Note on H.C.F and L.C.M

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Highest Common Factor (HCF)

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The highest common factor (HCF) of the algebraic expression is the largest number that divides evenly into both numbers. It can be said as largest of all common factors.

For example, HCF of 6x3y2 and 10x5y4 is 2x3y2 since

HCF of 6 and 10 is 2

HCF of x3 and x5 is x3

and HCF of y2 and y4 is y2

To find the HCF of compound expressions, first of all, resolve each expression into factors and then find HCF.

Example:

Find the HCF of 3x2- 6x and x2+ x - 6

Solution:

1st expression = 3x2- 6x

= 3x(x - 2)

2nd expression = x2+ x - 6

= x2+ 3x - 2x - 6

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

= (x + 3)(x - 2)

∴ = x - 2

Lowest Common Multiple (LCM)

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The lowest common multiple(LCM) is found by multiplying all the factors which appear on either list. LCM of any number is the smallest whole number which is multiple of both.

For example, LCM of 6x3yand 10x5yis 30 x5ysince

LCM of 6 and 10 is 30, LCM of xand xand LCM of yand yis y4.

To find the LCM of compound expressions, proceed as in the case of HCF and then find LCM.

Example

Find the LCM of 3x2- 6x

1st expression = 3x2- 6x

= 3x(x - 2)

2nd expression = x2+ x - 6

= x2+ 3x - 2x - 6

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

= (x + 3)(x - 2)

LCM = 3x(x - 2)(x + 3)

  • H.C.F is the largest number that divides every into both numbers.
  • H.C.F is useful when simplifying fraction.
  • L.C.M is the smallest number that is a common multiple of two or more numbers.
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Very Short Questions

Solution:

4x2y and xy2

Here, first expression = 4x2y = 4 × x × x × y

The second expression = xy2= x × y × y

Taking common of both expression = xy.

∴ H.C.F. = xy

Solution:

Here first expression = 9x2y3= 3 × 3 × x × x × y × y × y

Second expression = 15xy2= 3 × 5 × x × y × y

Taking common from both expression

= 3 × x × y × y

∴ H.C.F. = 3xy2

Solution:

Here, first expression = a2bc = a × a × b × c

Second expression=b2ac= b × b × a × c

Third expression= b2ac = b × b ×a× c

Taking common of the three expression=a × b × c

∴ H.C.F = abc

Solution:

Here given x2-4 and 3x+6

First expression = x2-4 = x2-22= (x-2)(x+2)

Second expression = 3x+6 = 3(x+2)

∴ H.C.F = x+2

Solution:

Given, x2-y2 and xy - y2

First expression = x2-y2= (x+y) (x-y)

Second expression = xy - y2= y(x-y)

Taking common from both expression = x-y

∴ H.C.F = x-y

Solution:

Here given,3x2-6x and x2+x-6

1st expression= 3x2-6x

= 3x(x-2)

2nd expression= x2+x-6

=x2+3x-2x-6

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

=(x+3)(x-2)

∴H.C.F = x-2

Solution:

Here given,3a+b and 15a +5 b

1st expression=3a+b

2nd expression=15a+5b=5(3a+b)

Taking common from both expression =3a+b

∴H.C.F= 3a+b

Solution:

Here given, 3x2-6x and x2+x-6

1st expression = 3x2-6x

= 3x(x-2)

2nd expression = x2 + x-6

= x2+ 3x - 2x-6

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

= (x+3)(x-2)

\(\therefore\) LCM = 3x(x-2)(x+3)

Solution:

Here given, 2x and 4

1st expression = 2x = 2 × x

2nd expression = 4 = 2×2

LCM= 2×2× x = 4x

 

 

Solution:

2 18,24
2 9,12
3 3,6
2 1,2
1,1

Lowest common multiple (L.C.M) of 18 and 24 = 2 × 2 × 3 × 2 = 24.

0%
  • Find the HCF of:

    576

    288
    150
    250
    123
  • Find the HCF of:

     625

    25
    20
    30
    35
  • Find the HCF of:

     496

    133
    149
    130
    124
  • Find the HCF of:

     1000

    130
    125
    135
    111
  • Find the HCF of:

     676

    46
    26
    56
    36
  • Finf the LCM of:

    1440

    2888
    2991
    1255
    2880
  • Find the LCM of:

     325

    8125
    8145
    8225
    8325
  • The product of two numbers is 120.  If their H.C.F is 6 what is their L.C.M.

    30
    35
    10
    20
  • Find the L.C.M.  of the following by listing their multiples.

    5, 10, 15

    25
    20
    30
    35
  • Find the L.C.M  of the following by listing their multiples.

     4, 10, 12

    10
    25
    60
    35
  • Find the L.C.M.  of the following by finding common prime factors.

    60, 75, 120

    600
    150
    268
    497
  • Find the L.C.M.  of the following by finding common prime factors.

     10, 15, 25

    140
    130
    150
    120
  • Find the L.C.M.  by division method.

    70, 110, 150

    11450
    11550
    11554
    11540
  • Find the L.C.M.  by division method.

    21, 49, 63

    541
    451
    242
    441
  • Find the lowest number which leaves 4 as remainder when divide by 9 and 12.

    44
    20
    25
    40
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