### Highest common factor (HCF)

Let's take two expressions xy and yz. Here, xy is the product of x and y and yz is the product of y and z. x and y are factors of xy and y and z are the factors of yz. y is factor of both the expressions. So, y is called the highest common factor (HCF) of the expressions xy and yz.

$$\boxed {Note: HCF\:\text {divides each of the given expression exactly} }$$

#### To find the HCF

• find the factors of the given expressions.
• choose the common factors of the expressions.
• express the HCF in the form of product.
 Note: If there is no any common factor in the given expression, HCF = 1 as 1 is the factor of any number.

### Lowest Common Multiple (LCM)

Let's take two expressions a2b - ab2and a3b - ab3.Factorizing the expressions,

Here,

\begin{align*} first \: expression &= a^2 b - ab^2 \\ &= ab (a -b) \end{align*}

\begin{align*} second \: expression &= a^3 b - ab^3 \\ &=ab (a^2 - b^2) \\ &= ab(a + b) \: (a - b)\\ \end{align*}

common factors = ab(a - b)
Remaining factors = (a + b)
\begin{align*} LCM &= Common \: factors \times Remaining \: factor \\ &= ab(a - b) \times (a + b)\\ &= ab(a^2 - b^2).\end{align*}

• The H.C.F of two or more numbers is smaller than or equal to the smallest number of given numbers.
• The L.C.M of two or more numbers is greater than or equal to the greatest number of given numbers.
• The smallest number which is exactly divisible by x, y and z are L.C.M of x, y, z.
• If the H.C.F of the numbers a, b, c is K, then a, b, c can be written as multiples of K (Kx, Ky, Kz, where x, y, z are some numbers). K divides the numbers a, b, c, so the given numbers can be written as the multiples of K.
• If the H.C.F of the numbers a, b is K, then the numbers (a + b), (a -b) is also divisible by K. The numbers a and b can be written as the multiples of K.

Solution:

\begin{align*} \text {First Expression} &= x^3-x^2+x-1\\ &= x^2(x-1)+1(x-1)\\&= (x-1) (x^2+1)\\ \text {Second Expression}&= 2x^3-x^2+x-2\\ &= 2x^3-2-x^2+x\\&=2(x^3-1)-x(x-1)\\&=2(x-1)(x^2-x-1)-x(x-1)\\&=(x-1)[2(x^2+x+1)-x]\\&=(x-1)(2x^2+2x+2-x)\\&=(x-1)(2x^2+x+2)\\ H.C.F = (x-1) \: \:_{Ans.}\end{align*}

Solution:

\begin{align*} First\:Expression &= 1+4x+4x^2-16x^4 \\ &= 1^2+2.1.2x + (2x)^2 -16x^4\\&=(1+2x)^2 - (4x^2)^2\\ &=(1+2x+4x^2) (1+2x-4x^2)\\ Second\:Expression&= 1+2x-8x^3-16x^4\\&=1(1+2x)-8x^3(1+2x)\\ &= (1+2x)(1-8x^3)\\ &= (1+2x)[(1)^3-(2x)^3]\\&=(1+2x)(1-2x)(1+2x+4x^2)\\\\ \end{align*}

\begin{align*} H.C.F &= (1+2x+4x^2) \:\:_{Ans.}\\ L.C.M &= (1+2x+4x^2)(1+2x-4x^2)(1+2x)(1-2x) \:\:_{Ans.} \end{align*}

Solution:

\begin{align*} First\:Exercise&=x^3 - 3x^2-x+3\\ &= x^2 (x-3) -1(x-3)\\ &= (x-3)(x^2 - 1^2 )\\ &= (x-3)(x-1) (x+1) \\\\Second \: Expression &= x^3 +-x^2-9x+9\\ &= x^2 (x -1) -9(x-1)\\&= (x-1)(x^2-9)\\&= (x-1)\{(x)^2 -(3)^2\} \\&= (x-1)(x+3)(x-3)\\ \end{align*}

$$\\ \therefore\:L.C.M. = (x-1)(x-3)(x+1)(x+3)$$

Solution:

\begin{align*} First\:Expression &= 4x^4+19x^2y^2+49y^4 \\ &= (2x^2)^2+2.2x^2.7y^2+(7y^2)^2-9x^2y^2\\&= (2x^2+y^2)^2-(3xy)^2\\ & = (2x^2-7y^2+3xy)(2x^2+7y^2-3xy) \\ & = (2x^2+3xy+7y^2)(2x^2-3xy+7y^2 \\ Second \: Expression&= 4x^3+14xy^2-6x^2y\\ &= 2x(2x^2+7y^2-3xy) \\ &=2x(2x^2-3xy+7y^2) \\ \end{align*}

$$H.C.F = (2x^2-3xy+7y^2) \: \:_{Ans.}$$
$$L.C.M = 2x(2x^2-3xy+7y^2)(2x^2+3xy+7y^2)\:\:\:_{Ans.}$$

Solution:

\begin{align*} First\:Expression &=6x^2-x-1\\ &= 6x^3-3x+2x-1\\&=3x(2x-1)+1(2x-1)\\ &=(2x-1)(3x+1)\\ Second \: Expression&= 54x^4+2x\\ &= 2x(27x^3+1) \\ &=2x[(3x)^3+(1)^3]\\&=2x(3x+1)[(3x)^2-3x.1+1^2]\\&=2x(3x+1)(9x^2-3x+1) \\ \end{align*}

$$L.C.M = 2x(3x+1)(2x=1)(9x^2-3x+1)\:\:\:_{Ans.}$$

Solution:

\begin{align*} First\:Expression &=16x^4-4x^2-4x-1\\ &= 16x^4-(4x^2+4x+1)\\&=16x^4-[(2x)^2+2.2x.1+(1)^2]\\ &=(4x^2)^2-(2x+1)^2\\&=(4x^2+2x+1)(4x^2-2x-1)\\ Second \: Expression &=8x^3-1 \\ &=(2x)^3-(1)^3\\&=(2x-1)(4x^2+2x+1) \\ \end{align*}

$$H.C.M = (4x^2+2x+1)\:\:\:_{Ans.}$$

First Expression:

8a3 +b3

= (2a)3 + (b)3

= (2a + b) (4a2 - 2ab + b2)

Second Expression:

16a4 + 4a2b2 + b4

= (4a2)2 + 2 . 4a2b2+ (b2)2 - 4a2b2

= (4a2 + b2)2 - (2ab)2

= (4a2 + b2 + 2ab) (4a2 + b2 - 2ab)

= (4a2 + 2ab + b2) (4a2 - 2ab + b2)

∴ HCF = (4a2- 2ab + b2)Ans

∴ LCM =(4a2- 2ab + b2) (4a2+ 2ab + b2) (2a + b)Ans

First Expression:

a2 + 2ab + b2 - c2

= (a + b)2 - c2

= (a + b + c) (a + b - c)

Second expression:

b2 + 2bc + c2 - a2

= (b + c)2 - a2

= (b + c + a) (b + c - a)

= (a + b + c) (b + c - a)

Third Expression:

c2 + 2ca + a2 - b2

= (c + a)2 - b2

= (c + a + b) (c + a - b)

= (a + b + c) (c + a - b)

∴ HCF = (a + b + c)Ans

First Expression:

9x2 - 4y2 - 8yz - 4z2

= 9x2 - (4y2 + 8yz + 4z2)

= 9x2 - [(2y)2 + 2 . 2y . 2z + (2z)2]

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

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

Second Expression:

4z2 - 4y2 - 9x2 - 12xy

= 4z2- 9x2 - 12xy - 4y2

= 4z2 - (9x2 + 12xy + 4y2)

= 4z2- [(3x)2 + 2 . 3x . 2y + (2y)2]

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

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

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

Third Expression:

9x2 + 12xy + 4z2- 4y2

= (3x)2 + 2 . 3x . 2z + (2z)2 + (2z)2 - (2y)2

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

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

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

∴ HCF = (3x + 2y + 2z)Ans

First Expression:

m2 - n2

= (m + n) (m - n)

Second Expression:

m - n

= m - n

Third Expression:

m3 - n3

= (m - n) (m2+ mn + n2)

∴ HCF = (m - n)

∴ LCM = (m - n) (m + n) (m2 + mn + n2)Ans

First expression:

x3 - 9x

= x (x2 - 9)

= x (x2 - 32)

= x (x + 3) (x - 3)

Second expression:

x4 - 2x3 - 3x2

= x2 (x2 - 2x - 3)

= x2 [x2 - 3x + x - 3]

= x2 [x(x - 3) +1 (x - 3)]

= x2 (x - 3) (x + 1)

Third expression:

x3 - 27

= (x)3 - (3)3

= (x - 3) (x2 + 3x + 9)

∴ LCM = x2 (x - 3) (x + 1) (x2 + 3x + 9)Ans

First Expression:

3a4 - 8a3 + 4a2

= a2 (3a2 - 8a + 4)

= a2 (3a2 - 6a - 2a + 4)

= a2 [3a (a - 2) - 2(a - 2)]

= a2 (a - 2) (3a - 2)

Second Expression:

a3 - 4a

= a (a2 - 4)

= a (a2 - 22)

= a (a + 2) (a - 2)

Third Expression:

4a3 - 10a2 + 4a

= 2a(2a2 - 5a + 2)

= 2a (2a2 - 4a - a + 2)

= 2a [2a (a - 2) - 1 (a - 2)]

= 2a (a - 2) (2a - 1)

∴ HCF = a (a - 2)Ans

∴ LCM = 2a2 (a - 2) (a + 2) (2a - 1) (3a - 2)Ans

First Expression:

p4 + 6p3 + 8p2

= p2 (p2 + 6p + 8)

= p2 (p2 + 4p + 2p + 8)

= p2[p(p + 4) + 2(p + 4)]

= p2 (p + 4) (p + 2)

Second Expression:

p3+ 8p2 + 16p

= p (p2 + 8p + 16)

= p (p2 + 4p + 4p + 16)

= p [p(p + 4) + 4(p + 4)]

= p (p + 4) (p + 4)

Third Expression:

p5- 16p3

= p3 (p2 - 16)

= p3 (p2 - 42)

= p3 (p + 4) (p - 4)

∴ HCF = p(p + 4)Ans

∴ LCM = p3 (p + 4) (p + 2) (p + 4) (p - 4)Ans

First Expression:

a4 + a2 + 1

= (a2)2 + 2a2 . 1 + (1)2 -a2

= (a2)2 + 2a2 . 1 + (1)2 - a2

= (a2+ 1)2 - (a)2

= (a2 + 1 + a) (a2 + 1 - a)

= (a2 + a + 1) (a2- a + 1)

Second Expression:

a3- 1

= a3 - 13

= (a - 1) (a2+ a + 1)

Third Expression:

a3-a2 - a

=a(a2 - a - 1)

∴ LCM =a (a - 1) (a2 - a - 1) (a2+ a + 1) (a2- a + 1)Ans

First expression:

m3 - n3

= (m - n) (m2 + mn + n2)

Second expression:

m4 + m2n2 + n4

= (m2)2 + 2m2n2 + (n2)2 - m2n2

= (m2 + n2) - (mn)2

= (m2 + n2 + mn) (m2 + n2 - mn)

= (m2 + mn + n2) (m2 - mn + n2)

Third expression:

m3 + m2n + mn2

= m(m2 + mn + n2)

∴ HCF = (m2 + mn + n2)Ans

0%

;i:1;s:10:
;i:3;s:10:
;i:2;s:22:

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

(a+3)
(a-2)
(a-3)
(a+2)

(a+1b)
(a-1b)
(a+2b)
(a-2b)

x-2
x+2
x-1
x+1

x+1

x-3

x+3

x-1

a-2

a+1

a-1

a+2

a-3
a-2
a+1
a-1
• ### m2-7m+12, m3--2m2-2m-3

(m+3)(m-4)(m2+m-1)

(m+3)(m-4)(m2+m+1)

(m+3)(m-4)(m2-m+1)

(m-3)(m-4)(m2+m+1)

• ### m2+3m-4,m3-2m2-2m+3

(m-1)(m+4)(m2-m-3)
(m-1)(m+4)(m2+m-3)
(m-1)(m+4)(m2-m+3)
(m-1)(m-4)(m2-m-3)
• ### t2+5t+6,t2-4,t2+t-6

(t+2)(t+2)(t+3)
(t+2)(t-2)(t-3)
(t+2)(t-2)(t+3)
(t-2)(t-2)(t+3)
• ### x3+5x2+6x,2x2+14x+24,x2+6x+8

2x(x-2)(x+3)(x+4)
2x(x+2)(x+3)(x+4)
2x(x+2)(x+3)(x-4)
2x(x+2)(x-3)(x+4)

a-1
a+2
a+1
a-2

a+2
a+1
a-2
a-1

a(b2-c2)
a(b3+c2)
a(b3-c2)
a(b2+c2)

b(a-c)3
b(a+b)3
a(a-b)3
b(a-b)3