Thread: [SOLVED] difference of two squares problem!!

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.

I think you should start from: . So you need to factor those numbers...

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.

. . . .