The LCM takes the highest power of each prime:
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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
- \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?
