4. Solve the system of equations:

$$
\begin{aligned}
\lfloor 2 x\rfloor+\lfloor x\rfloor+\lfloor 7 x\rfloor & =2017 \\
3\{x\}\{5 x\} & =4\{2 x\}\{6 x\}
\end{aligned}
$$

(With $\lfloor x\rfloor$ we denote the greatest integer not greater than $x$, and with $\{x\}$ we denote $x-\lfloor x\rfloor$.)
4. By adding the inequalities $x-1<\lfloor x\rfloor \leqslant x, 2 x-1<\lfloor 2 x\rfloor \leqslant 2 x$ and $7 x-1<\lfloor 7 x\rfloor \leqslant 7 x$, we obtain

$$
10 x-3<\lfloor x\rfloor+\lfloor 2 x\rfloor+\lfloor 7 x\rfloor \leqslant 10 x
$$

i.e., $10 x-3<2017 \leqslant 10 x$. From this, we conclude $201.7 \leqslant x<202$, so $\lfloor x\rfloor=201$ and $\lfloor 2 x\rfloor=403$, and we get $\lfloor 7 x\rfloor=2017-\lfloor x\rfloor-\lfloor 2 x\rfloor=1413$. This further gives

$$
1413=\lfloor 7(\lfloor x\rfloor+\{x\})\rfloor=\lfloor 7\lfloor x\rfloor+7\{x\}\rfloor=\lfloor 1407+7\{x\}\rfloor=1407+\lfloor 7\{x\}\rfloor
$$

i.e., $\lfloor 7\{x\}\rfloor=6$, and then we get $\frac{6}{7} \leqslant\{x\}<1$. Now we have $\lfloor 6 x\rfloor=\lfloor 6(\lfloor x\rfloor+\{x\})\rfloor=\lfloor 6\lfloor x\rfloor+6\{x\}\rfloor=6\lfloor x\rfloor+\lfloor 6\{x\}\rfloor$, and considering the inequality $\frac{36}{7} \leqslant 6\{x\}<6$, we have $\lfloor 6\{x\}\rfloor=5$. From this, we get

$$
\{6 x\}=6 x-\lfloor 6 x\rfloor=6(\lfloor x\rfloor+\{x\})-(6\lfloor x\rfloor+\lfloor 6\{x\}\rfloor)=6\{x\}-5
$$

Similarly, we get $\{5 x\}=5\{x\}-4$ and $\{2 x\}=2\{x\}-1$. Now, substitute all of this into the second equation. It reduces to

$$
3\{x\}(5\{x\}-4)=4(2\{x\}-1)(6\{x\}-5)
$$

which, after rearranging, becomes $33\{x\}^{2}-52\{x\}+20=0$. Solving this equation, we get

$$
\{x\}=\frac{52 \pm \sqrt{52^{2}-4 \cdot 33 \cdot 20}}{66}=\frac{52 \pm \sqrt{2704-2640}}{66}=\frac{52 \pm \sqrt{64}}{66}=\frac{52 \pm 8}{66}
$$

i.e., $\{x\}=\frac{52+8}{66}=\frac{60}{66}=\frac{10}{11}$ or $\{x\}=\frac{52-8}{66}=\frac{44}{66}=\frac{2}{3}$. We discard the second possibility because it does not satisfy the condition $\frac{6}{7} \leqslant\{x\}$. Therefore, $\{x\}=\frac{10}{11}$, and the only solution to the system is

$$
x=201+\frac{10}{11}=\frac{2221}{11}
$$