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📅 May 4, 2026 👤 scraface

May 04, 2026

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\boxed{\frac{640}{6561}}
$$Question: What is the greatest common divisor of $ 5^m - 1 $ and $ 5...
Solution: The greatest common divisor of $ 5^m - 1 $ and $ 5^n - 1 $ f...
\gcd(5^m - 1, 5^n - 1) = 5^{\gcd(m,n)} - 1.
We compute $ \gcd(24, 36) $. Using the Euclidean algorithm:
\gcd(36, 24) = \gcd(24, 12) = \gcd(12, 0) = 12.
\gcd(5^{24} - 1, 5^{36} - 1) = 5^{12} - 1.
We now compute $ 5^{12} - 1 $:
5^2 = 25, \quad 5^4 = (25)^2 = 625, \quad 5^8 = 625^2 = 390625,
5^{12} = 5^8 \cdot 5^4 = 390625 \cdot 625.

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