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HCF and LCM : Common Factor
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Common Factor

When two or more numbers have the same factor, that factor is called a common factor.

Lowest Common Multiple (L.C.M.).

The Lowest Common Multiple (L.C.M) is the smallest number that is a common multiple of two or more numbers.

Example: Find the LCM of
f(x) = 13(x - 1)(x - 2)2, g(x) = 7(x - 2)2(x + 3)2 and h(x)= (x - 1)2 (x + 3).

Solution:
The irreducible factors occurring in any of f(x), g(x) and h(x) are:
7, 13, (x - 1), (x - 2) and (x + 3).
The greatest exponents of these are 1, 1, 2, 2, and 2 respectively.

Hence, the LCM = 7 * 13 * (x - 1)2 * (x - 2)2 * (x + 3)2
= 91 (x - 1)2 (x - 2)2 (x + 3)2

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Highest Common Factor (H.C.F).

The Highest Common Factor (H.C.F) of two (or more) numbers is the largest number that divides evenly into both numbers, i.e. H.C.F is the largest of all the common factors.

Example: Find the value of b for which the HCF of x2 + 2bx + 3b + 3 and x2 + x - 5b is x + 5.

Solution:
Since x + 5 is the HCF of the given polynomials, this means -5 is a zero of both the given polynomials,

or    (-5)2 + 2b (-5) + 3b + 3 = 0
and  (-5)2 + (-5) - 5b = 0
or    28 - 7b = 0
and  20 - 5b = 0

Both the above equations give
              b = 4
Hence the required value of bis 4.

Verification: Substituting b = 4 in the given polynomials, we get
x2 + 8x + 15 and x2 + x - 20
or (x+ 3) (x + 5) and (x - 4) (x + 5)
Thus x + 5 is the HCF of the given polynomials when b= 4.
 
   
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