3. Determine for which values of $n$ and $k$ the equation $\sin x + \sin y = \frac{\pi k}{2017}$ is a consequence of the equation $x + y = \frac{\pi n}{48}$.
Answer: $k=0, n=96 m, m \in Z$.

Solution. The pair $\left(0, \frac{\pi n}{48}\right)$ is a solution to the second equation, hence $\sin \frac{\pi n}{48}=\frac{\pi k}{2017}$. The pair $\left(\pi ; \frac{\pi n}{48}-\pi\right)$ is also a solution to the second equation, hence $-\sin \frac{\pi n}{48}=\frac{\pi k}{2017}$. It is necessary that the conditions $k=0, \sin \frac{\pi n}{48}=0$ are satisfied, which means $n=48 l, l \in Z$.

Let's check the sufficiency of the obtained values. Substitute the pair $(x ; \pi l-x)$ into the first equation, we get: $\sin x+\sin (\pi l-x)=\sin x+(-1)^{l-1} \sin x$, which equals 0 only for even $l$, that is, $l=2 m$.

Answer to variant 172: $k=0, n=108 m, m \in Z$.

Answer to variant 173: $k=0, n=88 m, m \in Z$.

Answer to variant 174: $k=0, n=112 m, m \in Z$.