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28 Sep 2011 13:35:19 IST

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Number Theory

Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Algebra

K(N) denotes the number of ways in which N can be expressed as a difference of 2 perfect squares. Which of the following denotes the maximum number of ways:-

(a) K(110)

(b)K(105)

(c)K(216)

(d)K(384)

Comments (5)

Pritam Banerjee

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28 Sep 2011 17:32:48 IST

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Let the two numbers be A and B with A > B

Now K(N) = A² - B² = (A-B)(A+B) . Now we need to know the pair of factors of N which when multiplied gives N . Applying the symmetrical property of factors we need to look for factors upto sqtr(N).

Also both the factors must be either both even or both odd (as their sums must be even for integer values of A and B).

From the given options we get the pair of A+B and A-B

K(110) = (110, 1) , (55 , 2) ,( 5, 22), (11,10) .However for each pair the 2nd condition is not satisfied. So number of such pairs (A,B) = 0

Similarly for K(105) we get valid numbers (A,B) as (53,52),(19,16) and (13,8) = three

For K(216) we get four such pairs (satisfyingboth conditions)

K(384) we get 6 .

Hence the answer is K(384)

Manasi

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14 Feb 2012 15:06:09 IST

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beautifully solved pritam!

srinivasan

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14 Feb 2012 15:59:02 IST

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a² - b² = n(a + b)(a - b) = nEither both (a + b) and (a - b) are odd or both are even. So n is either an odd number or a multiple of 4.Number of ways in which an odd number n can be written as difference of two perfect squares = No of ways in which n can be written as product of two numbersNumber of ways in which n, a multiple of 4, can be written as difference of two mperfect squares = No of ways in which (n/4) can be written as product of two numbers.a) 110 - no solutionb) 105 - 3*5*7 - 4 waysc) 216 - 4*2*27 - 4 waysd) 384 - 4*32*3 - 6 wayse) 450 - no solution i think its 105 option( b )

srinivasan

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14 Feb 2012 16:00:07 IST

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I am getting 105 degrees;Approach: observe that P is the incenter of triangle BDC where angles of the triangle BDC are 20,130,30 degrees....... now its easy to calculate.

srinivasan

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14 Feb 2012 16:00:23 IST

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number of ways in which a number can be written as product of two numbers is = ( number of factors /2)105=3*5*7 so 8 factorsbasically 1*3*5*7*15*21*35*105105=1*105=3*35=5*21=7*15 so its 105

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