1. GCD and LCM relationship

2. 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

3. 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 = 2 x 2 x 2
Prime factorization is = 2 x 2 x 3
GCD = 2 x 2 = 4
LCM = 2 x 2 x 2 x 3 = 24

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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 If their GCD is 6, find their LCM.
If LCM of number is 290 and their product is 7250, what will be its GCD?