# A not-so-hard problem

Geometry Level 4

Let $$f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)$$. The intersection of the domain of $$f(x)$$ with the interval $$[0,1]$$ is a union of $$n$$ disjoint open intervals. What is $$n$$?

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