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Home » Halloween Cupcakes That Blow Your Mind—Creepy, Delicious, and Totally Unmissable! - 500apps

Halloween Cupcakes That Blow Your Mind—Creepy, Delicious, and Totally Unmissable! - 500apps

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Solution: Note $ x^2 - 2x + 1 = (x - 1)^2 $. Use polynomial division or remainder theorem for repeated roots. Let $ f(x) = x^5 - 3x^3 + 2x - 1 $. The remainder $ R(x) $ has degree < 2, so $ R(x) = ax + b $. Since $ (x - 1)^2 $ divides $ f(x) - R(x) $, we have $ f(1) = R(1) $ and $ f'(1) = R'(1) $. Compute $ f(1) = 1 - 3 + 2 - 1 = -1 $. $ f'(x) = 5x^4 - 9x^2 + 2 $, so $ f'(1) = 5 - 9 + 2 = -2 $. $ R(x) = ax + b $, so $ R(1) = a + b = -1 $, $ R'(x) = a $, so $ a = -2 $. Then $ -2 + b = -1 $ â $ b = 1 $. Thus, remainder is $ -2x + 1 $. Final answer: $ oxed{-2x + 1} $.Question: A plant biologist is studying a genetic trait that appears in every 12th plant in a rows of crops planted in a 120-plant grid. If the trait is expressed only when the plantâs position number is relatively prime to 12, how many plants in the first 120 positions exhibit the trait?

📌 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.

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