Removable sine factors keeping natural roots

High school · ●●●○○ 50 pts

The left-hand side of the equation

\sin \frac{\pi}{x} \sin \frac{2\pi}{x} \sin \frac{3\pi}{x} \cdots \sin \frac{2015\pi}{x}=0

is a product of factors

\sin \frac{n\pi}{x}

for

n=1,2,\ldots,2015

. What is the maximum number of these factors that can be crossed out so that the number of natural-number roots of the equation does not change?