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 6x^{3}y^{2} and 10x^{5}y^{4} is 2x^{3}y^{2} since
HCF of 6 and 10 is 2
HCF of x^{3} and x^{5} is x^{3}
and HCF of y^{2} and y^{4} is y^{2}
To find the HCF of compound expressions, first of all, resolve each expression into factors and then find HCF.
Example:
Find the HCF of 3x^{2}- 6x and x^{2}+ x - 6
Solution:
1st expression = 3x^{2}- 6x
= 3x(x - 2)
2nd expression = x^{2}+ x - 6
= x^{2}+ 3x - 2x - 6
= x(x + 3) - 2(x + 3)
= (x + 3)(x - 2)
∴ = x - 2
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 6x^{3}y^{2 }and 10x^{5}y^{4 }is 30 x^{5}y^{4 }since
LCM of 6 and 10 is 30, LCM of x^{3 }and x^{5 }and LCM of y^{2 }and y^{4 }is y^{4}.
To find the LCM of compound expressions, proceed as in the case of HCF and then find LCM.
Example
Find the LCM of 3x^{2}- 6x
1^{st} expression = 3x^{2}- 6x
= 3x(x - 2)
2^{nd} expression = x^{2}+ x - 6
= x^{2}+ 3x - 2x - 6
= x(x + 3) - 2(x + 3)
= (x + 3)(x - 2)
LCM = 3x(x - 2)(x + 3)
Solution:
4x^{2}y and xy^{2}
Here, first expression = 4x^{2}y = 4 × x × x × y
The second expression = xy^{2}= x × y × y
Taking common of both expression = xy.
∴ H.C.F. = xy
Solution:
Here first expression = 9x^{2}y^{3}= 3 × 3 × x × x × y × y × y
Second expression = 15xy^{2}= 3 × 5 × x × y × y
Taking common from both expression
= 3 × x × y × y
∴ H.C.F. = 3xy^{2}
Solution:
Here, first expression = a^{2}bc = a × a × b × c
Second expression=b^{2}ac= b × b × a × c
Third expression= b^{2}ac = b × b ×a× c
Taking common of the three expression=a × b × c
∴ H.C.F = abc
Solution:
Here given x^{2}-4 and 3x+6
First expression = x^{2}-4 = x^{2}-2^{2}= (x-2)(x+2)
Second expression = 3x+6 = 3(x+2)
∴ H.C.F = x+2
Solution:
Given, x^{2}-y^{2} and xy - y^{2}
First expression = x^{2}-y^{2}= (x+y) (x-y)
Second expression = xy - y^{2}= y(x-y)
Taking common from both expression = x-y
∴ H.C.F = x-y
Solution:
Here given,3x^{2}-6x and x^{2}+x-6
1st expression= 3x^{2}-6x
= 3x(x-2)
2nd expression= x^{2}+x-6
=x^{2}+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, 3x^{2}-6x and x^{2}+x-6
1st expression = 3x^{2}-6x
= 3x(x-2)
2nd expression = x^{2 }+ x-6
= x^{2}+ 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.
Find the HCF of:
576
Find the HCF of:
625
Find the HCF of:
496
Find the HCF of:
1000
Find the HCF of:
676
Finf the LCM of:
1440
Find the LCM of:
325
The product of two numbers is 120. If their H.C.F is 6 what is their L.C.M.
Find the L.C.M. of the following by listing their multiples.
5, 10, 15
Find the L.C.M of the following by listing their multiples.
4, 10, 12
Find the L.C.M. of the following by finding common prime factors.
60, 75, 120
Find the L.C.M. of the following by finding common prime factors.
10, 15, 25
Find the L.C.M. by division method.
70, 110, 150
Find the L.C.M. by division method.
21, 49, 63
Find the lowest number which leaves 4 as remainder when divide by 9 and 12.
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