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📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 Solution: We are to count how many integers from 1 to 120 are relatively prime to 12. This is given by Eulerâs totient function applied to the set of positions, but since the condition is relative primality with 12, we compute $ \phi(12) $, the number of integers from 1 to 12 that are coprime to 12, and then multiply by the number of full cycles in 120.

📌 First, factor 12:

📌 $ 12 = 2^2 imes 3 $.

📌 \phi(12) = 12 \left(1 - rac{1}{2}

📌 ight)\left(1 - rac{1}{3}

📌 ight) = 12 imes rac{1}{2} imes rac{2}{3} = 4.

📌 So, among every 12 consecutive integers, exactly 4 are relatively prime to 12.

📌 Now, the total number of full cycles of 12 in 120 is:

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