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 4a ^{2} b^{2} , 20ab^{2} .**

Solution :

First term = **4a ^{2} b^{2}** = 2 * 2 * a * a * b * b

Second term = **20ab ^{2} **= 5 * 2 * 2 * a * b * b

Required GCD = 2 * 2 * a * b * b= 4 ab^{2}

**2 ) Find G.C.D of 5p ^{2} q^{2}, 10p^{2} q^{2} , 25p^{4} q^{3} **

Solution :

First term : **5p ^{2} q^{2}** = 5 * 1 * p * p * q * q

Second term = **10p ^{2} q^{2}** = 5 * 2 * 1 * p * p * q * q

Third term = **25p ^{4} q^{3}** = 5 * 5 * 1 * p * p * p * p * q * q * q

Required GCD = 5 * 1 * p * p * q * q= **5p ^{2} q^{2}**

**3 ) Find G.C.D of 7y ^{3} z^{6} , 21y^{2} , 14z^{2}**

Solution :

First term = **7y ^{3} z^{6}** = 7 * 1 * y * y * y * z * z* z* z* z*z

Second term = **21y ^{2}** = 7 * 3 * y * y

Third term = **14z ^{2}** = 7 * 2 * 1 * z * z

Required GCD = 7 * 1 = 7

**4 ) Find G.C.D of 3a ^{2}b^{2}c , 12a^{2}b^{4}c^{2}, 9a^{5}b^{4}**

Solution:

First term = **3a ^{2}b^{2}c** = 3 * 1 * a * a * b * b * c

Second term = **12a ^{2}b^{4}c^{2}** = 3 * 2 * 2 * a * a * b * b * b * b * c * c

Third term = **9a ^{5}b^{4}** = 3 * 3 * 1 * a * a* a* a* a * b * b * b * b

Required GCD = 3 * a * a* b * b = 3a** ^{2}** b

^{2}**5 ) Find G.C.D of 2a ^{2}b, 4ab^{2}, 6a^{2}b^{2}**

Solution :

First term = **2a ^{2}b** = 2 * 1 * a * a * b

Second term = **4ab ^{2}** = 2 * 2 * 1 * a * b * b

Third term = **6a ^{2}b^{2}** = 3 * 2 * 1 * a * a * b * b

Required GCD = 2ab

**6 ) Find G.C.D of 6x ^{2}yz^{3} , 10x^{3}y^{3}z^{3} , 8x^{2}yz^{4}**

Solution :

First term = **6x ^{2}yz^{3}** = 3 * 2 * 1 * x * x * y * z * z * z

Second term = **10x ^{3}y^{3}z^{3}** = 5 * 2 * 1 * x * x * x * y * y * y * z * z * z

Third term = **8x ^{2}yz^{4}** = 2 * 2 * 2 * 2 * 1 * x * x * y * z * z * z * z

Required GCD = 2 * 1 * x * x * y * z * z * z =** 2x ^{2}yz^{3}**

**7 ) Find G.C.D of ax ^{2}, a^{2}x^{3}, a^{4}x**

Solution:

First term = **ax ^{2}** = a * x * x

Second term = **a ^{2}x^{3}** = a * a * x * x * x

Third term = **a ^{4}x** = a * a * a * a * x

Required GCD = ax

**8 ) Find G.C.D of 4xy ^{2}z, 6xyz^{2}, 10x^{2}yz**

Solution :

First term = **4xy ^{2}z** = 2 * 2 * x * y * y * z

Second term = **6xyz ^{2}** = 3 * 2 * 1 * x * y * z * z

Third term = **10x ^{2}yz** = 5 * 2 * 1 * x * x * y * z

Required GCD = 2 xyz

**9 ) Find G.C.D of 3ab, 9a ^{2}c, 12a^{2}c^{2}**

Solution :

First term = **3ab**

Second term = **9a ^{2}c** = 3 * 3 * a * a * c

Third term = **12a ^{2}c^{2}** = 3 * 2 * 2 * a * a * c * c

Required GCD = 3a

**10 ) Find the GCD of 11a ^{2}bc^{2}, 33a^{2}b^{2}c, 55a^{2}bc^{2}**

Solution :

First term = **11a ^{2}bc^{2}** = 11 * a * a * b * c * c

Second term = **33a ^{2}b^{2}c** = 11 * 3 * 1 * a * a * b * b * c

Third term = **55a ^{2}bc^{2}** = 11 * 5 * 1 * a * a * b * c * c

Required GCD = 11 a** ^{2}**bc

**11 ) Find the LCM of 2x ^{2}y^{3}, 10x^{3}y**

Solution :

First term = **2x ^{2}y^{3}** = 2 * x * x * x* y * y * y

Second term = **10x ^{3}y** = 5 * 2 * x * x * x * y

Required LCM = 2 * 5 * * x * x * x* y * y * y = 10 x** ^{3}**y

^{3}**12 ) Find the LCM of 7 p ^{2}q^{3} , 35p^{3}q , 42pq^{4}**

Solution :

First term = **7 p ^{2}q^{3}** = 7 * 1 * p * p * q * q * q

Second term = **35p ^{3}q** = 7 * 5 * 1 * p * p * p * q

Third term = **42pq ^{4}** = 7 * 3 * 2 * 1 * p * q* q* q * q

Required LCM = 7 * 5 * 3 * 2 * 1 * p * p * p * q * q * q * q = 210 p** ^{3}**q

^{4}**13 ) Find the LCM of 5a ^{5}b, 15ab^{2}c, 25a^{2}b^{2}c^{2}**

Solution :

First term = **5a ^{5}b** = 5 * 1 * a * a* a * a * a * a * b

Second term = **15ab ^{2}c** = 3 * 5 * 1 * a * b * b * c

Third term = **25a ^{2}b^{2}c^{2}** = 5 * 5 * 1 * a * a * b * b * c * c

Required LCM = 5 * 5 * 3 * a * a* a * a * a* b * b * c * c = 75 a** ^{5}**b

**c**

^{2}

^{2}14 ) Find the LCM of **11a ^{2}bc^{2}, 33a^{2}b^{2}c, 55a^{2}bc^{2}**

Solution :

First term = **11a ^{2}bc^{2}** = 11 * a * a * b * c * c

Second term = **33a ^{2}b^{2}c** = 11 * 3 * 1 * a * a * b * b * c

Third term = **55a ^{2}bc^{2}** = 11 * 5 * 1 * a * a * b * c * c

Required LCM = 11 * 3 * 5 * 1 * a * a * b * b * c * c = 165 a** ^{2}**b

**c**

^{2}

^{2}1**5 ) Find G.C.D of 2a ^{2}b, 4ab^{2}, 6a^{2}b^{2}**

Solution :

First term = **2a ^{2}b** = 2 * 1 * a * a * b

Second term = **4ab ^{2}** = 2 * 2 * 1 * a * b * b

Third term = **6a ^{2}b^{2}** = 3 * 2 * 1 * a * a * b * b

Required LCM = 2 * 2 * 3 * a * a* b * b = 12 a** ^{2}**b

^{2}1**6 ) Find G.C.D of 6x ^{2}yz^{3} , 10x^{3}y^{3}z^{3} , 8x^{2}yz^{4}**

Solution :

First term = **6x ^{2}yz^{3}** = 3 * 2 * 1 * x * x * y * z * z * z

Second term = **10x ^{3}y^{3}z^{3}** = 5 * 2 * 1 * x * x * x * y * y * y * z * z * z

Third term = **8x ^{2}yz^{4}** = 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 x** ^{3}**y

**z**

^{3}

^{4}1**7 ) Find G.C.D of ax ^{2}, a^{2}x^{3}, a^{4}x**

Solution:

First term = **ax ^{2}** = a * x * x

Second term = **a ^{2}x^{3}** = a * a * x * x * x

Third term = **a ^{4}x** = a * a * a * a * x

Required LCM = a * a * a * a * x * x* x = a** ^{4}**x

^{3}1**8 ) Find G.C.D of 4xy ^{2}z, 6xyz^{2}, 10x^{2}yz**

Solution :

First term = **4xy ^{2}z** = 2 * 2 * x * y * y * z

Second term = **6xyz ^{2}** = 3 * 2 * 1 * x * y * z * z

Third term = **10x ^{2}yz** = 5 * 2 * 1 * x * x * y * z

Required LCM = 2 * 2 * 5 * 3 * 1 * x * x * y * y * z * z = 60 x** ^{2}**y

**z**

^{2}

^{2}1**9 ) Find G.C.D of 3ab, 9a ^{2}c, 12a^{2}c^{2}**

Solution :

First term = **3ab**

Second term = **9a ^{2}c** = 3 * 3 * a * a * c

Third term = **12a ^{2}c^{2}** = 3 * 2 * 2 * a * a * c * c

Required LCM = 3 * 3 * 2 * 2 * a * a * c * c = 36 a** ^{2}**c

^{2}**20 )** **Find G.C.D of 7y ^{3} z^{6} , 21y^{2} , 14z^{2}**

Solution :

First term = **7y ^{3} z^{6}** = 7 * 1 * y * y * y * z * z* z* z* z*z

Second term = **21y ^{2}** = 7 * 3 * y * y

Third term = **14z ^{2}** = 7 * 2 * 1 * z * z

Required LCM = 7 * 2 * 3 * 1 * y * y* y * z * z * z* z* z*z = 42 y** ^{3}**z

^{6}## Level – 02

1 ) Find HCF of x^{2} + xy, xz + yz and x^{2} + 2xy + y^{2}

First expression = x^{2} + xy = x ( x + y )

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

Third Expression = x^{2} + 2xy + y^{2}

= ( x + y )^{2}

= ( x + y ) ( x + y )

Required HCF = ( x + y )

2 ) Find HCF of 8 (x^{2} – 4 ), 12 (x^{3 }+ 8 ) and 36 ( x^{2} – 3x – 10 )

First Expression = 8 (x^{2} – 4 )

= 8 { (x)^{2} – ( 2 )^{2} }

= 2 * 2 * 2 * ( x + 2 ) ( x – 2 )

Second Expression = 12 (x^{3 }+ 8 )

= 12 { x^{3} + ( 2 )^{3} }

= 2 * 3 * 2 (x+2)(x^{2} – x.2 + 2^{2} )

= 2 * 3* 2 ( x + 2 ) ( x^{2} – 2x + 4 )

Third Expression = 36 ( x^{2} – 3x – 10 )

= 36{ x^{2} – ( 5 – 2 )x – 10 }

= 36{ x^{2} – 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 ( x^{3} – 16x ), ( 2x^{3} + 9x^{2} + 4x ) and ( 2x^{3} + x^{2} – 28x )

First expression = (x^{3} – 16x)

= x ( x^{2} – 16 )

= x (x^{2} – 4^{2} )

= x ( x + 4 ) ( x – 4 )

Second Expression = 2x^{3} + 9x^{2} + 4x

= x ( 2x^{2} + 9x + 4 )

= x {2x^{2} + ( 8 + 1 )x + 4 }

= x{ 2x^{2} + 8x + x + 4 }

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

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

Third Expression = 2x^{3} + x^{2} – 28x

= x ( x^{2} + x – 28 )

= x { x^{2} + ( 8 – 7 )x – 28 }

= x { x^{2} + 8x – 7x – 28 }

= x { x ( x + 8 ) – 7( x + 8 ) }

= x ( x + 8 ) ( x – 7 )

Required HCF = x

4 ) Find HCF of ( x^{3} – 3x^{2} y ) and ( x^{2} – 9y^{2} )

First Expression = ( x^{3} – 3x^{2} y ) = x^{2} ( x – 3y )

Second Expression = x^{2} – 9y^{2}

= x^{2} – ( 3y )^{2}

= ( x + 3y ) ( x – 3y )

Required HCF = ( x – 3y )

5 ) Find HCF of ( x^{2} – 1 ), ( x^{2} – 2x + 1 ) and ( x^{3} + x^{2} – 2x )

First Expression = x^{2} – 1

= x^{2} – 1^{2}

= ( x + 1 ) ( x – 1)

Second Expression = x^{2} – 2x + 1

= x ^{2} – 2.x.1 + 1^{2}

= ( x – 1 )^{2}

= ( x – 1 ) ( x – 1)

Third Expression = x^{3} + x^{2} – 2x

= x ( x^{2} + x – 2 )

= x { x^{2} + ( 2 – 1 )x – 2 }

= x { x^{2} + 2x – x – 2 }

= x { x ( x + 2 ) – 1 (x + 2 ) }

= x ( x + 2 ) ( x – 1 )

Required HCF = ( x – 1 )

6 ) Find the HCF of a^{2} – 1, ( a^{3} – 1 ) , ( a^{2} + a – 2 )

First Expression = a^{2} – 1

= a^{2} – 1^{2}

= ( a + 1 ) ( a – 1 )

Second Expression = a^{3} – 1

= a^{3} – 1^{3}

= ( a – 1 ) ( a^{2} + a . 1 + 1^{2} )

= ( a – 1 ) ( a^{2} + a + 1 )

Third Expression = a^{2} + a – 2

= a^{2} + ( 2 – 1 )a – 2

= a^{2} + 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 = 4a^{2}x ( 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}

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