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