Last Updated on: December 19, 2023

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ToggleAs a math enthusiast, finding the least common multiple is important.The lowest positive integer and multiple of two or more numbers is known as the least common multiple (LCM). It is a fundamental mathematical concept that has many applications in real life. In this article, I will introduce you to the concept of LCM, show you how to find the least common multiple using different methods, provide examples of each method, and give you tips for finding the LCM efficiently.

## Understanding Factors and Multiples

Before we dive into finding the LCM, it is important to understand two important terms: factors and multiples.A full number that divides a given number evenly is called its factor. For instance, $$1, 2, 3, 4, 6, \text{and} 12$$ are factors of $$12$$. A multiple of a number is a number that can be obtained by multiplying the number by any other whole number. For example, the multiples 4 are 4, 8, 12, 16, etc.

## Method 1: How to find the least common multiple by Listing Multiples

Multiple techniques are listed, which is one of the easiest ways to calculate the LCM. This approach requires you to write down all of the multiples of the provided numbers until you identify the smallest multiple that all of them share. As an illustration, consider the integers 4 and 6.

Multiples of $$4$$:$$ 4, 8, 12, 16, 20, 24, 28, 32, 36, …$$

Multiples of $$6: 6, 12, 18, 24, 30, 36, …$$

The smallest multiple that is common to both 4 and 6 is 12. Therefore, the LCM of 4 and 6 is 12.

This method can be time-consuming and impractical when dealing with larger numbers.

## Method 2: How to find the least common multiple by Prime Factorization

Prime factorization is the second way to calculate LCM In Prime factorization, you have to express a number as the product of its prime factors.

Let’s take the numbers $$12$$ and $$20$$ as examples.

$$12 = 2 \times 2 \times3$$

$$20 = 2 \times 2 \times 5$$

We multiply the prime factors’ highest powers to find the LCM of 12 and 20.

$$2 \times 2 \times 3 \times 5 = 60$$

Therefore, the LCM of 12 and 20 is 60.

This method is more efficient than listing multiples, especially for larger numbers.

## Method 3: How to find the least common multiple by Using a Venn Diagram

The third method for finding the LCM is to use a Venn diagram. A Venn diagram is a graphical representation of sets that shows all possible logical relations between them.

Let’s take the numbers $$6$$ and $$9$$ as examples.

The LCM of 6 and 9 is the product of all the prime factors in the circles and the common factors outside the circles.

$$2 \times 3 \times 3 = 18$$

Therefore, the LCM of $$6$$ and $$9$$ is $$18$$.

This method can be helpful for visual learners.

## Method 4: How to find the least common multiple by Using the GCD (Greatest Common Divisor)

Utilising the GCD (greatest common divisor) is the fourth way to calculate the LCM. The largest positive integer that evenly divides each of the input numbers without leaving a remainder is known as the GCD of two or more numbers.

Let’s take the numbers $$8$$ and $$12$$ as examples.

The prime factors of $$8$$ are $$2 \times 2 \times 2 = 2^3$$ The prime factors of $$12$$ are $$2\ times 2 \times 3 = 2^2 times 3$$

The GCD of $$8$$ and $$12$$ is $$2^2 = 4$$.

The LCM of $$8$$ and $$12$$ can be found using the formula:

LCM = $$\frac{(8 \times 12)}{GCD(8, 12)} = \frac{96}{4} = 24$$

Therefore, the LCM of $$8$$ and $$12$$ is $$24$$.

This method can be useful when dealing with larger numbers and can be applied simultaneously to more than two numbers.

## Examples of Finding the LCM Using Each Method

### Method 1: Listing Multiples

**How to find the least common multiple of 15 and 25 by listing multiples?**

In listing methods, first we write down the multiples of both the number 15 and 25.

**15 Multiples: **$$15, 30, 45, 60, 75, 90, … $$

**25 Multiples****: **$$ 25, 50, 75, 100, 125, …$$

The smallest multiple that both 15 and 25 have in common is 75. The LCM of 15 and 25 is thus 75.

### Method 2: Prime Factorization

**How to find the least common multiple of 15 and 25 by prime factorization methods?**

Now the prime factors of 15 and 25 are $$15 = 3 \times 5;$$ $$25 = 5 \times 5$$

We need to multiply the highest powers of the prime elements to determine the LCM of 15 and 25.

$$3\times 5 \times 5 = 75$$

Therefore, the LCM of 15 and 25 is 75.

### Method 3: Using a Venn Diagram

**How to find the least common multiple of 15 and 25 by vein diagrams.**

To find the LCM of 15 and 25 using a Venn diagram, you can follow these steps ¹:

1. Write each number as a product of its prime factors without using index form. For 15 and 25, we have:

– 15 = 3 × 5

– 25 = 5 × 5

2. Draw a Venn diagram with two circles that overlap. Label one circle “3” and the other circle “5”.

3. Write the prime factors of each number in the appropriate circle. The prime factorization of 15 is 3 × 5, so write “3” in the left circle and “5” in the right circle. The prime factorization of 25 is 5 × 5, so write “5” in both circles.

4. To find the LCM, multiply all the prime factors that appear in either circle. In this case, we have:

– The left circle contains only one prime factor: 3.

– The right circle contains two prime factors: 5 and 5.

So the LCM of 15 and 25 is:

LCM(15,25)= $$3 \times 5 \times 5 = 75$$

Therefore, the LCM of 15 and 25 is 75.

### Method 4: Using the GCD (Greatest Common Divisor)

**How to find the least common multiple of 15 and 25 by ****using**** GCD method.**

The GCD of 15 and 25 is 5.

The LCM of 15 and 25 can be found using the formula:

$$LCM =\frac {(15 \times25)} { GCD(15, 25)}$$

$$ = \frac{375 }{ 5} = 75$$

Therefore, the LCM of $$15$$ and $$25$$ is $$75.$$

## Tips for Finding the LCM Efficiently

- Use prime factorization for larger numbers.
- Use a Venn diagram for visual learners.
- Use the GCD method for more than two numbers.
- Look for common factors in the given numbers.
- Use a calculator for larger numbers.

## Applications of Least Common Multiple in Real Life

The concept of LCM has many applications in real life, such as:

- Calculate the time it takes for two events to occur simultaneously, such as the time it takes for two trains to meet.
- Calculating the number of items needed to evenly distribute them into groups, such as the number of chairs needed for a certain number of tables.
- Calculate the number of ingredients needed for a recipe, such as doubling or tripling.

## Conclusion and Summary of Key Takeaways

In conclusion, finding the LCM is an important skill to have in mathematics. Different methods for finding the LCM include listing multiples, prime factorization, using a Venn diagram, and using the GCD (greatest common divisor). By applying these methods, you can find the LCM efficiently and accurately. The concept of LCM has many real-world applications, making it a useful tool to have in your mathematical toolkit.