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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 📰 Secure Your Familys Future With These Smart Estate Planning Secrets 7586704 📰 Unlock Fidelity 403B Phone Number Power Millions Awaityour Future Starts Here 9729733 📰 Unless Set X Y X Y Impossible 9864830 📰 These Numbers On The Map Will Change How You See Every Lanewhat They Really Mean 1886335 📰 Learn To Draw Ipad App 6921730 📰 A Chemist Must Select Gold Mining Stocks From Three Options Based On Projected Yields Stock X Gives 8 Return Stock Y Gives 12 And Stock Z Gives 15 She Wants To Achieve At Least 11 Overall Return By Investing In X And Y With The Total Return From X And Y Equal To Twice The Return From Z If She Invests 100000 Total How Much Goes To Stock Y 2411982 📰 How To Crack Horizontal Asymptotes Like A Math Pro No Tricks 7810627 📰 Allegany County Ny 3054933 📰 Fuw The Real Reason Everyones Stepping Over Privacy Now 2255508 📰 Privacy Data Protection News Today 131810 📰 You Wont Believe What Happened When Thefoodmarketba Launchedshocking Secrets Inside 6336537 📰 Bergen Pac 8093957 📰 This Rare Plant Is The Key To Mind Blowing Transformation 1173442 📰 Star Platinum The Crossroads Where Fame Fortune And Fame Collidewatch Now 3200354 📰 Unlock Oracle Identity Management The Ultimate Guide To Boss Level Security 4751875