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Chapter: 10th Mathematics : Algebra

Relationship between LCM and GCD

Relationship between Least Common Multiple (LCM) of Polynomials and Greatest Common Divisor (GCD) or Highest Common Factor (HCF).

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 2y2 ) and g (x ) = 8(x 3y3 )

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

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

From (1) and (2) we get,

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

= 24 ×(x 2y 2 )(x 2 + xy + y2 )

GCD f (x ), g(x) = 22 ×(xy ) = 4(xy)

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

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

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

= 96(x 2y2 )( x 3y3 )          ...(4)

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

 

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