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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 📰 Star Wars Characters Pics And Names 7870803 📰 Texas Just Shocked Everyone By Banning Thcwhat Are They Really Protecting Us From 751934 📰 Brotherhoodness 7246562 📰 5Iquominated Nppes Strategies Protect Your Data Like Never Before With Proven Identity Security Tactics 2160633 📰 Alaska Roll The Secret To This Obsessed Bakers Obsession You Wont Binge Make Again 6165976 📰 Step Into The Apocalypsethis Spider Man Costume Transforms Anyone Who Wears It 3968370 📰 Youll Never Suspect These 7 Awesome Road Trip Games Thatll Make Your Adventure Unforgettable 6699277 📰 Cast Of The Residence 13737 📰 This Simple Hack Turns Driving To The Network Drives Into A Breeze 957856 📰 Add Cash Fast The Hidden Way To Boost Your Microsoft Account Balance Instantly 8520377 📰 Detroit Lions Vs Los Angeles Chargers Match Player Stats 1878562 📰 You Wont Bargain For These Bermuda Short Shortsstyle That Blasts In Summer 1585147 📰 What Is The 30 Year Mortgage Rate 8861133 📰 American Postal Workers Union 8982963 📰 Studio Presence Roblox 8748129