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Home » Solution: Multiply numerator and denominator by $ \sqrt7 + \sqrt3 $: - DNSFLEX

Solution: Multiply numerator and denominator by $ \sqrt7 + \sqrt3 $: - DNSFLEX

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 Question: What is the greatest common divisor of $ 2^{12} - 1 $ and $ 2^{18} - 1 $?

📌 Solution: We use the identity:

📌 \gcd(2^a - 1, 2^b - 1) = 2^{\gcd(a,b)} - 1

📌 Here, $ a = 12 $, $ b = 18 $, so $ \gcd(12, 18) = 6 $.

📌 \gcd(2^{12} - 1, 2^{18} - 1) = 2^6 - 1 = 64 - 1 = 63

📌 Therefore, the greatest common divisor is $ \boxed{63} $.

📌 Question: What is the largest integer that must divide the product of any four consecutive positive integers?

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