Question 24
Let $$
\begin{gathered} f(x)=\log_{10}\big(\sin(\pi x) \\ {}\cdot\sin(2\pi x)\cdot\sin(3\pi x) \\ {}\cdot\sin(4\pi x)\cdot\sin(5\pi x) \\ {}\cdot\sin(6\pi x)\cdot\sin(7\pi x) \\ {}\cdot\sin(8\pi x)\big). \end{gathered}
$$ The intersection of the domain of $f(x)$ with the interval $[0,1]$ is a union of $n$ disjoint open intervals. What is $n?$
\begin{gathered} f(x)=\log_{10}\big(\sin(\pi x) \\ {}\cdot\sin(2\pi x)\cdot\sin(3\pi x) \\ {}\cdot\sin(4\pi x)\cdot\sin(5\pi x) \\ {}\cdot\sin(6\pi x)\cdot\sin(7\pi x) \\ {}\cdot\sin(8\pi x)\big). \end{gathered}
$$ The intersection of the domain of $f(x)$ with the interval $[0,1]$ is a union of $n$ disjoint open intervals. What is $n?$