3.3 H. C. F AND L. C. M.
Consider two expressions 6a^{4} b^{2} c^{3} and 9 a^{3} b^{3} c
Now 6a^{4} b^{2} c^{3} = 2 ´
3 ´ a^{4}
´ b^{2}
´ c^{3}
9 a^{3} b^{3} c = 32 ´
a^{3}´
b^{3} ´ c
The common factors of these two expressions are :
3, 3a, 3a^{2}, 3a^{3}, 3a^{3} b , 3a^{2} b^{2}, 3a^{3} b^{2} c.
Each one is a factor of the last one, viz. 3a^{3} b^{2} c.
Hence, 3a^{3} b^{2} c is called Highest Common Factor or H. C. F ( also called greatest common divisor i.e. G. C. D. ) of 6a^{4} b^{2} c^{3} and 9 a^{3} b^{3} c.
Notice that 3a^{3} b^{2} c is obtained by taking the lowest power of each of the number or letter which is common to both. Also 3 of H. C. F. is G. C. D. of 6 and 9.
Example Find the H. C. F. of 24x^{2}y
and 16xy3
Solution: 24x^{2}y = 23 ´
3 ´ x^{2}
´ y
16xy^{3} =
24 ´ x ´
y^{3}
Therefore, the H. C. F. is 23xy = 8xy
Example Find the G. C. D. of 12x^{2} y z^{2}
, 18 x y^{2}z , 24x^{2} y z
Solution: 12x^{2} y z^{2}
= 22 ´ 3 ´
x^{2} ´
y ´ z^{2} ,
18 x y^{2}z
= 2 ´ 32 ´
x ´
y^{2} ´ z and
24x^{2} y z
= 23 ´ 3 ´
x^{2} ´
y ´ z
\
The G. C. D. is 2 ´
3 ´ x
´ y ´
z = 6 xyz
Multiples :
A common multiple of two or more expressions is an expression each divided exactly. Thus 2x^{2} y^{2} , 3x^{2} yz and 18 xy^{2} are common multiple 36 x^{2} y^{2} z. We, therefore, call 36 x^{2} yz , the Lowest Common Multiple ( L. C. M.) of 2x^{2} y^{2}, x^{2} yz and 36x^{2} y^{2} z.
The L. C. M. may be obtained thus,
2x^{2} y^{2} = 2 ´
x^{2} ´ y^{2} ,
3x^{2} yz = 3 ´
x^{2} ´
y ´ z and
36 x^{2} y^{2} z = 22
´ 32 ´
x^{2} ´
y^{2} ´ z
Choose the highest powers of each number or letter which occurs in any of these expression.
Hence 22 ´
32 ´ x^{2}
´ y^{2}
´ z or 36 x^{2} y^{2}
z is the L. C. M.
Note that 36 is L. C. M. of 2, 3 and 36.
H.C.F. and L.C.M. of compound expressions:
1) Factorise each compound expression if possible.
2) H.C.F. is obtained by taking the lowest power of each common factor to all the
expressions.
3) L.C.M. is obtained by taking the highest power of every factor that occurs in any
of the given expressions.
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