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📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 For fixed \(mn = 506\), \(x = m+n\), \(y = n-m\) determined uniquely. So each divisor pair gives a unique \((x,y)\).

📌 But since \(m\) runs over all 16 integer divisors of 506 (8 positive, 8 negative), we get 16 solutions.

📌 However, consider symmetry: for example, \((m,n)\) and \((n,m)\) give:

📌 So symmetric about x-axis. But both are distinct unless \(y = 0\).

📌 But since 506 ≠ 0, no solution has \(y = 0\) (would require \(m = n\), \(m^2 = 506\), not square), so all solutions come in pairs \((x,y)\) and \((x,-y)\), except if \(y = 0\), which doesn’t occur.

📌 But each \((m,n)\) is distinct, so 16 solutions.

📌 But let’s test a small one: \(m = 1\), \(n = 506\):

📌 \(a = 2\), \(b = 1012\), then \(x = (2+1012)/2 = 507\), \(y = (1012 - 2)/2 = 505\).

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