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Home » \sum_k=1^24 3^k \equiv 4 \cdot 0 = 0 \mod 7 - Dachbleche24

\sum_k=1^24 3^k \equiv 4 \cdot 0 = 0 \mod 7 - Dachbleche24

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 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

📌 Therefore, both systems will align at the start after $ \boxed{75} $ seconds.

📌 Question: A geographer is studying a triangular grid of weather stations where true equilateral triangles have sides aligned to a coordinate system based on modulo $6$ symmetry. If the distance between adjacent stations in one direction is $60$ km, how many integer-coordinate points (lattice points) lie strictly inside a large equilateral triangle with side length $6$ units (in km), assuming the vertices are at lattice points satisfying modular constraints?

📌 Solution: We model the triangular grid using a coordinate system adapted to equilateral triangles, but the problem simplifies: we interpret lattice points inside a large equilateral triangle of side length 6 in a triangular lattice where vertices are at lattice points and side length corresponds to 6 units. However, for computational geometry in such grids, the classic approach uses area and Pick’s Theorem.

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