LCM

# How to Find LCM | Least Common Multiple, Definition, Meaning, Solved Problems

Written by Prerit Jain

Updated on: 26 May 2023

Contents

### How to Find LCM | Least Common Multiple, Definition, Meaning, Solved Problems

The full form of LCM is the Least Common Multiple. In mathematics, the least common multiple (LCM) is a method to find the lowest possible common number of two numbers that are divisible by both numbers. LCM can be calculated for two or more numbers. LCM is also commonly referred to as the Least Common Divisor (LCD).

There are three methods for determining the LCM of a given number: the listing method, prime factorization method, and division method. In this article, let us discuss everything about how to find the LCM of a given number with solved examples. Scroll down to find out more.

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## What is the Least Common Multiple (LCM) in Maths?

*The smallest positive number that is a multiple of two or more numbers*.LCM Definition

For example, the LCM of 4 and 9 is 2 * 2 * 3 * 3 = 36.

Here, 4 is expressed as 2 * 2, and 9 is expressed as 3 * 3.

If we consider the multiples of 4 and 9, we get:

- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 26,…
- Multiples of 9: 9, 18, 27, 36,…

The first common multiple for 4 and 9 is 36. Hence, the LCM of 4 and 9 is 36.

LCM is used for the addition and subtraction of two fractions. When the denominator value of the fractions is not the same, LCM is used to make the denominators equal. This makes the entire calculating process easy.

## Properties of LCM

There are certain properties of LCM that you should learn before knowing how to find LCM.

### 1. Associative Property of LCM

The associative property of LCM states that the LCM of A and B will be the same as the LCM of B and A.

**LCM (A, B) = LCM (B, A)**

For example, let’s consider A as 6 and B as 2. Here, 6 can be expressed as 2 * 3, and 2 can be expressed as 2.

So, 6 * 2 = 2 * 3 * 2.

Now, we will write the numbers in the exponent form and multiply the factors that have the highest power.

On writing the number in exponent form, we get:

6 = 2 * 3 = 2^{1 }* 3^{1}

2 = 2^{1}

So, the LCM of 6 and 2 is 2^{1 }* 3^{1 }= 6

And, LCM of 2 and 6 is also 6.

Hence, it is proved that LCM (6, 2) = LCM (2, 6) = 6.

### 2. Commutative Property of LCM

The commutative property is used while dealing with finding the LCM of 3 numbers. The commutative property of LCM states that,

**LCM (A, B, C) = LCM (LCM (A, B), C) = LCM (A, LCM (B, C))**

For example, let’s consider A as 3, B as 6, and C as 12. Here, 3 can be expressed as 3, 6 can be expressed as 2 * 3, and 12 can be expressed as 2 * 2 * 3.

Now, we will write the numbers in the exponent form and multiply the factors that have the highest power.

On writing the number in exponent form, we get:

3 = 3^{1}

6 = 2 * 3 = 2^{1 }* 3^{1}

12 = 2 * 2 * 3 = 2^{2} * 3^{1}

So, the LCM of 3, 6, and 12 is 2^{2} * 3^{1} = 2 * 2 * 3 = 12

Now, the LCM of A and B, that is, LCM 3 and 6 is 3^{1} * 2^{1} = 6 and

LCM of (A, B) and C, that is, LCM of 6 and 12 is 2^{2} * 3^{1 }= 2 * 2 * 3 = 12.

LCM (LCM (A, B), C) = LCM (LCM (3, 6), 12) = LCM (6, 12) = 12

LCM of B and C, that is, LCM of 6 and 12 is 2^{2} * 3^{1 }= 2 * 2 * 3 = 12 and

LCM of A and LCM of (B, C), that is, LCM of 3 and 12 is 2^{2} * 3^{1 }= 2 * 2 * 3 = 12.

LCM (A, LCM (B, C)) = LCM (3, LCM (6, 12)) = LCM (3, 12) = 12.

Hence, it is proved that LCM (3, 6, 12) = LCM (LCM (3, 6), 12) = LCM (3, LCM (6, 12)) = 12.

### 3. Distributive Property of LCM

The distributive property is also used while dealing with finding the LCM of 3 numbers. Distributive property of LCM states that,

LCM (dA, dB, dC) = d * LCM (A, B, C)

For example, let’s consider A as 5, B as 8, C as 13, and d to be any random variable.

Now, we will write the numbers in the exponent form and multiply the factors that have the highest power.

On writing the number in exponent form, we get:

5 = 5^{1}

8 = 2 * 2 * 2 = 2^{3}

13 = 13^{1}

LCM of 5, 8, and 13 is 5^{1} * 2^{3} * 13^{1} = 5 * 2 * 2 * 2 * 13 = 520.

So, LCM (5d, 8d, 13d) = d * LCM (5, 8, 13) = 520

Hence, it is proved that LCM (5d, 8d, 13d) = d * LCM (5, 8, 13)

## How to Find the LCM?

There are three major methods for finding the LCM of two or more numbers. The methods are:

### 1. Division Method

To find the LCM using the division method, divide the given numbers by the smallest prime number, which is divisible by any of the given numbers. Then, the prime factors further obtained will be used to calculate the final LCM.

You can follow the following steps to find the LCM using the division method:

- Step 1: Write all the given numbers for which you have to find the LCM, separated by commas.
- Step 2: Now, find the smallest prime number which is divisible for any of the given two numbers.
- Step 3: If any number is not divisible, write that number in the next row just below it and proceed further.
- Step 4: Continue dividing the numbers obtained after each step by the prime numbers, until you get the result as 1 in the entire row.
- Step 5: Now, multiply all the prime numbers and the final result will be the LCM of the given numbers.

For example, you have to find the LCM of 12 and 5 using the division method.

Prime Factors | First Number | Second Number |

2 | 12 | 5 |

2 | 6 | 5 |

3 | 3 | 5 |

5 | 1 | 5 |

1 | 1 |

So, LCM of 12 and 5 = 2 * 2* 3 * 5 = 50

### 2. Prime Factorization Method

To find the LCM of the given numbers using the prime factorization method, follow the steps given below:

- Step 1: Find the prime factors of the given numbers using the repeated division method explained above.
- Step 2: Write the prime factors in their exponent forms. Then multiply the prime factors having the highest power.
- Step 3: The final result after multiplication will be the LCM of the given numbers.

For example, you have to find the LCM of 18, 10, and 7 using the prime factorization method.

- Prime factorization of 18 can be expressed as 2 * 3 * 3 = 21 * 32
- Prime factorization of 10 can be expressed as 2 * 5 = 21 * 51
- Prime factorization of 7 can be expressed as 71

So, the LCM of 18, 10, and 7 = 2^{1} * 3^{2 }* 5^{1 }* 7^{1} = 2 * 3 * 3 * 5 * 7 = 630.

### 3. Listing Method

To find the LCM of the given numbers using the listing method, you can follow the following steps:

- Step 1: Write down the first few multiples of the given numbers separately.
- Step 2: Out of all the multiples of the numbers focus on the multiples which are common to all the given numbers.
- Step 3: Now, find out of all the common multiples, and take out the smallest common multiple. That will be the LCM of the given numbers

For example, you have to find the LCM of 8 and 5 using the listing method.

- Multiples of 8 are 8, 16, 24, 32, 40, 48, 64,…
- Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45,…

Here, it is clear that the least common multiple is 40.

So, the LCM of 8 and 5 is 40.

## Important LCM Formulas

There are two major LCM formulas, one for finding the LCM of integers and the other for finding the LCM of fractions.

Before moving forward and knowing the formulas, you should know about the HCF (Highest Common Factor).

HCF is the highest factor which is common among the factors of all the given numbers. It is also known as the greatest common divisor (GCD).

### 1. Formula for finding the LCM of the given integers

Let A and B be two given integers. So, the LCM of A and B can be calculated using the formula:

LCM (A, B) = (A * B) / HCF (A, B),

where HCF is the highest common factor or the greatest common divisor of A and B.

Another formula for finding the LCM of the given integers is:

A * B = LCM (A, B) * HCF (A, B), that is,

The product of the two given integers is equal to the product of their LCM and HCF.

### 2. Formula for finding the LCM of the given fractions

LCM = LCM of the numerator / HCF of the denominator

LCM List

## Solved Example Problems Based on LCM

**Question 1: Find the LCM of 9 and 4 using the listing method.****Solution:**

Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90,…

Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40,…

Here, it is clear that the least common multiple is 36.

So, the LCM of 9 and 4 is 36.

**Question 2: What is the LCM of 16 and 21 using the prime factorization method?Solution: **

Prime factorization of 16 can be expressed as 2 * 2 * 2 * 2 = 2

^{4}

Prime factorization of 21 can be expressed as 3 * 7 = 3

^{1}* 7

^{1}

So, the LCM of 16 and 21 = 2

^{4}* 3

^{1 }* 7

^{1}= 2 * 2 * 2 * 2 * 3 * 7 = 336.

**Question 3: If the LCM and HCF of two numbers, 5 and B, are 45 and 1, respectively. Find B.Solution: **

As we know,

Product of two numbers = LCM * HCF

We are given that,

One of the numbers = 5, LCM = 45, and HCF = 1

So, 5 * B = 45 * 1

B = (45 * 1) / 5

B = 9

Hence, the other number is 9.

**Question 4: Find the LCM of 24 and 45 using the division method.Solution:**

Prime Factors | First Number | Second Number |

2 | 24 | 45 |

2 | 12 | 45 |

2 | 6 | 45 |

3 | 3 | 45 |

3 | 1 | 15 |

5 | 1 | 5 |

1 | 1 |

So, the LCM of 24 and 45 = 2 * 2 * 2 * 3 * 3 * 5 = 360.

**Question 5: Find the LCM of 14, 22, and 18 using the prime factorization method.Solution: **

Prime factorization of 14 can be expressed as 2 * 7 = 2

^{1}* 7

^{1}

I can express the prime factorization of 22 as 2 * 11 = 2

^{1}* 11

^{1}

Prime factorization of 18 can be expressed as 2 * 3 * 3 = 2

^{1}* 3

^{2}

So, the LCM of 14, 22, and 18 = 2

^{1}* 7

^{1}* 11

^{1}* 3

^{2 }= 2 * 7 * 11 * 3 * 3 = 1386.

**Question 6: If the HCF of two numbers, 42 and 9, is 3. Find the LCM.Solution: **

As we know,

LCM (A, B) = (A * B) / HCF (A, B)

We are given that,

Numbers = 42 and 9 and HCF = 3

So, LCM (42, 9) = (42 * 9) / HCF (42, 9)

= 378 / 3

= 126

Hence, the LCM of 42 and 9 is 126.

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## FAQs on LCM

**What do you mean by LCM?**

LCM is the least common multiple. It is used to find the lowest possible common number that is divisible by all the numbers, for which you have to find the LCM.

**How do LCM and HCF are connected?**

LCM (A, B) = (A * B) / HCF (A, B),

where A and B are two integers.

This formula is used to find the LCM of the given integers.

**Can LCM be calculated for only 2 numbers?**

No, LCM can be calculated for more than two numbers as well. There should be at least 2 numbers for finding the LCM.

**What are the methods to find the LCM?**

There are three major methods for finding the LCM of two or more numbers. The methods are:

1. Division Method

2. Prime Factorization Method

3. Listing Method

**What is the LCM of 2 and 8?**

LCM of 2 and 8 is 8.

**What are the properties of LCM?**

There are three major properties of LCM. They are:

1. Associative property

2. Commutative property

3. Distributive property

**What is the formula for finding the LCM of the fraction?**

The formula for finding the LCM of a given fraction is:

LCM = LCM of the numerator / HCF of the denominator

We hope this article on LCM is helpful to you.

Written by by

Prerit Jain