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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)

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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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Very Short Questions

0%
  •  x3- x2-x+1,x4-2x3+2x-1

    ;i:1;s:10:
    ;i:3;s:10:
    ;i:2;s:22:
  • X2+3x-4,x3-2x2-x+2

    (x+2)
    (x-1)
    (x-2)
    (x+1)
  • (a+3)2-9a-27,a5-13a3+36a

    (a-3)
    (a-2)
    (a+3)
    (a+2)
  • a2-ab-2b2,a3-a2b-4ab2+4b3

    (a+1b)
    (a+2b)
    (a-2b)
    (a-1b)
  • x2+5x+6,x2+3x+2,x2-4

    x+2
    x+1
    x-2
    x-1
  • x2+7x+12,x2+4x+3,x2-9

    x+1


    x-3


    x+3


    x-1


  • 3a2-8a+4,2a2-5a+2,a4-8a

    a-2


    a-1


    a+1


    a+2


  • 2a2-5a+2,3a2-8a+4,a4-8a

    a-3
    a+1
    a-1
    a-2
  • 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)
  • The product of two expressions is (a+1)3(a-1) and LCM is (a+1)2(a-1).Find their HCF.

    a-1
    a+2
    a-2
    a+1
  • The product of two expressions is (a+1)3(a-1) and LCM is (a+1)2(a-1).Find their HCF.

    a+1
    a-1
    a+2
    a-2
  • The product of two expressions is  a(b+c)2(b-c) and HCF is (b+c).Find the LCM.

    a(b3-c2)
    a(b3+c2)
    a(b2+c2)
    a(b2-c2)
  • The HCF and LCM of two expressions are (a-b)and b(a-b)2.Find the product of the expressions.

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