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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 📰 Premalu Cast 1284869 📰 Verizon York Sc 9219893 📰 Ugma Custodial Accounts Revealed How To Protect And Grow Your Assets Like A Pro 7275960 📰 Ryan Gosling Music 8533536 📰 Shocked By Your Horoscope This 2024 Forecast Will Change How You See Yourself Forever 5398860 📰 This Shocking 90S Makeup Look Is Trendingget Ready For The Glam That Rules Social Media Now 706181 📰 Unlock The Sandbox Universe Game Now And Build Your Own Epic Adventure 6953207 📰 Grow Juicy Tomatoes Without Chaosthis Ultimate Cage Works Like A Tiny Giant 5605247 📰 Stage 1 Lymphoma Rash 7893974 📰 Download Vmware On Mac 6078694 📰 San Sebastian Hotels 5513009 📰 Spamsieve Mac 1293215 📰 Discover The Cutest Animal Coloring Pages Instant Joy For Kids Ninjas Alike 9561145 📰 Indexof Java Explained This One Method Will Change How You Handle Strings Forever 1141887 📰 The Shocking Truth About Costume Monster Inc Boo You Wont Believe The Price 5058257