[ Trigonometric equations ] [ Divisibility of numbers. General properties ]

Authors: Begun $\underline{\text { A.V. }}, \underline{\text { Goryashin D.V. }}$

What is the maximum number of factors of the form $\sin \frac{n \pi}{x}$ that can be crossed out in the left-hand side of the equation

$\sin \frac{\pi}{x} \sin \frac{2 \pi}{x} \sin \frac{3 \pi}{x} \ldots \sin \frac{2015 \pi}{x}=0$ so that the number of its natural roots does not change?
Note that the factor $\sin n \pi / x$ becomes zero only when $n \pi / x = \pi k$, that is, $x = n / k$, where $k \in \mathbf{Z}$. Among these values, the natural numbers will only be the number $n$ itself and all its divisors.

Let's divide the factors of the form $\sin { }^{n \pi} / x$ on the left side of the equation into two groups: the first group will include factors corresponding to numbers $n$ from 1 to 1007, and the second group will include numbers $n$ from 1008 to 2015. When any factor $\sin n \pi / x$ from the second group is removed, the natural root of the original equation, equal to $n$, does not make any other remaining factor zero, so such factors cannot be removed without changing the number of natural roots. However, when any factor from the first group is removed, the number of natural roots does not change, because for any natural $n$ not exceeding 1007, there will be a factor $\sin ^{m \pi} / x$ from the second group where $m$ is a multiple of $n$, so it also becomes zero for those natural values of $x$ for which $\sin ^{n \pi} / x = 0$. Therefore, all such factors can be removed without changing the number of natural roots of the original equation. Thus, the number of factors we are looking for is 1007.

## Answer

1007.