Now compute $\gcd(15,9) = 3$.
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- Solution: Use the property:
- \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.
- \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:
