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Home » Solving for \( x \), we find \( x = 8 \). - Decision Point

Solving for \( x \), we find \( x = 8 \). - Decision Point

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 The number of positive divisors is $ (4+1)(2+1) = 15 $, so there are 15 positive divisor pairs $ (a, b) $ with $ a > 0, b > 0 $, and another 15 with $ a < 0, b < 0 $, giving 30 total integer divisor pairs (since $ a $ and $ b $ can both be negative).

📌 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.

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