HCF(Highest Common Factor) AND LCM(Least Common Multiple)


For discussing HCF AND LCM , first we learn the concept of prime number and composite numbers.

हम HCF(मा० सा०) और LCM (ला० सा०)  निकालने से पहले यह समझेंगे की अभाज्य संख्या और भाज्य संख्या क्या होती है। 


Prime numbers(अभाज्य संख्या )

The number which has exactly two factors 1 and itself are called prime number. For example - 2, 3, 5, 7, 11, etc.
अभाज्य संख्या वह संख्या होती है जो केवल 1 से या फिर स्वयं से विभाजित होती है।  अभाज्य संख्या कहलाती है।  जैसे = 2, 3, 5, 7, 11, आदि। 


Composite numbers( भाज्य संख्या )

The numbers which has more than 2 factors 1 , itself and the other ones are called composite numbers. For example - 4, 6, 8, 10, 12, etc.
भाज्य संख्या वह संख्या होती है , जो स्वयं से , 1 से और किसी अन्य संख्या से भी विभाजित होती है। जैसे - 4, 6, 8, 10, 12, आदि। 

Mathematics Section 


Co-Prime numbers( सह- अभाज्य संख्या )

Co-prime numbers are those numbers which have only one common factor. Example - (3,4) , ( 5,6) , (10,11) , etc.


How to find HCF(Highest Common Factors) ?

For finding HCF their are three ways , 
#1  
If we take two number for example 6 and 12.
First we prime factorized both the numbers 
6 = 2〤3
12 = 2〤2〤3

so,  
6 = ❷〤❸
12 = ❷〤2〤❸

After prime factorization the highest common value we get from both the number and multiply that numbers we get the  HCF(highest Common Factor), from 6 and 12 common numbers are 2 and 3 .

Therefore HCF(6, 12) = 2〤3 = 6

 #2

We discuss the second method to find the HCF of any numbers. we take the same above example and the HCF(6,12) by factors method.
In this method we find the factors of 6 and 12.

Factor of 6 = 1, 2, 3, 6.
Factor of 12 = 1, 2, 3, 4, 6, 12.

The common factor of 6 and 12 is 1, 2, 3, and 6.
then the highest of these factor is 6.

 Therefore HCF (6,12) = 6


#3  Continuous Division method

This is also a method to find HCF of any two or more then two numbers. When we get a very large numbers to find the HCF then it is difficult to use above methods to find the answer. So this is one of the important method to find the HCF of a number. 
I explain the concept with one example so you can understand easily. 
For example if we find the HCF(24, 36) ?

24 and 36 are the two numbers and we divide the larger number 36 with the smaller number 24 . and we do this process till we not get the remainder zero.

so, we divide 36 by 24


After divide we get the remainder 12 and quotient 1, we didn't get the remainder 0.
so, we again do the same process in this case 24 is divided by 12


Here we get the remainder zero , quotient is 2 and divisor is 12.
The divisor of last division in which we  get the remainder 0 is called the HCF of these following numbers.

Therefore HCF(24,36) = 12


How to find the LCM(Least Common Multiples)?

The least common multiple of any two number or more than two numbers is the lowest of their common multiples.

We take the same example , what we take in finding of HCF , so you can understand the concept easily.

Find the LCM of 6 and 12 by finding their multiples.

Multiple of 6 = 6, 12, 24, …
Multiple of 12 = 12, 24, 36, …

The common multiple of 6 and 12 are 12 and 24, … so on.
we take the lowest common multiple of their numbers. which is 12.

Therefore LCM of 6 and 12 is 12.

#2

For finding LCM by prime factorization method.
In this method we prime factorized both the number and what are the terms we get, multiply all the terms and we get the LCM.

For example if we find the LCM of 24 and 36 by prime factorization method.