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\( 6w + 8 = 68 \) - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 \boxed{\frac{640}{6561}}

📌 $$Question: What is the greatest common divisor of $ 5^m - 1 $ and $ 5^n - 1 $, where $ m = 24 $ and $ n = 36 $?

📌 Solution: The greatest common divisor of $ 5^m - 1 $ and $ 5^n - 1 $ for positive integers $ m $ and $ n $ is given by the identity:

📌 \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 $:

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