Relationship between LCM and GCD

Relationship between LCM and GCD

Let us consider two numbers 12 and 18.

We observe that, LCM (12, 18) = 36, GCD (12, 18) = 6 .

Now, LCM (12,18) × GCD(12,18) = 36 ×6 = 216 = 12 ×18

Thus LCM × GCD is equal to the product of two given numbers.

Similarly, the product of two polynomials is the product of their LCM and GCD, 

That is, f (x) × g (x) = LCM [ f (x ), g(x )] ×GCD[ f (x), g (x)]

Illustration

Consider     f(x) = 12(x ² −y² ) and g (x ) = 8(x ³ −y³ )

Now          f(x) = 12(x ² −y² ) = 2² × 3 ×(x + y )(x −y)    ...(1)

And          g(x) = 8(x ³ −y³ ) = 2 ³ ×(x − y )(x ² + xy + y² ) ...(2)

From (1) and (2) we get,

LCM[f(x), g(x)] = 2 ³ ×3 × (x + y)(x − y )(x ² + xy + y² )

= 24 ×(x ² − y ² )(x ² + xy + y² )

GCD f (x ), g(x) = 2² ×(x − y ) = 4(x −y)

LCM ×GCD = 24 ×4 × (x ² −y² )× (x ² + xy + y ² )×(x −y)

LCM ×GCD = 96(x ³ −y³ )(x ² −y² )         ...(3)

product of f(x) and g(x) = 12(x ² −y² )× 8(x ³ −y³ )

= 96(x ² −y² )( x ³ −y³ )          ...(4)

From (3) and (4) we obtain LCM × GCD = f (x )×g (x )