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Home » Sea el ancho \( w \). Entonces la longitud es \( 3w \). - Baxtercollege

Sea el ancho \( w \). Entonces la longitud es \( 3w \). - Baxtercollege

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 But we only count pairs where $ a $ and $ b $ have the same parity. Since 2025 is odd, all its divisors are odd. Therefore, $ a $ and $ b $ are both odd in every factor pair, and the sum and difference are even. So all 30 pairs yield integer $ (x, y) $.

📌 However, each solution $ (x, y) $ corresponds to a unique pair $ (a, b) $, and since $ x $ and $ y $ are determined uniquely, and the hyperbola is symmetric, we must check for duplicates. But since $ (a, b) $ and $ (b, a) $ would give different $ x $ and $ y $, and both are included, all 30 pairs produce valid, distinct lattice points.

📌 For example:

📌 If $ a = 1, b = 2025 $, then $ x = 1013, y = 1012 $

📌 If $ a = 2025, b = 1 $, then $ x = 1013, y = -1012 $

📌 These are distinct points.

📌 Thus, all 30 factor pairs produce valid lattice points.

📌 Hence, the total number of lattice points is $ \boxed{30} $.

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