Class 8 Maths Solution | Lesson 10 HCF and LCM | Curriculum Development Centre (CDC)

Highest Common Factor (HCF) of Algebraic Expressions

Exercise 10.2.1

1. Find the highest common factor (HCF):

(a) 4x²y and xy²

Solution:

Here,

First Expression = 4x²y

= 2 × 2 × x × x × y

Second Expression = xy²

= x × y × y

∴ HCF = x × y = xy

(b) 25x²y³ and 15xy²

Solution:

Here,

First Expression = 25x²y³

= 5 × 5 × x × x × y × y × y

Second Expression = 15xy²

= 5 × 3 × x × y × y

∴ HCF = 5 × x × y × y = 5xy²

(c) a²bc, b²ac and c²ab

Solution:

Here,

First Expression = a²bc

= a × a × b × c

Second Expression = b²ac

= b × b × a × c

Third Expression = c²ab

= c × c × a × b

∴ HCF = a × b × c = abc

(d) x² – 4 and 3x + 6

Solution:

Here,

First Expression = x² – 4

= (x + 2)(x – 2)

Second Expression = 3x + 6

= 3(x + 2)

∴ HCF = (x + 2)

(e) x² – y² and xy – y²

Solution:

Here,

First Expression = x² – y²

= (x + y)(x – y)

Second Expression = xy – y²

= y(x – y)

∴ HCF = (x – y)

(f) p²q – q²p and 2p² – 2pq

Solution:

Here,

First Expression = p²q – q²p

= pq(p – q)

Second Expression = 2p² – 2pq

= 2p(p – q)

∴ HCF = p(p – q)

(g) 3a + b and 15a + 5b

Solution:

Here,

First Expression = 3a + b

= 3a + b

Second Expression = 15a + 5b

= 5(3a + b)

∴ HCF = (3a + b)

(h) x² + 4x + 4 and x² – 4

Solution:

Here,

First Expression = x² + 4x + 4

= x² + 2x + 2x + 4

= x (x + 2) + 2 (x + 2)

= (x + 2)(x + 2)

Second Expression = x² – 4

= (x + 2)(x – 2)

∴ HCF = (x + 2)

(i) x² – 11x + 30 and x² – 36

Solution:

Here,

First Expression = x² – 11x + 30

= x² – 5x - 6x + 30

= x (x – 5) - 6 (x - 5)

= (x – 5)(x – 6)

Second Expression = x² – 36

= (x – 6)(x + 6)

∴ HCF = (x – 6)

(j) x² – 9 and x² – 6x + 9

Solution:

Here,

First Expression = x² – 9

= (x – 3)(x + 3)

Second Expression = x² – 6x + 9

= x² – 3x - 3x + 9

= x (x – 3) - 3 (x - 3)

= (x – 3)(x – 3)

∴ HCF = (x – 3)

(k) x² + 16x + 60 and x² + 20x + 100

Solution:

Here,

First Expression = x² + 16x + 60

= x² + 10x + 6x + 60

= x(x + 10) + 6 (x + 10)

= (x + 6)(x + 10)

Second Expression = x² + 20x + 100

= (x + 10)(x + 10)

∴ HCF = (x + 10)

(l) a² + 5a + 6 and a² + a – 6

Solution:

Here,

First Expression = a² + 5a + 6

= a² + 3a + 2a + 6

= a (a + 3) + 2 (a + 3)

= (a + 3)(a + 2)

Second Expression = a² + a – 6

= (a + 3)(a – 2)

∴ HCF = (a + 3)

(m) x² – 11x + 10 and x³ – x

Solution:

Here,

First Expression = x² – 11x + 10

= x² – 10x - x + 10

= x (x – 10) - 1 (x - 10)

= (x – 10)(x – 1)

Second Expression = x³ – x

= x(x – 1)(x + 1)

∴ HCF = (x – 1)

(n) a² – 2ab + b² and a⁴ – b⁴

Solution:

Here,

First Expression = a² – 2ab + b²

= (a – b)(a – b)

Second Expression = a⁴ – b⁴

= (a² + b²)(a + b)(a – b)

∴ HCF = (a – b)

(o) x² – x²y² and y² – y⁴

Solution:

Here,

First Expression = x² – x²y²

= x²(1 – y²)

= x²(1 – y)(1 + y)

Second Expression = y² – y⁴

= y²(1 – y²)

= y²(1 – y)(1 + y)

∴ HCF = (1 – y)(1 + y)

(p) x² – a² and x² – 2ax + a²y

Solution:

Here,

First Expression = x² – a²

= (x – a)(x + a)

Second Expression = x² – 2ax + a²

= (x – a)(x – a)

∴ HCF = (x – a)

(q) x² – y² and x²y – y²x

Solution:

Here,

First Expression = x² – y²

= (x – y)(x + y)

Second Expression = x²y – y²x

= y(x² – yx)

= y(x – y)(x)

∴ HCF = (x – y)

(r) a³ – ab² and a²b + ab²

Solution:

Here,

First Expression = a³ – ab²

= a(a² – b²)

= a(a – b)(a + b)

Second Expression = a²b + ab²

= ab(a + b)

∴ HCF = a(a + b)

(s) x² + 5x + 6 and x² + x – 6

Solution:

Here,

First Expression = x² + 5x + 6

= (x + 3)(x + 2)

Second Expression = x² + x – 6

= (x + 3)(x – 2)

∴ HCF = (x + 3)

(t) a² + 2a – 3 and a² – 3a + 2

Solution:

Here,

First Expression = a² + 2a – 3

= (a + 3)(a – 1)

Second Expression = a² – 3a + 2

= (a – 1)(a – 2)

∴ HCF = (a – 1)

(u) x² + 7x + 10 and x² – x – 6

Solution:

Here,

First Expression = x² + 7x + 10

= (x + 5)(x + 2)

Second Expression = x² – x – 6

= (x – 3)(x + 2)

∴ HCF = (x + 2)

(v) x² – 7x + 12 and 3x² – 27

Solution:

Here,

First Expression = x² – 7x + 12

= (x – 4)(x – 3)

Second Expression = 3x² – 27

= 3(x – 3)(x + 3)

∴ HCF = (x – 3)

(w) a² – 3a + 2 and 2a² – 9a + 10

Solution:

Here,

First Expression = a² – 3a + 2

= (a – 2)(a – 1)

Second Expression = 2a² – 9a + 10

= (2a – 5)(a – 2)

∴ HCF = (a – 2)

(x) a² + 5a + 6 and a² – 4

Solution:

Here,

First Expression = a² + 5a + 6

= (a + 3)(a + 2)

Second Expression = a² – 4

= (a – 2)(a + 2)

∴ HCF = (a + 2)

Lowest Common Factor (LCM) of Algebraic Expressions

Exercise 10.3.1

2. Find the lowest common multiple (LCM):

(a) 2x and 4

Solution:

Here,

First Expression = 2x

= 2 × x

Second Expression = 4

= 2 × 2

∴ LCM = 2 × 2 × x = 4x

(b) 3x²y and 6xy²

Solution:

Here,

First Expression = 3x²y

= 3 × x × x × y

Second Expression = 6xy²

= 2 × 3 × x × y × y

∴ LCM = 2 × 3 × x × x × y × y = 6x²y²

(c) 5xy and 10y²

Solution:

Here,

First Expression = 5xy

= 5 × x × y

Second Expression = 10y²

= 2 × 5 × y × y

∴ LCM = 2 × 5 × x × y × y = 10xy²

(d) 6a²b and 6ab²

Solution:

Here,

First Expression = 6a²b

= 2 × 3 × a × a × b

Second Expression = 6ab²

= 2 × 3 × a × b × b

∴ LCM = 2 × 3 × a × a × b × b = 6a²b²

(e) 2a and 2a + 4

Solution:

Here,

First Expression = 2a

Second Expression = 2a + 4

= 2(a + 2)

∴ LCM = 2a(a + 2)

(f) 3x² − 3 and x² − 1

Solution:

Here,

First Expression = 3x² − 3

= 3(x² − 1)

Second Expression = x² − 1

∴ LCM = 3(x² − 1) = 3(x − 1)(x + 1)

(g) x + y and x² + xy

Solution:

Here,

First Expression = x + y

Second Expression = x² + xy

= x(x + y)

∴ LCM = x(x + y)

(h) x² + 4x + 4 and x² + 2x

Solution:

Here,

First Expression = x² + 4x + 4

= x² + 2x + 2x + 4

= x (x + 2) + 2 (x + 2)

= (x + 2)(x + 2)

Second Expression = x² + 2x

= x(x + 2)

∴ LCM = x(x + 2)(x + 2)

(i) 5x − 20 and x² − 16

Solution:

Here,

First Expression = 5x − 20

= 5(x − 4)

Second Expression = x² − 16

= (x − 4)(x + 4)

∴ LCM = 5(x − 4)(x + 4)

(j) p² − pq and pq − q²

Solution:

Here,

First Expression = p² − pq

= p(p − q)

Second Expression = pq − q²

= q(p − q)

∴ LCM = pq(p − q)

(k) 3x³ + 15x² and 2x³ − 50x

Solution:

Here,

First Expression = 3x³ + 15x²

= 3x²(x + 5)

Second Expression = 2x³ − 50x

= 2x(x² − 25)

= 2x(x − 5)(x + 5)

∴ LCM = 6x²(x − 5)(x + 5)

(l) x³ − 4x and x² + 7x + 10

Solution:

Here,

First Expression = x³ − 4x

= x(x² − 4)

= x(x − 2)(x + 2)

Second Expression = x² + 7x + 10

= x² + 5x + 2x + 10

= x (x + 5) + 2 (x + 5)

= (x + 5)(x + 2)

∴ LCM = x(x − 2)(x + 2)(x + 5)

(m) 3x² + 7x + 2 and 2x² + 3x − 2

Solution:

Here,

First Expression = 3x² + 7x + 2

= 3x² + 6x + x + 2

= 3x (x + 2) + 1 (x + 2)

= (x + 2) (3x + 1)

Second Expression = 2x² + 3x − 2

= 2x² + 4x - x − 2

= 2x (x + 2) - 1 (x + 2)

= (x + 2) (2x − 1)

∴ LCM = (3x + 1)(2x − 1)(x + 2)

(n) y² + 2y − 48 and y² − 9y + 18

Solution:

Here,

First Expression = y² + 2y − 48

= y² + 2y − 48

= y² + (8 - 6)y − 48

= y² + 8y - 6y − 48

= y(y + 8) - 6(y + 8)

= (y + 8)(y − 6)

Second Expression = y² − 9y + 18

= (y − 6)(y − 3)

∴ LCM = (y + 8)(y − 6)(y − 3)

(o) a² + 4ab + 4b² and a² − 4b²

Solution:

Here,

First Expression = a² + 4ab + 4b²

= (a + 2b)(a + 2b)

Second Expression = a² − 4b²

= (a − 2b)(a + 2b)

∴ LCM = (a + 2b)(a + 2b)(a − 2b)

(p) 9x² − 24xy + 16y² and 3x² − xy − 4y²

Solution:

Here,

First Expression = 9x² − 24xy + 16y²

= (3x − 4y)(3x − 4y)

Second Expression = 3x² − xy − 4y²

= 3x² − 4xy+ 3xy − 4y²

= x(3x − 4y) + y(3x − 4y)

= (3x - 4y)(x + y)

∴ LCM = (3x − 4y)(3x - 4y)(x − y)

(q) a² − 1 and a² + a − 2

Solution:

Here,

First Expression = a² − 1

= (a − 1)(a + 1)

Second Expression = a² + a − 2

= (a − 1)(a + 2)

∴ LCM = (a − 1)(a + 1)(a + 2)

(r) x² − 4 and x² + 3x + 2

Solution:

Here,

First Expression = x² − 4

= (x − 2)(x + 2)

Second Expression = x² + 3x + 2

= (x + 1)(x + 2)

∴ LCM = (x − 2)(x + 2)(x + 1)

(s) x² + x − 6 and x² + 2x − 3

Solution:

Here,

First Expression = x² + x − 6

= (x + 3)(x − 2)

Second Expression = x² + 2x − 3

= (x + 3)(x − 1)

∴ LCM = (x + 3)(x − 2)(x − 1)

(t) 4x² + 12xy + 9y² and 4x² − 12xy + 9y²

Solution:

Here,

First Expression = 4x² + 12xy + 9y²

= (2x + 3y)(2x + 3y)

Second Expression = 4x² − 12xy + 9y²

= (2x − 3y)(2x − 3y)

∴ LCM = (2x + 3y)(2x + 3y)(2x − 3y)(2x − 3y)

(u) 6x³ + 5x² − 6x and 3x³ − 5x² + 2x

Solution:

Here,

First Expression = 6x³ + 5x² − 6x

= x(3x − 2)(2x + 3)

Second Expression = 3x³ − 5x² + 2x

= x(3x − 2)(x − 1)

∴ LCM = x(3x − 2)(2x + 3)(x − 1)

(v) x³ − x² − 42x and x⁴ + 4x³ − 12x²

Solution:

Here,

First Expression = x³ − x² − 42x

= x (x² − x − 42)

= x {(x² - 7x + 6x − 42)}

= x {(x(x - 7) + 6 (x − 7)}

= x(x − 7)(x + 6)

Second Expression = x⁴ + 4x³ − 12x²

= x² {(x² + 4x − 12)}

= x² {(x² + 6x - 2x − 12)}

= x² {(x (x + 6) - 2 (x + 6)}

= x² { (x + 6) (x − 2)

∴ LCM = x²(x − 7)(x + 6)(x − 2)