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📰 a^3 + b^3 = 7^3 - 3(10)(7) = 343 - 210 = 133 📰 Thus, the value is $ oxed{133} $.Question: How many lattice points lie on the hyperbola $ x^2 - y^2 = 2025 $? 📰 Solution: The equation $ x^2 - y^2 = 2025 $ factors as $ (x - y)(x + y) = 2025 $. Since $ x $ and $ y $ are integers, both $ x - y $ and $ x + y $ must be integers. Let $ a = x - y $ and $ b = x + y $, so $ ab = 2025 $. Then $ x = rac{a + b}{2} $ and $ y = rac{b - a}{2} $. For $ x $ and $ y $ to be integers, $ a + b $ and $ b - a $ must both be even, meaning $ a $ and $ b $ must have the same parity. Since $ 2025 = 3^4 \cdot 5^2 $, it has $ (4+1)(2+1) = 15 $ positive divisors. Each pair $ (a, b) $ such that $ ab = 2025 $ gives a solution, but only those with $ a $ and $ b $ of the same parity are valid. Since 2025 is odd, all its divisors are odd, so $ a $ and $ b $ are both odd, ensuring $ x $ and $ y $ are integers. Each positive divisor pair $ (a, b) $ with $ a \leq b $ gives a unique solution, and since 2025 is a perfect square, there is one square root pair. There are 15 positive divisors, so 15 such factorizations, but only those with $ a \leq b $ are distinct under sign and order. Considering both positive and negative factor pairs, each valid $ (a,b) $ with $ a 📰 Top Rated Ice Makers 3877080 📰 Dove Tattoo Designs That Tap Into Soulful Memoriestrend Alert 9999273 📰 How To Master Peptide Calculations In Seconds Start With This Revolutionary Calculator 4587360 📰 Meteor Shower 5927023 📰 5 As A Factor 2888136 📰 Fifa World Cup 26 Tickets 1697842 📰 Iphone Se 2020 4483085 📰 This Simple Dollar Figure Reveals The True Poverty Line In The Usashocking Facts Inside 5049739 📰 Wtm Stock Forecast Will It Crash Or Soar In 2025 Heres What You Need To Know Now 7116845 📰 From Hooper House To Heart The Real Story Of Marge Simpson Revealed 9946534 📰 American Residential Services 5444050 📰 Mileage Rate 2025 5173651 📰 Water Dispense 485588 📰 Unlock The Secrets Of X Men Legendstop 5 Characters That Define The Team 9924738 📰 Papa Louie Vs The Pizza Inferno Channel This Wild Attackevery Second Counts 7460119