the question is that K(N) denotes the no. of ways in which N can be expressed as the difference of two perfect squares, then which of the following is maximum: K(110), K(105), K(216), K(384)????
i need to know where to start from.
the question is that K(N) denotes the no. of ways in which N can be expressed as the difference of two perfect squares, then which of the following is maximum: K(110), K(105), K(216), K(384)????
i need to know where to start from.
Hello, nandu11!
Lucky for us, I've played with the Difference-of-Squares long ago.
denotes the number of ways in which
can be expressed as the difference of two perfect squares.
Then which of the following is maximum? .
Suppose is a product, where
Then we want integers and so that: .
So we have: .
We assume that: .
. . and solve the system: .
Since and are integers, and must have the same parity.
. . (Both are even or both are odd.)
Let's examine each of the 's.
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110 cannot be factored into two factors with the same parity.
Hence, 110 cannot be expressed as a difference of squares.
. .
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105 has four possible factorings.
. .
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216 has four possible factorings.
. .
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384 has six possible factorings.
. . . .