H.C.F and L.C.M of Algebraic Expression Class 8 Mathematics

G.C.D = Greatest Common Divisior

H.C.F = Higest Common Factor

L.C.M = Least Common Multiple

Level – 01

1 ) Find G.C.D of 4a² b² , 20ab² .

Solution :

First term = 4a² b² = 2 * 2 * a * a * b * b

Second term = 20ab² = 5 * 2 * 2 * a * b * b

Required GCD = 2 * 2 * a * b * b= 4 ab²

2 ) Find G.C.D of 5p² q², 10p² q² , 25p⁴ q³

Solution :

First term : 5p² q² = 5 * 1 * p * p * q * q

Second term = 10p² q² = 5 * 2 * 1 * p * p * q * q

Third term = 25p⁴ q³ = 5 * 5 * 1 * p * p * p * p * q * q * q

Required GCD = 5 * 1 * p * p * q * q= 5p² q²

3 ) Find G.C.D of 7y³ z⁶ , 21y² , 14z²

Solution :

First term = 7y³ z⁶ = 7 * 1 * y * y * y * z * z* z* z* z*z

Second term = 21y² = 7 * 3 * y * y

Third term = 14z² = 7 * 2 * 1 * z * z

Required GCD = 7 * 1 = 7

4 ) Find G.C.D of 3a²b²c , 12a²b⁴c², 9a⁵b⁴

Solution:

First term = 3a²b²c = 3 * 1 * a * a * b * b * c

Second term = 12a²b⁴c² = 3 * 2 * 2 * a * a * b * b * b * b * c * c

Third term = 9a⁵b⁴ = 3 * 3 * 1 * a * a* a* a* a * b * b * b * b

Required GCD = 3 * a * a* b * b = 3a² b²

5 ) Find G.C.D of 2a²b, 4ab², 6a²b²

Solution :

First term = 2a²b = 2 * 1 * a * a * b

Second term = 4ab² = 2 * 2 * 1 * a * b * b

Third term = 6a²b² = 3 * 2 * 1 * a * a * b * b

Required GCD = 2ab

6 ) Find G.C.D of 6x²yz³ , 10x³y³z³ , 8x²yz⁴

Solution :

First term = 6x²yz³ = 3 * 2 * 1 * x * x * y * z * z * z

Second term = 10x³y³z³ = 5 * 2 * 1 * x * x * x * y * y * y * z * z * z

Third term = 8x²yz⁴ = 2 * 2 * 2 * 2 * 1 * x * x * y * z * z * z * z

Required GCD = 2 * 1 * x * x * y * z * z * z = 2x²yz³

7 ) Find G.C.D of ax², a²x³, a⁴x

Solution:

First term = ax² = a * x * x

Second term = a²x³ = a * a * x * x * x

Third term = a⁴x = a * a * a * a * x

Required GCD = ax

8 ) Find G.C.D of 4xy²z, 6xyz², 10x²yz

Solution :

First term = 4xy²z = 2 * 2 * x * y * y * z

Second term = 6xyz² = 3 * 2 * 1 * x * y * z * z

Third term = 10x²yz = 5 * 2 * 1 * x * x * y * z

Required GCD = 2 xyz

9 ) Find G.C.D of 3ab, 9a²c, 12a²c²

Solution :

First term = 3ab

Second term = 9a²c = 3 * 3 * a * a * c

Third term = 12a²c² = 3 * 2 * 2 * a * a * c * c

Required GCD = 3a

10 ) Find the GCD of 11a²bc², 33a²b²c, 55a²bc²

Solution :

First term = 11a²bc² = 11 * a * a * b * c * c

Second term = 33a²b²c = 11 * 3 * 1 * a * a * b * b * c

Third term = 55a²bc² = 11 * 5 * 1 * a * a * b * c * c

Required GCD = 11 a²bc

11 ) Find the LCM of 2x²y³, 10x³y

Solution :

First term = 2x²y³ = 2 * x * x * x* y * y * y

Second term = 10x³y = 5 * 2 * x * x * x * y

Required LCM = 2 * 5 * * x * x * x* y * y * y = 10 x³y³

12 ) Find the LCM of 7 p²q³ , 35p³q , 42pq⁴

Solution :

First term = 7 p²q³ = 7 * 1 * p * p * q * q * q

Second term = 35p³q = 7 * 5 * 1 * p * p * p * q

Third term = 42pq⁴ = 7 * 3 * 2 * 1 * p * q* q* q * q

Required LCM = 7 * 5 * 3 * 2 * 1 * p * p * p * q * q * q * q = 210 p³q⁴

13 ) Find the LCM of 5a⁵b, 15ab²c, 25a²b²c²

Solution :

First term = 5a⁵b = 5 * 1 * a * a* a * a * a * a * b

Second term = 15ab²c = 3 * 5 * 1 * a * b * b * c

Third term = 25a²b²c² = 5 * 5 * 1 * a * a * b * b * c * c

Required LCM = 5 * 5 * 3 * a * a* a * a * a* b * b * c * c = 75 a⁵b²c²

14 ) Find the LCM of 11a²bc², 33a²b²c, 55a²bc²

Solution :

First term = 11a²bc² = 11 * a * a * b * c * c

Second term = 33a²b²c = 11 * 3 * 1 * a * a * b * b * c

Third term = 55a²bc² = 11 * 5 * 1 * a * a * b * c * c

Required LCM = 11 * 3 * 5 * 1 * a * a * b * b * c * c = 165 a²b²c²

15 ) Find G.C.D of 2a²b, 4ab², 6a²b²

Solution :

First term = 2a²b = 2 * 1 * a * a * b

Second term = 4ab² = 2 * 2 * 1 * a * b * b

Third term = 6a²b² = 3 * 2 * 1 * a * a * b * b

Required LCM = 2 * 2 * 3 * a * a* b * b = 12 a²b²

16 ) Find G.C.D of 6x²yz³ , 10x³y³z³ , 8x²yz⁴

Solution :

First term = 6x²yz³ = 3 * 2 * 1 * x * x * y * z * z * z

Second term = 10x³y³z³ = 5 * 2 * 1 * x * x * x * y * y * y * z * z * z

Third term = 8x²yz⁴ = 2 * 2 * 2 * 2 * 1 * x * x * y * z * z * z * z

Required LCD = 2 * 2 * 2 * 5 * 3* x * x * x * y * y * y *z * z * z * z = 120 x³y³z⁴

17 ) Find G.C.D of ax², a²x³, a⁴x

Solution:

First term = ax² = a * x * x

Second term = a²x³ = a * a * x * x * x

Third term = a⁴x = a * a * a * a * x

Required LCM = a * a * a * a * x * x* x = a⁴x³

18 ) Find G.C.D of 4xy²z, 6xyz², 10x²yz

Solution :

First term = 4xy²z = 2 * 2 * x * y * y * z

Second term = 6xyz² = 3 * 2 * 1 * x * y * z * z

Third term = 10x²yz = 5 * 2 * 1 * x * x * y * z

Required LCM = 2 * 2 * 5 * 3 * 1 * x * x * y * y * z * z = 60 x²y²z²

19 ) Find G.C.D of 3ab, 9a²c, 12a²c²

Solution :

First term = 3ab

Second term = 9a²c = 3 * 3 * a * a * c

Third term = 12a²c² = 3 * 2 * 2 * a * a * c * c

Required LCM = 3 * 3 * 2 * 2 * a * a * c * c = 36 a²c²

20 ) Find G.C.D of 7y³ z⁶ , 21y² , 14z²

Solution :

First term = 7y³ z⁶ = 7 * 1 * y * y * y * z * z* z* z* z*z

Second term = 21y² = 7 * 3 * y * y

Third term = 14z² = 7 * 2 * 1 * z * z

Required LCM = 7 * 2 * 3 * 1 * y * y* y * z * z * z* z* z*z = 42 y³z⁶

Level – 02

1 ) Find HCF of x² + xy, xz + yz and x² + 2xy + y² 

First expression = x² + xy = x ( x + y ) 

Second Expression = xz + yz = z ( x + y ) 

Third Expression = x² + 2xy + y² 

               = ( x + y )² 

               = ( x + y ) ( x + y )

Required HCF = ( x + y )

2 ) Find HCF of 8 (x² – 4 ), 12 (x³ + 8 ) and 36 ( x² – 3x – 10 )

First Expression = 8 (x² – 4 )

              = 8 { (x)² – ( 2 )² }

            = 2 * 2 * 2 * ( x + 2 ) ( x – 2 )

Second Expression = 12 (x³ + 8 )

    = 12 { x³ + ( 2 )³ }

= 2 * 3 * 2 (x+2)(x² – x.2 + 2² )

= 2 * 3* 2 ( x + 2 ) ( x² – 2x + 4 )

Third Expression = 36 ( x² – 3x – 10 )

= 36{ x² – ( 5 – 2 )x – 10 }

= 36{ x² – 5x + 2x – 10 }

= 36 {x ( x – 5 ) + 2 ( x – 5 )}

= 3 * 3 * 2 *2 ( x – 5 ) ( x + 2 )

Required HCF = 2 * 2 * ( x + 2 ) = 4(x + 2)

3 ) Find the HCF of ( x³ – 16x ), ( 2x³ + 9x² + 4x ) and ( 2x³ + x² – 28x )

First expression = (x³ – 16x)

= x ( x² – 16 )

= x (x² – 4² )

= x ( x + 4 ) ( x – 4 )

Second Expression = 2x³ + 9x² + 4x

= x ( 2x² + 9x + 4 )

= x {2x² + ( 8 + 1 )x + 4 }

= x{ 2x² + 8x + x + 4 }

= x { 2x( x + 4 ) + 1 ( x + 4 ) }

= x ( x + 4 ) ( 2x + 1 )

Third Expression = 2x³ + x² – 28x

= x ( x²  + x – 28 )

= x { x²  + ( 8 – 7 )x – 28 }

= x { x² + 8x – 7x – 28 }

= x { x ( x + 8 ) – 7( x + 8 ) }

= x ( x + 8 ) ( x – 7 )

Required HCF = x

4 ) Find HCF of ( x³ – 3x² y ) and ( x² – 9y² )

First Expression = ( x³  – 3x² y ) = x² ( x – 3y )

Second Expression = x² – 9y² 

= x² – ( 3y )² 

= ( x + 3y ) ( x – 3y )

Required HCF = ( x – 3y )

5 ) Find HCF of ( x² – 1 ), ( x² – 2x + 1 ) and ( x³ + x² – 2x )

First Expression = x²  – 1

= x² – 1² 

= ( x + 1 ) ( x – 1)

Second Expression = x²  – 2x + 1

= x ² – 2.x.1 + 1² 

= ( x – 1 )² 

= ( x – 1 ) ( x – 1)

Third Expression = x³ + x² – 2x

= x ( x² + x – 2 )

= x { x² + ( 2 – 1 )x – 2 }

= x { x² + 2x – x – 2 }

= x { x ( x + 2 ) – 1 (x + 2 ) }

= x ( x + 2 ) ( x – 1 )

Required HCF = ( x – 1 )

6 ) Find the HCF of a² – 1, ( a³ – 1 ) , ( a² + a – 2 )

First Expression = a² – 1

= a² – 1² 

= ( a + 1 ) ( a – 1 )

Second Expression = a³ – 1

= a³ – 1³

= ( a – 1 ) ( a² + a . 1 + 1² )

= ( a – 1 ) ( a² + a + 1 )

Third Expression = a² + a – 2

= a²  + ( 2 – 1 )a – 2

= a² + 2a – a – 2

= a ( a + 2 ) – 1(a + 2 )

= ( a + 2 ) ( a – 1 )

Required HCF = ( a – 1 )

7 ) Find the HCF of

First expression= 2ax ( a- x )²

                         = 2ax ( a – x ) ( a – x )

Second Expression = 4a²x ( a – x )³

 = 2 * 2 * a * a* x * ( a – x ) ( a- x) ( a – x )

Required HCF = 2 * a * x * ( a- x ) ( a – x )

             = 2ax(a – x )²