Relation Between HCF and LCM

In this article, before discussing the relation between HCF and LCM of two numbers, first we will learn the definition of the H.C.F. (Highest Common Factor) and the L.C.M (Least Common Multiple) and how to find the HCF and LCM of given two numbers.


H.C.F. (Highest Common Factor)

The HCF of two or more numbers is the greatest number that divides each of the numbers exactly. 

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 so obtained is HCF of given numbers.

For example: Find HCF of  9 and 21. 

                       Factors of 9 = 3 x 3 = 3

                                      Factors of 21 = 3 x 7 

                       Product of common factors of 9 and 21 = 3. 

                       So, HCF(9, 12) = 3


L.C.M (Least Common Multiple)

The L.C.M. of two or more numbers is the smallest number which is a multiple of all the given numbers.

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 are the relations between HCF and LCM of two numbers:

  1.  The product of HCF and LCM of any two given natural numbers is equal to the product of the given numbers. i.e. 

                  HCF × LCM = Product of the two numbers

Suppose A and B are two numbers, then.

                  HCF (A & B) × LCM (A & B) = A × B

 

For example:  Find HCF and LCM of 9 and 12. And verify HCF × LCM = Product of the two numbers.

Factors of 9 = 3 x 3 = 32

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

So, HCF = 3 and 

      LCM = 22 x 32 = 36

And HCF x LCM = 36 x 3 = 108

Product of 9 and 12 = 9 x 12 = 108.

Hence, HCF(9, 12) × LCM(9, 12) = Product of 9 and 12.

verified.

 

  1.   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.

 

  1. 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, 4LCM 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, 4HCF of  3, 4, 5 = 121 = 12

 

 

Solved Examples:

 

Q.1. The HCF of two numbers is 27 and their LCM is 2079. If one of the numbers is 297, find the other.

 

Solution: Given that, HCF = 27 , LCM = 2079 and one number = 297.

Since, we know that: HCF × LCM = Product of the two numbers

Putting the respective values in the above formula, we get:

      27 × 2079 = 297 × other number

⇒ other number = (27 x 2079) / 297

⇒ other number = 189.

Q.2. The ratio of two numbers is 3: 4 and their H.C.F. is 5. Find their L.C.M.

Solution: Let the two numbers be 3x and 4x.

Given that their H.C.F. = 5

Thus, x = 5

Therefore, the two numbers are 3 × 5 = 15 and 4 × 5 = 20.

Now, Prime factors of 15 = 3 × 5

And, Prime factors of 20 = 2 × 2 × 5 = 22  × 5

So, L.C.M. of 15 and 20 = 22 × 3 × 5

  L.C.M. of 15 and 20 = 60