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But when we list all divisor pairs \((m,n)\) with \(mn = 506\), we include both positive and negative, and all 16 are valid and give distinct \((x,y)\)?
No: for each factorization, we get a point. But since \(m\) and \(n\) are determined, and \(x = m+n\), \(y = n-m\), and since \(m\) runs through all 16 divisors (positive and negative), we get 16 values of \(x\), 16 of \(y\), but not necessarily distinct points.
But each pair \((m,n)\) with \(mn = 506\) gives a unique \((x,y)\)?

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