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LCM

# How To Find LCM of 3 and 7? | Listing, Division, and Prime Factorization Method

Written by Prerit Jain

Updated on: 18 Feb 2023

### How To Find LCM of 3 and 7? | Listing, Division, and Prime Factorization Method

LCM of 3 and 7 is 21. LCM also known as the Least Common Multiple or Lowest Common Multiple of 3 and 7 is the lowest possible common number that is divisible by 3 and 7.

Now, let’s see how to find the LCM of 3 and 7. Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54,…and multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63,…Here, both 21 and 42 are the common numbers in the multiples of 3 and 7, respectively, or that are divisible by 3 and 7. But, when you have to find the LCM, you must focus on the lowest common number. So, 21 is the lowest common number among all the multiples that is divisible by 3 and 7, and hence the LCM of 3 and 7 is 21.

## Methods to Find the LCM of 3 and 7

There are three different methods for finding the LCM of 3 and 7. They are:

1. Division Method
2. Prime Factorization Method
3. Listing Method

## Finding LCM of 3 and 7 Using the Division Method

The division method is one of the methods for finding the LCM. To find the LCM of 3 and 7 using the division method, divide 3 and 7 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 3 and 7.

Follow the following steps to find the LCM of 3 and 7 using the division method:
Step 1: Write the numbers for which you have to find the LCM, that is 3 and 7 in this case, separated by commas.
Step 2: Now, find the smallest prime number which is divisible by either 3 or 7.
Step 3: If any of the numbers between 3 and 7 is 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 3 and 7.

LCM of 3 and 7 can be obtained using the division method:

So, the LCM of 3 and 7 = 3 * 7 = 21

## LCM of 3 and 7 Using the Prime Factorization Method

The prime factorization method is one of the methods for finding the LCM. To find the LCM of 3 and 7 using the prime factorization method, follow the following steps:

• Step 1: Find the prime factors of 3 and 7 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 3 and 7.

LCM of 3 and 7 can be obtained using the prime factorization method:

• Prime factorization of 3 can be expressed as 3 = 31
• Prime factorization of 7 can be expressed as 7 = 71

So, the LCM of 3 and 7 = 31 * 71= 3 * 7 = 21

## Finding LCM of 3 and 7 Using the Listing Method

The listing method is one of the methods for finding the LCM. To find the LCM of 3 and 7 using the listing method, follow the following steps:

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

LCM of 3 and 7 can be obtained using the listing method as

• 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, 63,…

Here, it is clear that the least common multiple is 21. So, the LCM of 3 and 7 is 21.

## What Is the Formula for Finding the LCM of 3 and 7?

LCM of 3 and 7 can be calculated using the formula:

LCM (3, 7) = (3 * 7) / HCF (3, 7),
where HCF is the highest common factor or the greatest common divisor of 3 and 7.

Another formula, using which the LCM of 3 and 7 can be found:
3 * 7 = LCM (3, 7) * HCF (3, 7), that is,
the product of 3 and 7 is equal to the product of its LCM and HCF.

## Problems Based on LCM of 3 and 7

Question 1: What is the least perfect square divisible by 3 and 7?
Solution:
The least number divisible by 3 and 7 is the LCM of 3 and 7, that is, 21.
Using prime factorization, we can expand and write the LCM of 3 and 7 as 3 * 7.
Here, we didn’t get complete pairs of all numbers, so to make the pairs complete, we will multiply 3 and 7 with them.
Hence, the least perfect square divisible by 3 and 7 is
LCM(3, 7) * 3 * 7 = 3 * 7 * 3 * 7 = 441

Question 2: Find the LCM of 3 and 7 using the division method.
Solution:
To find the LCM of 3 and 7 using the division method, first, we will find the smallest prime number which is divisible by either 3 or 7. If any of the numbers between 3 and 7 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 3 and 7.

So, the LCM of 3 and 7 = 3 * 7 = 21

Question 3: If the LCM of two numbers is 21, HCF is 1, and one of the numbers is 7, find the other number.
Solution:
As we know,
product of two numbers = LCM * HCF
It is given that,
one of the numbers = 7, LCM = 21, and HCF = 1
Let the other number be x.
So, 7 * x = 21 * 1
x = (21 * 1) / 7
x = 21 / 7
x = 3
Hence, the other number is 3.

Question 4: Find the LCM of 3 and 7 using the listing method.
Solution:
To find the LCM of 3 and 7 using the listing method, first, we will write down the first few multiples of 3 and 7 separately. Out of all the multiples of 3 and 7, we will focus on the multiples which are common to both numbers.
Then, out of all the common multiples, we will take out the smallest common multiple. That will be the LCM of 3 and 7.
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, 63,…
Here, it is clear that the least common multiple is 21. So, the LCM of 3 and 7 is 21.

Question 5: What are the other two numbers having the LCM as 21? Show the representation using the prime factorization method.
Solution:
Other than 3 and 7, LCM of 1 and 21 is also 21. We will prove this using the prime factorization method.
To find the LCM of 3 and 7 using the prime factorization method, first, we will find the prime factors of 3 and 7 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 3 and 7.

• Prime factorization of 3 can be expressed as 3 = 31
• Prime factorization of 7 can be expressed as 3 = 71

So, the LCM of 3 and 7 = 3* 71= 3 * 7 = 21

Now, to find the LCM of 1 and 21 using the prime factorization method, first, we will find the prime factors of 1 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 1 and 21.

• Prime factorization of 1 can be expressed as 1 = 11
• Prime factorization of 21 can be expressed as 3 * 7= 3* 71

So, the LCM of 1 and 21 = 3* 7* 11 = 3 * 7 * 1 = 21

Are LCM and HCF of 3 and 7 the same?
LCM of 3 and 7 is 21 and HCF of 3 and 7 is 1. So, LCM and HCF of 3 and 7 are not the same.

Is 42 also considered as the LCM of 3 and 7?
42 isn’t considered as the LCM of 3 and 7. 42 is a common multiple of 3 and 7. But, it is not the least common number which is divisible by 3 and 7, and while finding the LCM, you must focus on the lowest common number. So, 21 is the lowest common number divisible by 3 and 7.

What is the LCM of 3 and 7?
LCM of 3 and 7 is 21.

What are the methods to find the LCM of 3 and 7?
There are 3 major methods for finding the LCM of 3 and 7:
1. Division Method
2. Prime Factorization Method
3. Listing Method

Are the LCM of 3 and 7 the same as the LCM of 1, 3, and 7?
LCM of 3 and 7 is 21 and LCM of 1, 3, and 7 is also 21. So, the LCM of 3 and 7 are the same as the LCM of 1, 3, and 7.

We hope you understand all the basics about how to find the LCM of 3 and 7.

Written by by

Prerit Jain

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