LCM

# How to Find the LCM of 7, 8, 14, and 21? | Listing, Division, and Prime Factorization Method

Written by Prerit Jain

Updated on: 24 Feb 2023

Contents

### How to Find the LCM of 7, 8, 14, and 21? | Listing, Division, and Prime Factorization Method

LCM of 7, 8, 14, and 21 is 168. LCM of 7, 8, 14, and 21, also known as Least Common Multiple or Lowest Common Multiple of 7, 8, 14, and 21 is the lowest possible common number that is divisible by 7, 8, 14, and 21.

Now, let’s see how to find the LCM of 7, 8, 14, and 21.

Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189,…

Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192,…

Multiples of 14 are 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196, 210,…

Multiples of 21 are 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231,…

Here, 168 is the common number in the multiples of 7, 8, 14, and 21, respectively, or that are divisible by 7, 8, 14, and 21. So, 168 is the lowest common number among all the multiples that is divisible by 7, 8, 14, and 21, and hence the LCM of 7, 8, 14, and 21 is 168.

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## Methods to Find the LCM of 7, 8, 14, and 21

There are three different methods for finding the LCM of 7, 8, 14, and 21. They are:

1. Listing Method

2. Prime Factorization Method

3. Division Method

## LCM of 7, 8, 14, and 21 using the Prime Factorization Method

The prime factorization method is one of the methods for finding the LCM. To find the LCM of 7, 8, 14, and 21 using the prime factorization method, follow the following steps:

**Step 1:**Find the prime factors of 7, 8, 14, and 21 using the repeated division method.**Step 2:**Write all 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 7, 8, 14, and 21.

LCM of 7, 8, 14, and 21 can be obtained using the prime factorization method as

- Prime factorization of 7 can be expressed as 7 = 7
^{1 } - Prime factorization of 8 can be expressed as 2 * 2 * 2 = 2
^{1}* 2^{1}* 2^{1 }= 2^{3} - Prime factorization of 14 can be expressed as 2 * 7 = 2
^{1}* 7^{1} - Prime factorization of 21 can be expressed as 3 * 7 = 3
^{1}* 7^{1}

So, the LCM of 7, 8, 14, and 21 = 2^{3} * 3^{1} * 7^{1} = 2 * 2 * 2 * 3 * 7 = 168

## LCM of 7, 8, 14, and 21 Using the Listing Method

The listing method is one of the methods for finding the LCM. To find the LCM of 7, 8, 14, and 21 using the listing method, follow the following steps:

**Step 1:**Write down the first few multiples of 7, 8, 14, and 21 separately.**Step 2:**Out of all the multiples of 7, 8, 14, and 21, focus on the multiples that are common to all the numbers, that is, 7, 8, 14, and 21.**Step 3:**Now, out of all the common multiples, take out the smallest common multiple. That will be the LCM of 7, 8, 14, and 21.

LCM of 7, 8, 14, and 21 can be obtained using the listing method as

- Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189,…
- Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192,…
- Multiples of 14 are 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196, 210,…
- Multiples of 21 are 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231,…

Here, it is clear that the least common multiple is 168. So, the LCM of 7, 8, 14, and 21 is 168.

## LCM of 7, 8, 14, and 21 Using the Division Method

The division method is one of the methods for finding the LCM. To find the LCM of 7, 8, 14, and 21 using the division method, divide 7, 8, 14, and 21 by the smallest prime number, which is divisible by any of them. Then, the prime factors further obtained will be used to calculate the final LCM of 7, 8, 14, and 21.

Follow the following steps to find the LCM of 7, 8, 14, and 21 using the division method:

**Step 1:**Write the numbers for which you have to find the LCM, that is 7, 8, 14, and 21 in this case, separated by commas.**Step 2:**Now, find the smallest prime number, which is divisible by either 7, 8 or 14, or 21.**Step 3:**If any of the numbers among 7, 8, 14, and 21 are not divisible by the respective prime number, 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 7, 8, 14, and 21.

LCM of 7, 8, 14, and 21 can be obtained using the division method:

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

2 | 7 | 8 | 14 | 21 |

2 | 7 | 4 | 7 | 21 |

2 | 7 | 2 | 7 | 21 |

3 | 7 | 1 | 7 | 21 |

7 | 7 | 1 | 7 | 7 |

1 | 1 | 1 | 1 |

So, the LCM of 7, 8, 14, and 21 = 2 * 2 * 2 * 3 * 7 = 168

## Problems Based on the LCM of 7, 8, 14, and 21

**Question 1: What are the other numbers having the LCM as 168? Show the representation using the prime factorization method.****Solution:**

Besides, 7, 8, 14, and 21, LCM of 7 and 24 is also 168. We will prove this using the prime factorization method.

To find the LCM of 7, 8, 14, and 21 using the prime factorization method, first, we will find the prime factors of 7, 8, 14, and 21 using the repeated division method. Then, we will write all the prime factors in their exponent forms and multiply the prime factors having the highest power. The final result after multiplication will be the LCM of 7, 8, 14, and 21.

- Prime factorization of 7 can be expressed as 7 = 7
^{1 } - Prime factorization of 8 can be expressed as 2 * 2 * 2 = 2
^{1}* 2^{1}* 2^{1 }= 2^{3} - Prime factorization of 14 can be expressed as 2 * 7 = 2
^{1}* 7^{1} - Prime factorization of 21 can be expressed as 3 * 7 = 3
^{1}* 7^{1}

So, the LCM of 7, 8, 14, and 21 = 2^{3} * 3^{1} * 7^{1} = 2 * 2 * 2 * 3 * 7 = 168

Now, to find the LCM of 7 and 24 using the prime factorization method, first, we will find the prime factors of 7 and 24 using the repeated division method. Then, we will write all the prime factors in their exponent forms and multiply the prime factors having the highest power. The final result after multiplication will be the LCM of 7 and 24.

- Prime factorization of 7 can be expressed as 7 = 7
^{1 }

- Prime factorization of 24 can be expressed as 2 * 2 * 2 * 3 = 2
^{3}* 3^{1}

So, the LCM of 7 and 24 = 2^{3} * 3^{1} * 7^{1} = 2 * 2 * 2 * 3 * 7 = 168

**Question 2: Find the LCM of 7, 8, 14, and 21 using the division method.Solution:**

To find the LCM of 7, 8, 14, and 21 using the division method, first, we will find the smallest prime number, which is divisible by either 7 or 8 or 14 or 21. If any of the numbers among 7, 8, 14, and 21 is not divisible by the respective prime number, we will write that number in the next row just below it and proceed further. We will continue dividing the numbers obtained after each step by the prime numbers until we get the result as 1 in the entire row. We will multiply all the prime numbers and the final result will be the LCM of 7, 8, 14, and 21.

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

2 | 7 | 8 | 14 | 21 |

2 | 7 | 4 | 7 | 21 |

2 | 7 | 2 | 7 | 21 |

3 | 7 | 1 | 7 | 21 |

7 | 7 | 1 | 7 | 7 |

1 | 1 | 1 | 1 |

So, the LCM of 7, 8, 14, and 21 = 2 * 2 * 2 * 3 * 7 = 168

**Question 3: Find the smallest number divisible by 7, 8, 14, and 21.Solution:**

The smallest number divisible by 7, 8, 14, and 21 is the LCM of 7, 8, 14, and 21. Using the listing method, we can write the LCM of 7, 8, 14, and 21:

- Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189,…
- Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192,…
- Multiples of 14 are 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196, 210,…
- Multiples of 21 are 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231,…

Here, the smallest number divisible by 7, 8, 14, and 21 is 168.

**Question 4: Find the LCM of 7, 8, 14, and 21 using the prime factorization method.Solution:**

To find the LCM of 7, 8, 14, and 215 using the prime factorization method, first, we will find the prime factors of 7, 8, 14, and 21 using the repeated division method. Then, we will write all the prime factors in their exponent forms and multiply the prime factors having the highest power. The final result after multiplication will be the LCM of 7, 8, 14, and 21.

- Prime factorization of 7 can be expressed as 7 = 7
^{1 } - Prime factorization of 8 can be expressed as 2 * 2 * 2 = 2
^{1}* 2^{1}* 2^{1 }= 2^{3} - Prime factorization of 14 can be expressed as 2 * 7 = 2
^{1}* 7^{1} - Prime factorization of 21 can be expressed as 3 * 7 = 3
^{1}* 7^{1}

So, the LCM of 7, 8, 14, and 21 = 2^{3} * 3^{1} * 7^{1} = 2 * 2 * 2 * 3 * 7 = 168

**Question 5: What is the smallest square number that is divisible by each of the numbers, 7, 8, 14, and 21?Solution:**

The smallest square number that is divisible by each of the numbers, 7, 8, 14, and 21 is the LCM of 7, 8, 14, and 21, that is, 168.

Prime factors of 168 = 2 * 2 * 2 * 3 * 7. Here, factors 2, 3, and 7 have no pair. So, to make 168 a perfect square, we will multiply 168 by 2, 3, and 7.

Hence, the smallest square number that is divisible by each of the numbers, 7, 8, 14, and 21 is (168 * 2 * 3 * 7) = 7056

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## Frequently Asked Questions (FAQs)

**What is the LCM of 7, 8, 14, and 21?**

LCM of 7, 8, 14, and 21 is 168.

**What are the first three common numbers divisible by 7, 8, 14, and 21?**

The first three common numbers divisible by 7, 8, 14, and 21 are 168, 336, and 504.

**What are the methods to find the LCM of 7, 8, 14, and 21?**

There are 3 major methods for finding the LCM of 7, 8, 14, and 21. These methods are:

1. Prime Factorization Method

2. Listing Method

3. Division Method

**Are LCM and HCF of 7, 8, 14, and 21 the same?**

LCM of 7, 8, 14, and 21 is 168, and HCF of 7, 8, 14, and 21 is 1. So, LCM and HCF of 7, 8, 14, and 21 are not the same.

**Are the LCM of 7 and 14 the same as the LCM of 7 and 21?**

LCM of 7 and 14 is 14 and LCM of 7 and 21 is 21. So, the LCM of 7 and 14 are not the same as the LCM of 7 and 21.

We hope you understand all the basics on how to find the LCM of 7, 8, 14, and 21.

Written by by

Prerit Jain