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Home » Solution: We begin by factoring the numerator: - ECD Germany

Solution: We begin by factoring the numerator: - ECD Germany

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 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.

📌 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.

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