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Home » The least common multiple is \boxed72. - wispro

The least common multiple is \boxed72. - wispro

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 + 2 + 6 + 4 + 5 + 1 = 21 \equiv 0 \mod 7

📌 Since $24 = 4 \times 6$, the total sum modulo $7$ is:

📌 \sum_{k=1}^{24} 3^k \equiv 4 \cdot 0 = 0 \mod 7

📌 Question: A robotics engineer designs a conveyor belt system where sensors trigger every time the belt completes a full cycle. If the belt completes one cycle every $15$ seconds and another subsystem cycles every $25$ seconds, after how many seconds will both systems align at the start for the first time?

📌 Solution: We seek the least common multiple (LCM) of $15$ and $25$, which represents the first time both cycles coincide. Factor both numbers:

📌 = 3 \cdot 5, \quad 25 = 5^2

📌 The LCM takes the highest power of each prime:

📌 \text{LCM}(15, 25) = 3 \cdot 5^2 = 3 \cdot 25 = 75

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