Latest MBA Aptitude Question SOLUTION: K(N) denotes the number of ways in which N can be expressed as a difference of two perfect squares. which of the following is maximum?

1) K(110)
2) K(105)
3) K

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K(N) denotes the number of ways in which N can be expressed as a difference of two perfect squares. which of the following is maximum?

1) K(110)
2) K(105)
3) K(216)
4) K(384)

Asked In MBA (11 years ago)

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Gugulothu  (6 years ago)

Suppose n is a product, PQ, where P≥Q.
Then we want integers a and b so that: .a^2−b^2=PQ
so (a+b)(a-b)=PQ
We assume that a+b=P
a-b=Q
And by solving we get a=(P+Q)/2
b=(P-Q)/2
since a and b are integers, P and Q must be Parity.
Both are Even or Both are Odd
Now,
n=110=2⋅5⋅11
110 cannot be factored into two factors with the same parity.
Hence, 110 cannot be expressed as a difference of squares.
K(110)=0

n=105=3⋅5⋅7
105 has four possible factorings.
P Q a b
105 1 53 52
35 3 19 16
21 5 13 8
7 15 11 4

n=216=2^3⋅3^3
216 has four possible factorings.
P Q a b
108 2 55 53
54 4 29 25
36 6 21 15
18 12 15 3

n=384=2^7⋅3
384 has six possible factorings.
P Q a b
192 2 97 95
96 4 50 46
64 6 35 29
48 8 28 20
32 12 22 10
24 16 20 4
Therefore K(384) has Max number of possibilities

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