Relation Between HCF and LCM Explained Clearly

HCF and LCM are two basic functions in Mathematics that can be used for a number of applications. HCF is an abbreviation for Highest Common Factor, while LCM is an abbreviation for Lowest Common Multiple. 

What are HCF And LCM?

HCF is the highest factor of two or more than two numbers which will divide the number completely and leave no remainder. LCM of two or more than two numbers refers to the lowest number that will divide the given number and leave no remainder. In rough terms, this is what the two terms indicate. 

How to find HCF of Given Two or More Numbers

Firstly, the given number is resolved into its prime factors. Then, Common prime factors of given numbers are multiplied. And the product obtained is HCF of given numbers.

For example: Find HCF of  9 and 21.                        

Factors of 9 = 3 x 3 = 32                                       

Factors of 21 = 3 x 7                        

Product of common factors of 9 and 21 = 3.                        

So, HCF(9, 12) = 3

How to find LCM of Given Numbers

Firstly, the given number is resolved into its prime factors. Then, L.C.M. is given by the product of the factors of the resolved expressions, each factor considered once with the maximum exponent which appears in it. 

For example: Find LCM of  12 and 18 .                       

Factors of 12 = 2 x 2 x 3 =22 x 3                      

Factors of 18 = 2 x 3 x 3 = 2 x 32

Since LCM is given by the product of the maximum exponent of each factor which has appeared in the prime factorisation of each of the given numbers.

So, LCM(12, 18) =  22 x 32 = 36. 

Relation between HCF and LCM

The relation between HCF and LCM provides an easy way to solve the problem. Following ar e the relations between HCF and LCM of two numbers: 

• The HCF of Co-prime numbers is equal to 1. LCM of Co-prime numbers is equal to the product of the Co-prime numbers.  

For example: 10 and 11 are coprime numbers.                        

So, HCF(10, 11) = 1 and                         

LCM (10, 11) = 10 x 11 = 110. 

• H.C.F. and L.C.M. of Fractions             

HCF of fractions = HCF of Numerators / LCM of Denominators                  

LCM of fractions = LCM of Numerators  / HCF of Denominators   

For example: Find HCF and LCM of \[\frac{2}{3}\] ,\[\frac{3}{4}\] and \[\frac{4}{5}\] .                

First, Find prime factors of 2, 3, 4 and 5.          

2 = 1 x 2          

3 = 1 x 3         

 4 = 1 x 2 x 2 = 1 x 22          

5 = 1 x 5

So, HCF of given fractions  23 , 34 and 45

HCF of 2, 3, 4 = 1

LCM of 3, 4, 5 = 22 x 3 x 5 = 60 

HCF(  23 , 34 and 45) = HCF of 2, 3, 4

LCM of 3, 4, 5= 160 

And LCM of given fractions  23 , 34 and 45

HCF of 3, 4, 5 = 1

LCM of 2, 3, 4 = 22 x 3 = 12 

LCM(  23 , 34 and 45) = LCM of 2, 3, 4

HCF of  3, 4, 5 = 121 = 12 

• HCF and LCM of positive integers

Positive integers refer to any number that is greater than 0 and lies on the right side of zero when graphed on a number line. The relationship here will be as follows:

If the positive integers are x and y, then

HCF (x,y) * LCM (x,y) = x*y

To demonstrate this, we can say that if we take the numbers 12 and 8, 

HCF of 12 and 8 = 4

LCM of 12 and 8 = 24

Therefore, HCF(4) * LCM (24) = 96

This is also equal to 4*24. 

Some Special Cases Of HCF And LCM

In some cases where the numbers may not be whole numbers, it is important to know the rules and the relationships for HCF and LCM. these cases may include:

• HCF and LCM of more than 2 numbers:

In case there is a need to find out the HCF and LCM for more than two numbers, this method can be employed. Here we have used three numbers and will find out the HCF and LCM for them. 

Suppose the three numbers are x,y and z. 

To find the LCM of these, we need to multiply the product of x,y and z with their HCF and divide that with HCF of x and y, HCF of y and z and HCF of x and z. 

Therefore, 

LCM = (x*y*z) * (HCF of x,y,z)/ HCF (x,y)* HCF (y,z) * HCF (x,z)

To find the HCF, the inverse formula needs to be used. 

HCF (x,y and z) = (x*y*z) * (LCM of x,y,z)/ LCM (x,y)* LCM (y,z) * LCM (x,z)