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Consider two expressions 6a4 b2 c3 and 9 a3 b3 c Now 6a4 b2 c3 = 2 ´ 3 ´ a4 ´ b2 ´ c3 9 a3 b3 c = 32 ´ a3´ b3 ´ c The common factors of these two expressions are : 3, 3a, 3a2, 3a3, 3a3 b , 3a2 b2, 3a3 b2 c. Each one is a factor of the last one, viz. 3a3 b2 c. Hence, 3a3 b2 c is called Highest Common Factor or H. C. F ( also called greatest common divisor i.e. G. C. D. ) of 6a4 b2 c3 and 9 a3 b3 c. Notice that 3a3 b2 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 24x2y and 16xy3 Solution: 24x2y = 23 ´ 3 ´ x2 ´ y 16xy3 = 24 ´ x ´ y3 Therefore, the H. C. F. is 23xy = 8xy Example Find the G. C. D. of 12x2 y z2 , 18 x y2z , 24x2 y z Solution: 12x2 y z2 = 22 ´ 3 ´ x2 ´ y ´ z2 , 18 x y2z = 2 ´ 32 ´ x ´ y2 ´ z and 24x2 y z = 23 ´ 3 ´ x2 ´ 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 2x2 y2 , 3x2 yz and 18 xy2 are common multiple 36 x2 y2 z. We, therefore, call 36 x2 yz , the Lowest Common Multiple ( L. C. M.) of 2x2 y2, x2 yz and 36x2 y2 z. The L. C. M. may be obtained thus, 2x2 y2 = 2 ´ x2 ´ y2 , 3x2 yz = 3 ´ x2 ´ y ´ z and 36 x2 y2 z = 22 ´ 32 ´ x2 ´ y2 ´ z Choose the highest powers of each number or letter which occurs in any of these expression. Hence 22 ´ 32 ´ x2 ´ y2 ´ z or 36 x2 y2 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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