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Home » First, solve the first two congruences: - Upplift

First, solve the first two congruences: - Upplift

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 $ 18 = 2 imes 3^2 $,

📌 $ 30 = 2 imes 3 imes 5 $.

📌 ext{lcm}(18, 30) = 2 imes 3^2 imes 5 = 90.

📌 Thus, they will first align again after $ oxed{90} $ seconds.

📌 Question: In a nanotechnology sensor, signal pulses are emitted every 7 nanoseconds, and calibration checks occur every 13 nanoseconds. If both events start at time zero, what is the smallest positive time in nanoseconds when both events occur simultaneously?

📌 Solution: We seek the smallest positive time $ t $ such that $ t $ is divisible by both 7 and 13 â that is, $ t = ext{lcm}(7, 13) $. Since 7 and 13 are distinct primes, they are coprime and

📌 ext{lcm}(7,

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