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# Least Common Multiple|Definition & Meaning

## Definition

In mathematics, the least common multiple is sometimes referred to as** LCM** or the lowest common multiple. The smallest number among all the common multiples of the provided numbers is the **least common multiple** of two or more numbers.

The Least Common Multiple is the meaning of the abbreviation **LCM**. The lowest number that may be divided by both numbers is known as the **LCM** of two numbers. It can also be computed using two or more numbers. Finding the LCM of any particular number can be done in a variety of ways.

Applying the prime factorization for each number and then calculating the product of the greatest powers of the shared prime factors is one of the quickest techniques to determine the LCM of two numbers.

Figure 1 – LCM of 2 and 5

In the above figure, the common multiple of 2 and 5 are 10 and 20 but the smallest is 10 so the LCM of 2 and 5 is 10.

## Methods To Find LCM

There are different methods to find LCM. Mainly there are three methods to find the LCM of any two numbers.

- Prime Factorization Method
- Listing Method
- Division Method

## Prime Factorization Method

The LCM of the supplied numbers can be determined by applying the **prime factorization method**. The following steps are used to determine the LCM by the prime factorization method:

**Step 1**

By using the repeated division method, determine the given numbers’ prime factors.

**Step 2**

The numbers should be expressed in exponentiation. Only those prime factors’ products with the **highest power** should be sought.

**Step 3**

The LCM of the provided numbers is the result of these factors’ highest powers combined.

## Listing Method

By listing the multiples of two or more numbers, we can determine the common multiples of those numbers. The LCM among these frequent multiples is taken into account, allowing the LCM of any two values to be determined. The steps below are used to determine the Least Common Multiple of the two numbers by using the listing approach.

### Step 1

Write some first multiples of two given numbers of which we must find LCM.

### Step 2

Mark the common multiples of both numbers.

### Step 3

For LCM, choose the smallest multiple of both numbers. That common number will be the LCM of both numbers.

The below figure shows the LCM of 10 and 50 by the Listing Method.

Here, the first few multiples of 10 and 50 have been written. The common multiple in both numbers is 50 and 100, but LCM means the least common multiple. We can see from Figure 3 that the smallest common multiple for 10 and 50 is just 50, so we write LCM(10, 50) = 50.

## Division Method

To find the LCM using the method of division, we first write the supplied numbers in a row separated by commas. Then, we divide each of those numbers by a **common prime number**. As soon as we reach one of the prime numbers, we stop dividing. The least significant factor of a set of numbers is the product of its** common prime factors** and its **uncommon prime factors**.

**Step 1**

Create a line with the given numbers, separating them with **commas**.

**Step 2**

**Divide** them by an appropriate **prime number**, one that perfectly divides at least two of the numbers that have been provided.

**Step 3**

We wrote the **quotient** directly **beneath** the values in the row that followed it. In the event that the number cannot be divided exactly, the **remainder** is taken **down** in the next row.

**Step 4**

**Step 2** and also **step 3** are repeated as necessary until there are no more co-prime numbers in the bottom row.

**Step 5**

After that, we multiply all of the prime numbers with which we’ve divided as well as the co-prime numbers that are still in the last row. The offered numbers have the fewest possible multiples in common with this product.

The above figure shows the **LCM** of **12** and **18** by the division method. From the figure, we can really see that the **LCM** of **12** and **18** is** 36**.

## Examples of Finding the Least Common Multiple

### Example 1

What is LCM of 10 and 20? Use the prime factorization method.

### Solution

First of all, we will find the prime factors of 10 and 20.

**10 = 2 x 5**

**20 = 2 x 2 x 5**

Here:

**Common Multiples of 10 and 20 = 2, 5 **

**Non-Common Multiples of 10 and 20 = 2**

So by looking at common and non-common multiples of 10 and 20, the LCM will be 20:

**LCM(10,20) = 20**

### Example 2

Find the LCM of 25 and 50 by listing method.

### Solution

Here in this example, we will write the multiples of the given two numbers that is 25 and 50.

**25 = 25, 50, 75, 100, 125, 150, 175, 200**

**50 = 50, 100, 150, 200, 250, 300**

We can see here the common multiples are 50,100,150,200. Since the least common multiple is **50,** so:

**LCM(25,50) = 50**

### Example 3

Find the LCM of 20 and 30 by the division method.

### Solution

The two numbers, 20 and 30, will be divided up to 1, and then the divisor will be multiplied to get LCM.

The two numbers are **20** and **30**. First will see the lowest multiple, which is **2**. Both the numbers are divisible by** 2**. When **20** is divided by **2,** the answer is** 10,** which is written below **20.** When** 30** is divided by **2,** the answer is **15,** which is written below **30**.

Now,** 10** and** 15** are divided by the number. Here** 10** is divisible by** 2,** but **15** is not divisible by **2,** so we will divide just 10, so the answer is** 5,** which will be written below **10,** and **15** will be written as it is.

Now we got** 5** and **15,** which are divisible by **5. **When **5** is divided by **5,** the answer is 1, which is written below **5. **When **15 **is divided by **5,** the answer is **3,** which is written below **15**.

In the end, only **3** is left, which will be divided by **3**. When we multiply the prime numbers, we will get **LCM**. The LCM of **2**0 and **30** is **60**. It will be written as **LCM(20, 30) = 60**.

*All the figures above are created on GeoGebra.*