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Home » Finally, a Simple Way to Break Free from Debt—See How Traceloans.com Works Like Magic! - Tacotoon

Finally, a Simple Way to Break Free from Debt—See How Traceloans.com Works Like Magic! - Tacotoon

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Set this equal to 150:

📌 \frac{n}{2} (4n + 10) = 150 \implies n(4n + 10) = 300 \implies 4n^2 + 10n - 300 = 0

📌 Divide by 2: \( 2n^2 + 5n - 150 = 0 \). Use the quadratic formula:

📌 n = \frac{-5 \pm \sqrt{5^2 - 4(2)(-150)}}{2(2)} = \frac{-5 \pm \sqrt{25 + 1200}}{4} = \frac{-5 \pm \sqrt{1225}}{4} = \frac{-5 \pm 35}{4}

📌 Take the positive root: \( n = \frac{30}{4} = 7.5 \). Since \( n \) must be an integer, the largest possible \( n \) is 7. Check \( S_7 \):

📌 S_7 = \frac{7}{2} (4(7) + 10) = \frac{7}{2} (38) = 7 \cdot 19 = 133 \leq 150

📌 \( S_8 = \frac{8}{2} (4(8) + 10) = 4 \cdot 42 = 168 > 150 \), so maximum is 7.

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We compute $ 390625 \cdot 625 $:
Break $ 625 = 600 + 25 $, so
\cdot 600 = 390625 \cdot 6 \cdot 100 = (2,343,750) \cdot 100 = 234,375...
\cdot 25 = (390625 \cdot 100)/4 = 39,062,500 / 4 = 9,765,625.
Add: $ 234,375,000 + 9,765,625 = 244,140,625 $.
So $ 5^{12} = 244,140,625 $, and
5^{12} - 1 = 244,140,624.
Thus, the greatest common divisor is $ \boxed{244140624} $.
Question: How many lattice points lie on the hyperbola $ x^2 - y^2 = 2...
Solution: We seek integer solutions $ (x, y) $ to $ x^2 - y^2 = 2025 $...

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