#FutureSTEMLeaders - Wiingy's $2400 scholarship for School and College Students

Apply Now

LCM

How To Find LCM of 45, 60, and 75? | Listing, Division, and Prime Factorization Method

Written by Prerit Jain

Updated on: 17 Feb 2023

How To Find LCM of 45, 60, and 75? | Listing, Division, and Prime Factorization Method

How To Find LCM of 45, 60, and 75? | Listing, Division, and Prime Factorization Method

LCM of 45, 60, and 75 is 900. LCM of 45, 60, and 75, also known as the Least Common Multiple or Lowest Common Multiple of 45, 60, and 75 is the lowest possible common number that is divisible by 45, 60, and 75.

To find the LCM of 45, 60, and 75 let’s list out the multiples of 45, 60, and 70 till we find the least or lowest common multiple:

Multiples of 45 are 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, 495, 540, 585, 630, 675, 720, 765, 810, 855, 900, 945, 990,… 
Multiples of 60 are 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720, 780, 840, 900, 960, 1020,… 
Multiples of 75 are 75, 150, 225, 300, 375, 450, 525, 600, 675, 750, 825, 900, 975, 1050,… 
Here, 900 is the common number in the multiples of 45, 60, and 75, respectively, or that are divisible by 45, 60, and 75. So, 900 is the lowest common number among all the multiples that is divisible by 45, 60, and 75, and hence the LCM of 45, 60, and 75 is 900.

Methods to Find the LCM of 45, 60, and 75

There are three different methods for finding the LCM of 45, 60, and 75. These methods are:

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

LCM of 45, 60, and 75 Using the Listing Method

The listing method is one of the methods for finding the LCM. To find the LCM of 45, 60, and 75 using the listing method, follow the following steps:

  • Step 1: Write down the first few multiples of 45, 60, and 75 separately.
  • Step 2: Out of all the multiples of 45, 60, and 75, focus on the multiples that are common to both the numbers, that is, 45, 60, and 75.
  • Step 3: Now, out of all the common multiples, take out the smallest common multiple. That will be the LCM of 45, 60, and 75.

LCM of 45, 60, and 75 can be obtained using the listing method:

  • Multiples of 45 are 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, 495, 540, 585, 630, 675, 720, 765, 810, 855, 900, 945, 990,…
  • Multiples of 60 are 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720, 780, 840, 900, 960, 1020,… 
  • Multiples of 75 are 75, 150, 225, 300, 375, 450, 525, 600, 675, 750, 825, 900, 975, 1050,… 

Here, it is clear that the least common multiple is 900. So, the LCM of 45, 60, and 75 is 900.

LCM of 45, 60, and 75 Using the Prime Factorization Method

The prime factorization method is one of the methods for finding the LCM. To find the LCM of 45, 60, and 75 using the prime factorization method, follow the following steps:

  • Step 1: Find the prime factors of 45, 60, and 75 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 45, 60, and 75.

LCM of 45, 60, and 75 can be obtained using the prime factorization method:

  • Prime factorization of 45 can be expressed as 3 * 3 * 5 = 31 * 31 * 5= 32 * 51 
  • Prime factorization of 60 can be expressed as 2 * 2 * 3 * 5 = 21 * 21 * 3* 51 = 22 * 3* 5
  • Prime factorization of 75 can be expressed as 3 * 5 * 5 = 31 * 51  * 51 = 3* 52

So, the LCM of 45, 60, and 75 = 22 * 32 * 52 = 2 * 2 * 3 * 3 * 5 * 5 = 900

LCM of 45, 60, and 75 Using the Division Method

The division method is one of the methods for finding the LCM. To find the LCM of 45, 60, and 75 using the division method, divide 45, 60, and 75 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 45, 60, and 75.

Follow the following steps to find the LCM of 45, 60, and 75 using the division method:

  • Step 1: Write the numbers for which you have to find the LCM, that is 45, 60, and 75 in this case, separated by commas.
  • Step 2: Now, find the smallest prime number which is divisible by either 45 or 60, or 75.
  • Step 3: If any of the numbers among 45, 60, and 75 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 45, 60, and 75.

LCM of 45, 60, and 75 can be obtained using the division method:

Prime FactorsFirst NumberSecond NumberThird Number
2456075
2453075
3451575
315525
55525
5115
 111

So, the LCM of 45, 60, and 75 = 2 * 2 * 3 * 3 * 5 * 5 = 900

What Is the Formula for Finding the LCM of 45, 60, and 75?

LCM of 45, 60, and 75 can be calculated using the formula:

LCM (45, 60, 75) = [(45 * 60 * 75) * HCF (45, 60, 75)] / [HCF (45, 60) * HCF (60, 75) * HCF (45, 75)]
where HCF is the highest common factor or the greatest common divisor.

Problems Based on LCM of 45, 60, and 75

Q 1: What are the other numbers having the LCM as 900? Show the representation using the prime factorization method.
Solution:

Other than 45, 60, and 75, LCM of 300 and 450 is also 900. We will prove this using the prime factorization method.

To find the LCM of 45, 60, and 75 using the prime factorization method:

  • Step 1: First, we will find the prime factors of 45, 60, and 75 using the repeated division method.
  • Step 2: 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 45, 60, and 75.

  • Prime factorization of 45 can be expressed as 3 * 3 * 5 = 31 * 31 * 5= 32 * 51 
  • Prime factorization of 60 can be expressed as 2 * 2 * 3 * 5 = 21 * 21 * 3* 51 = 22 * 3* 5
  • Prime factorization of 75 can be expressed as 3 * 5 * 5 = 31 * 51  * 51 = 3* 52

So, the LCM of 45, 60, and 75 = 22 * 32 * 52 = 2 * 2 * 3 * 3 * 5 * 5 = 900

Now, to find the LCM of 300 and 450 using the prime factorization method:

  • Step 1: First, we will find the prime factors of 300 and 450 using the repeated division method.
  • Step 2: 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 300 and 450.

  • Prime factorization of 300 can be expressed as 2 * 2 * 3 * 5 * 5 = 21 * 21 * 3* 51 * 51 = 22 * 3* 52
  • Prime factorization of 450 can be expressed as 2 * 3 * 3 * 5 * 5 = 21 * 31 * 3* 51 * 51 = 21 * 3* 52

So, the LCM of 300 and 450 = 2* 3* 52 = 2 * 2 * 3 * 3 * 5 * 5 = 900

 

Q 2: Find the LCM of 45, 60, and 75 using the listing method.
Solution:

To find the LCM of 45, 60, and 75 using the listing method:

  • Step 1: First, we will write down the first few multiples of 45, 60, and 75 separately.
  • Step 2: Out of all the multiples of 45, 60, and 75, we will focus on the multiples which are common to all numbers.
  • Step 3: Then, out of all the common multiples, we will take out the smallest common multiple. That will be the LCM of 45, 60, and 75.

 

  • Multiples of 45 are 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, 495, 540, 585, 630, 675, 720, 765, 810, 855, 900, 945, 990,…
  • Multiples of 60 are 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720, 780, 840, 900, 960, 1020,… 
  • Multiples of 75 are 75, 150, 225, 300, 375, 450, 525, 600, 675, 750, 825, 900, 975, 1050,… 

Here, it is clear that the least common multiple is 900. So, the LCM of 45, 60, and 75 is 900.

 

Q 3: If the product of three numbers is 202500, their HCF is 15, and the product of the HCF of the 1st and 2nd number and 2nd and 3rd number, and 1st and 3rd number among three numbers is 3375, find the LCM.
Solution:

As we know,
LCM of three numbers = (Product of three numbers * HCF) / [(HCF of 1st and 2nd number) * (HCF of 2nd and 3rd number) * (HCF 1st and 3rd number)]
It is given that,
product of the numbers = 202500, HCF = 15, and the product of the HCF of the 1st and 2nd number and 2nd and 3rd number, and 1st and 3rd number = 3375
So, LCM = (202500 * 15) / 3375
LCM = 3037500 / 3375
LCM = 900 
Hence, the LCM is 900.

 

Q 4: Find the smallest number divisible by 45, 60, and 75.
Solution:

The smallest number divisible by 45, 60, and 75 is the LCM of 45, 60, and 75.
Using the listing method, we can write the LCM of 45, 60, and 75 as:

  • Multiples of 45 are 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, 495, 540, 585, 630, 675, 720, 765, 810, 855, 900, 945, 990,…
  • Multiples of 60 are 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720, 780, 840, 900, 960, 1020,… 
  • Multiples of 75 are 75, 150, 225, 300, 375, 450, 525, 600, 675, 750, 825, 900, 975, 1050,… 

Here, the smallest number divisible by 45, 60, and 75 is 900.

 

Q 5: Find the LCM of 45, 60, and 75 using the division method.
Solution:

To find the LCM of 45, 60, and 75 using the division method, first, we will find the smallest prime number which is divisible by either 45 or 60, or 75. If any of the numbers among 45, 60, and 75 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 45, 60, and 75.

Prime FactorsFirst NumberSecond NumberThird Number
2456075
2453075
3451575
315525
55525
5115
 111

So, the LCM of 45, 60, and 75 = 2 * 2 * 3 * 3 * 5 * 5 = 900

 

Frequently Asked Questions (FAQs)

What is the LCM of 45, 60, and 75?
LCM of 45, 60, and 75 is 900.

 

Are LCM and HCF of 45, 60, and 75 the same?
LCM of 45, 60, and 75 is 900, and HCF of 45, 60, and 75 is 15. So, LCM and HCF of 45, 60, and 75 are not the same.

 

What are the methods to find the LCM of 45, 60, and 75?
There are 3 major methods for finding the LCM of 45, 60, and 75:

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

 

Are the LCM of 45 and 60 the same as the LCM of 60 and 75?
LCM of 45 and 60 is 180 and LCM of 60 and 75 is 300. So, the LCM of 45 and 60 are not the same as the LCM of 60 and 75.

 

What are the first three common numbers divisible by 45, 60, and 75?
The first three common numbers divisible by 45, 60, and 75 are 900, 1800, and 2700. 

We hope you understand all the basics of how to find the LCM of 45, 60, and 75.

Written by

Prerit Jain

Share article on

tutor Pic
tutor Pic