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b_4 = F\left(rac38 - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 5Question: What is the remainder when the sum of the first 10 terms of the arithmetic sequence starting at 1 with common difference 4 is divided by 9?

📌 Solution: The sequence is $1, 5, 9, 13, 17, 21, 25, 29, 33, 37$. The sum of an arithmetic sequence is given by $ S = \frac{n}{2}(2a + (n-1)d) $. Here, $ n = 10 $, $ a = 1 $, $ d = 4 $.

📌 S = \frac{10}{2}(2 \cdot 1 + 9 \cdot 4) = 5(2 + 36) = 5 \cdot 38 = 190

📌 Now, compute $ 190 \mod 9 $. Since $ 9 \cdot 21 = 189 $, the remainder is $ 190 - 189 = 1 $.

📌 Question: A museum curator is cataloging early computing devices and notes that the number of components in a mechanical computer from 1850 is a three-digit number divisible by both 12 and 15. What is the smallest such number that ends in 0?

📌 Solution: A number divisible by both 12 and 15 must be divisible by their least common multiple. Since $ \text{lcm}(12, 15) = 60 $, we are looking for the smallest three-digit number divisible by 60 and ending in 0.

📌 The smallest three-digit multiple of 60 is 120, which ends in 0.

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