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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 📰 Hercules Labor 12 7099172 📰 Canvas Nau 1169649 📰 Httpresponse 9743316 📰 Whats Behind Those Red Lines On Your Pc Screen You Wont Believe The Troubles They Cause 5629546 📰 Best Futures Trading Broker 1782967 📰 These Crazy Driving Games On Crazy Games Are Taking Over Global Playlists 3016306 📰 Discover The Secret Character Hub That Every Aspiring Writer Needs 5574739 📰 What Episode Does George Omalley Die In Greys Anatomy 4785795 📰 Ready To Beat Risk Master Hazard Vulnerability Analysis Today 2157713 📰 R Sqrt3216 6 891221 📰 Best Unbiased News App 8403201 📰 Best Feeding Bra 8458248 📰 Discover Mario Golfs Ultimate Rookie Tip Thatll Change How You Play Forever 6488160 📰 Marvel Angela The Hidden Princess Everyones Reacting Toyou Wont Believe Her Original Story 3300839 📰 Yapyap Steam 8237449