LCM

# How To Find LCM of Three Numbers? | Listing, Division, and Prime Factorization Method

Written by Prerit Jain

Contents

### How To Find LCM of Three Numbers? | Listing, Division, and Prime Factorization Method

**LCM of Three Numbers: **In mathematics, the least common multiple (LCM) is a method of finding the lowest possible common number that is divisible by all the given numbers, for which LCM has to be found. LCM can be calculated for two or more two numbers. It is also known as the Least Common Divisor (LCD).

For example, LCM of 3, 6, and 8 is 24. Here, 3 is expressed as 3, 6 is expressed as 3 * 2, and 8 is expressed as 2 * 2 * 2. If we consider the prime factors of 3, 6, and 8, we get:

3: 3^{1}

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

8: 2 * 2 * 2 = 2^{1} * 2^{1} * 2^{1 }= 2^{3}

We took 2^{3} and 3^{1} for calculating the LCM of 3, 6, and 8 as 2^{3} * 3^{1} = 24. Hence, LCM of 3, 6, and 8 is 24.

LCM is used for the addition and subtraction of 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.

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## Properties of LCM of Three Numbers

There are two properties of LCM that you should keep in mind while solving problems of three numbers.

### 1. Commutative Property

The commutative property is used while dealing with the problems of LCM of three 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 2, B as 12, and C as 14. Here, 2 can be expressed as 2, 12 can be expressed as 2 * 2 * 3, and 14 can be expressed as 2 * 7.

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:

2 = 2

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

14 = 2 * 7 = 2^{1} * 7^{1}

So, the LCM of 2, 12, and 14 is 2^{2} * 3^{1} * 7^{1 }= 2 * 2 * 3 * 7 = 84

Now, LCM of A and B, that is, LCM 2 and 12 is 2^{2} * 3^{1} = 12 and LCM of (A, B) and C, that is, LCM of 12 and 14 is 2^{2} * 3^{1 }* 7^{1 }= 2 * 2 * 3 * 7= 84.

LCM (LCM (A, B), C) = LCM (LCM (2, 12), 14) = LCM (12, 14) = 84

LCM of B and C, that is, LCM of 12 and 14 is 2^{2} * 3^{1 }* 7^{1 }= 2 * 2 * 3 * 7= 84 and LCM of A and LCM of (B, C), that is, LCM of 2 and 84 is 2^{2} * 3^{1 }* 7^{1 }= 2 * 2 * 3 * 7= 84.

LCM (A, LCM (B, C)) = LCM (2, LCM (12, 14)) = LCM (2, 84) = 84.

Hence, it is proved that LCM (2, 12, 14) = LCM (LCM (2, 12), 14) = LCM (2, LCM (12, 14)) = 84.

### 2. Distributive Property

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

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

For example, let’s consider A as 3, B as 6, C as 18, 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:

3 = 3

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

18 = 2 * 3 * 3 = 2^{1} * 3^{1 }* 3^{1 }= 2^{1} * 3^{2}

LCM of 3, 6, and 18 is 2^{1} * 3^{2} = 2 * 3 * 3 = 18.

So, LCM (3d, 6d, 18d) = d * LCM (3, 6, 18) = 18

Hence, it is proved that LCM (3d, 6d, 18d) = d * LCM (3, 6, 18)

## How To Find the LCM?

There are three major methods for finding the LCM of two or more numbers. These 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 by at least one of the given 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 4, 9, and 16 using the division method.

Prime Factors | First Number | Second Number | Third Number |

2 | 4 | 9 | 16 |

2 | 2 | 9 | 8 |

2 | 1 | 9 | 4 |

2 | 1 | 9 | 2 |

3 | 1 | 9 | 1 |

3 | 1 | 3 | 1 |

1 | 1 | 1 |

So, LCM of 4, 9, and 16 = 2 * 2 * 2 * 2 * 3 * 3 = 144

### 2. Prime Factorization Method

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

**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 6, 15, and 21 using the prime factorization method.

- Prime factorization of 6 can be expressed as 2 * 3 = 2
^{1}* 3^{1} - Prime factorization of 15 can be expressed as 3 * 5 = 3
^{1}* 5^{1} - Prime factorization of 21 can be expressed as 3 * 7 = 3
^{1}* 7^{1}

So, the LCM of 6, 15, and 21 = 2^{1} * 3^{1 }* 5^{1 }* 7^{1} = 2 * 3 * 5 * 7 = 210.

### 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, out of all the common multiples, take out the smallest common multiple. That will be the LCM of the given numbers.

For example, you have to find the LCM of 2, 3, and 9 using the listing method.

- Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18,…
- Multiples of 3 are 3, 6, 9, 12, 15, 18, 21,…
- Multiples of 9 are 9, 18, 27, 36,…

Here, it is clear that the least common multiple is 18.

So, the LCM of 2, 3, and 9 is 18.

## What Is the Formula for Finding the LCM of Three Numbers?

Before moving forward and knowing the formula for finding the LCM of three numbers, you must know about what is 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).

So, the formula for finding the LCM of three numbers is:

Let A, B, and C be three given integers.

So, the LCM of A, B, and C can be calculated using the formula:

LCM (A, B, C) = [(A * B * C) * HCF (A, B, C)] / [HCF (A, B) * HCF (B, C) * HCF (A, C)], where HCF is the highest common factor or the greatest common divisor.

## Problems Based on LCM

**Question 1: What is the LCM of 8, 14, and 25 using the prime factorization method?****Solution:**

- Prime factorization of 8 can be expressed as 2 * 2 * 2 = 2
^{3} - Prime factorization of 14 can be expressed as 2 * 7 = 2
^{1}* 7^{1} - Prime factorization of 25 can be expressed as 5 * 5 = 5
^{2}

So, the LCM of 8, 14, and 25 = 2^{3} * 5^{2 }* 7^{1} = 2 * 2 * 2 * 5 * 5 * 7 = 1400.

**Question 2: Find the LCM of 4, 6, and 12 using the listing method.Solution: **

- Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40,…
- Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54,…
- Multiples of 12 are 12, 24, 36, 48,…

Here, it is clear that the least common multiple is 12. So, the LCM of 4, 6, and 12 is 12.

**Question 3: Find the LCM of 9, 16, and 34 using the division method.****Solution:**

Prime Factors | First Number | Second Number | Third Number |

2 | 9 | 16 | 34 |

2 | 9 | 8 | 17 |

2 | 9 | 4 | 17 |

2 | 9 | 2 | 17 |

3 | 9 | 1 | 17 |

3 | 3 | 1 | 17 |

17 | 1 | 1 | 17 |

1 | 1 | 1 |

So, the LCM of 24 and 45 = 2 * 2 * 2 * 2 * 3 * 3 * 17 = 2448.

**Question 4: Find the LCM of 7, 21, and 35 using the prime factorization method.****Solution: **

- Prime factorization of 7 can be expressed as 7 = 7
^{1} - Prime factorization of 21 can be expressed as 3 * 7 = 3
^{1}* 7^{1} - Prime factorization of 35 can be expressed as 5 * 7 = 5
^{1}* 7^{1}

So, the LCM of 7, 21, and 35 = 3^{1} * 5^{1} * 7^{1} = 3 * 5 * 7 = 105.

**Question 5: Find the LCM of 3, 7, and 14 using the listing method.Solution: **

- Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54,…
- Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56,…
- Multiples of 14 are 14, 28, 42, 56, 70, 84,…

Here, it is clear that the least common multiple is 42. So, the LCM of 3, 7, and 14 is 42.

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## Frequently Asked Questions (FAQs)

**What is the formula for finding the LCM of three numbers?**

The formula for finding the LCM of three numbers is

LCM (A, B, C) = [(A * B * C) * HCF (A, B, C)] / [HCF (A, B) * HCF (B, C) * HCF (A, C)], where A, B, and C are the given integers.

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

No, LCM can be calculated for more than two numbers as well. At least 2 numbers are required for calculating the 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.

**What is the LCM of 5, 9, and 18?**

LCM of 5, 9, and 18 is 90.

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

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

- Division Method
- Prime Factorization Method
- Listing Method

**What are the properties of the LCM of three numbers?**

There are two major properties of the LCM of three numbers. They are:

- Commutative property
- Distributive property

Written by

Prerit Jain