If x is an integer, then without even reading the rest of the question none of the answer choices could possibly be right ('between' means 'strictly between' unless you add the word 'inclusive').
I don't know if there's a typo in the original question - I guess so, considering how the OE reads - but the question makes no sense as written. I suppose it makes sense if you take out the word 'integers' and just require x and y to be positive.
edit: or perhaps the question means to say "If \(x^2\) and \(y^2\) are positive integers..." Then the solution is right.
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