## Note on Highest Common Factor and Lowest Common Factor Multiple

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### 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.

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

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

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

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

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

x+1

x-1

x-3

x+3

a-1

a+1

a+2

a-2

a-2
a-3
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-2
a+1
a+2
a-1

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

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

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