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📅 July 6, 2026 👤 scraface

Jul 06, 2026

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Question: Find the remainder when $ x^5 - 3x^3 + 2x - 1 $ is divided b...
Solution: Note $ x^2 - 2x + 1 = (x - 1)^2 $. Use polynomial division o...
Solution: We are to count how many integers from 1 to 120 are relative...
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 prim...
Now, the total number of full cycles of 12 in 120 is:

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