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LCM

How To Find LCM of 30 and 35? | Listing, Division, and Prime Factorization Method

Written by Prerit Jain

Updated on: 17 Feb 2023

How To Find LCM of 30 and 35? | Listing, Division, and Prime Factorization Method

How To Find LCM of 30 and 35? | Listing, Division, and Prime Factorization Method

LCM of 30 and 35 is 210. LCM of 30 and 35, also known as Least Common Multiple or Lowest Common Multiple of 30 and 35 is the lowest possible common number that is divisible by 30 and 35.

Now, let’s see how to find the LCM of 30 and 35. Multiples of 30 are 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480,… and multiples of 35 are 35, 70, 105, 140, 175, 210, 245, 280, 315, 385, 420, 455, 490,… Here, both 210 and 420 are the common numbers in the multiples of 30 and 35, respectively, or that are divisible by 30 and 35. But, when you have to find the LCM, you must focus on the lowest common number. So, 210 is the lowest common number among all the multiples that is divisible by 30 and 35, and hence the LCM of 30 and 35 is 210.

How to Find the LCM of 30 and 35?

There are three different methods for finding the LCM of 30 and 35. They are

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

LCM of 30 and 35 Using the Prime Factorization Method

The prime factorization method is one of the methods for finding the LCM. To find the LCM of 30 and 35 using the prime factorization method, follow the following steps:

  • Step 1: Find the prime factors of 30 and 35 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 30 and 35.

LCM of 30 and 35 can be obtained using the prime factorization method as

  • Prime factorization of 30 can be expressed as 2 * 3 * 5 = 21 * 31 * 51
  • Prime factorization of 35 can be expressed as 5 * 7 = 51 * 71 

So, the LCM of 30 and 35 = 21 * 31 * 51 * 71 = 2 * 3 * 5 * 7 = 210

LCM of 30 and 35 Using the Division Method

The division method is one of the methods for finding the LCM. To find the LCM of 30 and 35 using the division method, divide 30 and 35 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 30 and 35.

Follow the following steps to find the LCM of 30 and 35 using the division method:

  • Step 1: Write the numbers for which you have to find the LCM, that is 30 and 35 in this case, separated by commas.
  • Step 2: Now, find the smallest prime number which is divisible by either 30 or 35.
  • Step 3: If any of the numbers among 30 and 35 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 30 and 35.

LCM of 30 and 35 can be obtained using the division method:

Prime FactorsFirst NumberSecond Number
23035
31535
5535
717
 11

So, the LCM of 30 and 35 = 2 * 3 * 5 * 7 = 210

LCM of 30 and 35 Using the Listing Method

The listing method is one of the methods for finding the LCM. To find the LCM of 30 and 35 using the listing method, follow the following steps:

  • Step 1: Write down the first few multiples of 30 and 35 separately.
  • Step 2: Out of all the multiples of 30 and 35 focus on the multiples that are common to both the numbers, that is, 30 and 35.
  • Step 3: Now, out of all the common multiples, take out the smallest common multiple. That will be the LCM of 30 and 35.

LCM of 30 and 35 can be obtained using the listing method:

  • Multiples of 30 are 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480,…
  • Multiples of 35 are 35, 70, 105, 140, 175, 210, 245, 280, 315, 385, 420, 455, 490,… 

Here, it is clear that the least common multiple is 210. So, the LCM of 30 and 35 is 210.

What Is the Formula for Finding the LCM of 30 and 35

LCM of 30 and 35 can be calculated using the formula:

LCM (30, 35) = (30 * 35) / HCF (30, 35),
where HCF is the highest common factor or the greatest common divisor of 30 and 35.

Another formula, using which the LCM of 30 and 35 can be found:
30 * 35 = LCM (30, 35) * HCF (30, 35), that is,
the product of 30 and 35 is equal to the product of its LCM and HCF.

Problems Based on LCM of 30 and 35

Question 1: What is the LCM of 30, 35, and 42?
Solution:

There are 3 methods of calculating LCM, we will find the LCM of 30, 35, and 42 using the prime factorization method:

To find the LCM of 30, 35, and 42 using the prime factorization method, first, we will find the prime factors of 30, 35, and 42 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 30, 35, and 42.

  • Prime factorization of 30 can be expressed as 2 * 3 * 5 = 21 * 31 * 51
  • Prime factorization of 35 can be expressed as 5 * 7 = 51 * 71 
  • Prime factorization of 42 can be expressed as 2 * 3 * 7 = 21 * 31 * 71 

So, the LCM of 30, 35, and 42 = 21 * 31 * 51 * 71 = 2 * 3 * 5 * 7 = 210

 

Question 2: Find the LCM of 30 and 35 using the division method.
Solution:

To find the LCM of 30 and 35 using the division method, first, we will find the smallest prime number which is divisible by either 30 or 35. If any of the numbers among 30 and 35 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 30 and 35.

Prime FactorsFirst NumberSecond Number
23035
31535
5535
717
 11

So, the LCM of 30 and 35 = 2 * 3 * 5 * 7 = 210

 

Question 3: If the product of two numbers is 1050 and their HCF is 5, find the LCM.
Solution:

As we know,
product of two numbers = LCM * HCF
It is given that,
product of the numbers = 1050 and HCF = 5
So, 1050 = LCM * 5
LCM = 1050 / 5
LCM = 210 
Hence, the LCM is 210.

Question 4: Find the LCM of 30 and 35 using the listing method.
Solution:

To find the LCM of 30 and 35 using the listing method, first, we will write down the first few multiples of 30 and 35 separately. Out of all the multiples of 30 and 35, 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 30 and 35.

  • Multiples of 30 are 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480,…
  • Multiples of 35 are 35, 70, 105, 140, 175, 210, 245, 280, 315, 385, 420, 455, 490,… 

Here, it is clear that the least common multiple is 210. So, the LCM of 30 and 35 is 210.

 

Question 5: What is the least perfect square divisible by 30 and 35?
Solution:

The least number divisible by 30 and 35 is the LCM of 30 and 35, that is, 210.
Using prime factorization, we can expand and write the LCM of 30 and 35 as 2 * 3 * 5 * 5 * 7. Here, we didn’t get complete pairs of all numbers, so to make the pairs complete, we will multiply 2, 7, and 3 with it.
Hence, the least perfect square divisible by 30 and 35 is LCM(30, 35) * 2 * 7 * 3 = 2 * 3 * 5 * 5 * 7 * 2 * 7 * 3 = 44100.

Frequently Asked Questions (FAQs)

What is the LCM of 30 and 35?
LCM of 30 and 35 is 210.

 

What are the methods to find the LCM of 30 and 35?
There are 3 major methods for finding the LCM of 30 and 35:

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

 

Are LCM and HCF of 30 and 35 the same?
LCM of 30 and 35 is 210 and HCF of 30 and 35 is 5. So, LCM and HCF of 30 and 35 are not the same.

 

Are the LCM of 30 and 35 the same as the LCM of 30, 35, and 42?
LCM of 30 and 35 is 210 and LCM of 30, 35, and 42 is also 210. So, the LCM of 30 and 35 are the same as the LCM of 30, 35, and 42.

 

Is 420 also considered as the LCM of 30 and 35?
No, 420 is not considered as the LCM of 30 and 35. 420 is a common multiple of 30 and 35. But, it is not the least common number which is divisible by 30 and 35, and while finding the LCM, you must focus on the lowest common number. So, 210 is the lowest common number divisible by 30 and 35.

We hope you understand all the basics of how to find the LCM of 30 and 35.

Written by

Prerit Jain

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