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

• Note
• Things to remember
• Exercise
• Quiz

#### Highest Common Factor (HCF)

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)

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.
.

### 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.

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