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Home » Divide by 2: \( x^2 + x - 42 = 0 \) - Aurero

Divide by 2: \( x^2 + x - 42 = 0 \) - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Question: A science educator is designing a modular arithmetic activity and asks: How many of the first 200 positive integers leave a remainder of 3 when divided by 7?

📌 Solution: We are looking for integers $ n $ such that $ n \equiv 3 \pmod{7} $ among the first 200 positive integers. These numbers form an arithmetic sequence: $ 3, 10, 17, \ldots $, with first term 3 and common difference 7.

📌 Let the last term be less than or equal to 200. Solve:

📌 + (n-1) \cdot 7 \leq 200 \Rightarrow (n-1) \cdot 7 \leq 197 \Rightarrow n-1 \leq 28.14 \Rightarrow n \leq 29

📌 Thus, there are 29 such numbers.

📌 Question: A community garden coordinator in Brooklyn is planning a layout where plots are labeled in base 7. If a plot is labeled $ 314_7 $, what is its label in base 10?

📌 Solution: Convert $ 314_7 $ to base 10:

📌 314_7 = 3 \cdot 7^2 + 1 \cdot 7^1 + 4 \cdot 7^0 = 3 \cdot 49 + 1 \cdot 7 + 4 = 147 + 7 + 4 = 158

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