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\( 200 \times 1.25^6 \) - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 \gcd(a^m - 1, a^n - 1) = a^{\gcd(m,n)} - 1, \quad ext{for } a > 1.

📌 \gcd(3^{15} - 1, 3^9 - 1) = 3^{\gcd(15,9)} - 1.

📌 Now compute $\gcd(15,9) = 3$.

📌 \gcd(3^{15} - 1, 3^9 - 1) = 3^3 - 1 = 27 - 1 = 26.

📌 Therefore, the greatest common divisor is $oxed{26}$.**Question:** An archaeologist discovers an ancient Egyptian artifact with markings that represent two numbers \(x\) and \(y\), which are believed to encode a relationship through the equation: \[rac{x+y}{x-y} + rac{x-y}{x+y} = 3.\] Find all possible pairs \((x, y)\) of real numbers satisfying this equation.

📌 Solution:** Start with the given equation:

📌 rac{x+y}{x-y} + rac{x-y}{x+y} = 3.

📌 Let \( u = rac{x+y}{x-y} \), so the second term becomes \( rac{1}{u} \). Then the equation becomes:

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