GCD and LCM relationship

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Presentation transcript:

GCD and LCM relationship

GCD and LCM relationship states that the product of two numbers is equal to the product of their LCM and GCD. 1st number x 2nd number = LCM x GCD

Proof : Consider the numbers 8 and 12. Let us find the L.C.M And G.C.D of 8 and 12. To do that we need to first find prime factorization of 8 and 12 Prime factorization is 8 = 2 x 2 x 2 Prime factorization is 12 = 2 x 2 x 3 GCD = 2 x 2 = 4 LCM = 2 x 2 x 2 x 3 = 24

8 x 12 24 x 4 = 96 96 1st number x 2nd number = LCM x GCD ∴ It is proved product of two number is product of its GCD and LCM

We know that 1st number x 2nd number = LCM x GCD On the basis of this relation, we find that: 1st number = (LCM x GCD) / 2nd number 2nd number = (LCM x GCD) / 1st number LCM = (1st number x 2nd number) / GCD GCD = (1st number x 2nd number) / LCM

Example1 : G.C.D and L.C.M of two numbers are 18 and 1782 respectively. One number is 162, find the other. Solution:- We know, G.C.D . × L.C.M. = First number × Second number then we get, 18 × 1782 = 162 × Second number Second number=18 × 1782/162 =198 Therefore, the second number = 198

Example2 : Find the LCM of the two numbers, if their product is 140 and GCD is 2 Solution:- Given:- product of two number is 140 and GCD is 2 We know that LCM = (1st number x 2nd number) / GCD ∴ LCM= 140 /2=70

Example3 : The LCM of 60 and 75 is 300, find their GCD. Solution:- Given :- the numbers are 60 and 75 and LCM is 300 We know that GCD= (1st number x 2nd number) / LCM ∴ GCD= 60x75/300 = 15

Try these The GCD and LCM of two numbers are 25 and 1750 respectively, if one of the number is 175, find the other. The product of two numbers are 3276. If their GCD is 6, find their LCM. If LCM of number is 290 and their product is 7250, what will be its GCD?