HCF and LCM

What is the Definition of LCM and HCF?

What Does HCF Mean?

  • The largest common factor of all the given numbers is known as the Highest Common Factor of the numbers.

  • The highest number that can be divided exactly into two or more numbers without any remainders.

  • It is also known as the Greatest Common Divisor (GCD).

  • The most easy way to find the HCF of two or more given numbers is to create a factor tree.

Here are a few highest common factor examples : 


(image will be uploaded soon)


The above picture shows how you can calculate the H.C.F. of 90 and 30 using the prime factorization method.


How is HCF Calculated?

There are three methods of how to find the highest common factor of any two or more given numbers:

  1.  Factorization Method

  2. Prime Factorization Method

  3. Division Method


How to Calculate LCM and HCF?

This how to calculate LCM & HCF :-


  1. FACTORIZATION METHOD 

In the hcf by factorization method,we find the greatest common factor by listing down the factors of the numbers.


Step 1: List down the factors of all the given numbers.


Step 2: Look for the highest factor among the listed factors which are common for both the given numbers.


Let us take an example:


Problem :Find the H.C.F. of 36 and 45.


Solution: 36 = 1,2,3,4,6.9,12,18,36


45 =1,3,5,9,45


Since, the highest common factor of 36 and 45 is 9, therefore H.C.F of 36 and 45 is 9.


  1. PRIME FACTORIZATION METHOD (Also known as the factor tree method)


Let us consider an example,


Step 1: In calculating the hcf by prime factorization, we factorize the numbers into prime numbers, which is known as the prime factors.


Step 2: Start by dividing the given numbers by 2(the first prime number),go on dividing till you can’t divide the number any further.


Step 3: Finally, then write the numbers as a product of the prime numbers. The product  of these common factors is the highest common factor of the given numbers.


Here’s an example for better understanding.


Find the hcf by prime factorization of the numbers 24 and 36.


Solution: To find the hcf by prime factorization ,

 

(image will be uploaded soon)


24 = 2×2×2×3×1

36 = 2×2×3×3×1

           

Since, the common factors of 24 and 36 are 2×3×2×1= 12


Therefore, hcf of 24 and 36 is 12.


DIVISION METHOD:

 In division method, we follow the following steps:


Step 1:  Treat the smallest number as the divisor,and the bigger number as the   dividend.


Step 2:  Further divide the divisor by the remainder.


Step 3: Stop dividing when the remainder equals zero.


Here’s an example,

Find the H.C.F. of 12,18.


Solution:

                                     12) 18 (1

                               - 12  

                                      6   )12 (2

                                        -12

                                            -------------

                                                    0

                                           --------------

         Here, 6 is the H.C.F.


How to Find the HCF of 3 Numbers or How to Find the Highest Common Factor?

We can find the hcf of 3 numbers either by Prime Factorisation Method or by Division Method. However, the steps for finding the highest common factor remains the same as  above.


Here are a few highest common factor examples of how to find the highest common factor, 


1.Find the hcf of 3 numbers 15, 30 and 90 using the Prime Factorization method.

Solution:


3

15

5

5


1


2

30

3

15

5

5


1


2

90

3

45

3

15

5

5


1


15 = 5 × 3 × 1

30 = 5 × 3 × 2 × 1

90 = 3 × 3 × 2 × 5 × 1

The common factors here are 1,3,5.

Therefore, the highest common factor of the numbers,15,30 and 90 is 5×3×1=15.


How is LCM Calculated?

LCM by Listing Multiples -

Step 1) You need to list the multiples of each number until at least one of the multiples appears on all the lists.

Step 2)Now find the smallest number that is on all of the lists

Step 3)This number is the Least Common Multiple.

Example: Let’s find the LCM of (6,7,21)

Write down the multiples of 6:  6, 12, 18, 24, 30, 36, 42, 48, 54, 60

Write down the multiples of 7:  7, 14, 21, 28, 35, 42, 56, 63

Write down the multiples of 21: 21, 42, 63

Now you need to find the smallest number that is present on all of the lists. So the LCM(6, 7, 21) is 42.


How to Find LCM by Prime Factorization?

Step 1) Firstly, find all the prime factors of each given number.

Step 2) Now, list down all the prime numbers found, as many times as these numbers occur most often for any one given number.

Step 3) In step 3, you need to multiply the list of prime factors together in order to find the LCM.

Step 4) The LCM (a,b) can be calculated by finding the prime factorization of both the numbers a and b. We can do the same process for the LCM of more than 2 numbers.

For example, Let’s find the LCM (12,30) we find:

First find the prime factorization of 12 = 2 × 2 × 3

Second, the prime factorization of 30 = 2 × 3 × 5

Using all prime numbers we have found as often as each occurs most often we take 2 × 2 × 3 × 5 = 60

Therefore, the LCM (12,30) = 60.


How to Find LCM by Prime Factorization Using the Concept of Exponents?

Step 1) Firstly, you need to find all the prime factors of each of the given numbers and write all the prime factors in exponent form.

Step 2) Now you need to list all the prime numbers found, using the highest exponent found for each of them.

Step 3) Lastly, multiply the list of prime factors you have with exponents together to find the Least common multiple.

For example: Find the LCM(12,18,30)

List down the prime factors of 12 = 2 × 2 × 3 = 22 × 31

List down the prime factors of 18 = 2 × 3 × 3 = 21 × 32

List down the prime factors of 30 = 2 × 3 × 5 = 21 × 31 × 51

You need to list all the prime numbers found, the number of times as they occur most often for any one given number and you need to multiply them together to find the Least common multiple.

After multiplying, 2 × 2 × 3 × 3 × 5 = 180

Using the concept of exponents instead, multiply together each of the prime numbers with the highest power

In exponential form, 22 × 32 × 51 = 180

So, the LCM(12,18,30) = 180.


Where is HCF and LCM Used?

Highest Common Factor

Least Common Multiple

Highest Common Factor helps us to figure out how many people can be invited?

Least Common multiple is about an event that is or will be repeating over and over.

Highest Common Factor helps to arrange something into rows or groups.

Least Common multiple helps to figure out when something will happen again simultaneously.


Questions to be Solved:

Question 1) Find out the LCM of(24,300)

Prime factors of 24 = 2 × 2 × 2 × 3 = 23 × 31

Prime factors of 300 = 2 × 2 × 3 × 5 × 5 = 22 × 31 × 52

List all the prime numbers found, as many times as they occur most often for any one given number and multiply them together to find the LCM

2 × 2 × 2 × 3 × 5 × 5 = 600.

Using exponents instead, multiply together each of the prime numbers with the highest power

23 × 31 × 52 = 600.

So LCM (24,300) = 600.

FAQ (Frequently Asked Questions)

Question 2) Find the Highest Common Factor of 184,230 and 276 by Using the Division Method.

Answer) Let’s find the highest common factor of 184,230 and 276-

184) 230 (1

       - 184

           46    )184 (4

                   -184

                -----------   

                      0

                 -----------


The Highest Common Factor of 184 and 230 is 46.

Now we will repeat the process again, to find the final solution.


       46) 276 (6

             -276

           -----------   

                0

           -----------


The Highest Common Factor of 276 and 46 is 46.

Therefore , the highest common factor (H.C.F.) of 184,230 and 276 is 46.