3. (22 points) Find all solutions to the equation $\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]+\left[\frac{x}{7}\right]=x$ ( $[a]$ denotes the greatest integer not exceeding the real number $a$).


3. From the equation, we know that the solution $x$ is an integer.

Let $x=42 p+q(p \in \mathbf{Z}, q \in\{0,1, \cdots, 41\})$.
Then $\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]+\left[\frac{x}{7}\right]=x$
$$
\Leftrightarrow p=\left[\frac{q}{2}\right]+\left[\frac{q}{3}\right]+\left[\frac{q}{7}\right]-q .
$$

Therefore, the solution set of the equation is
$$
\begin{aligned}
\{42 & \left.\left.\left(\left[\frac{q}{2}\right]+\left[\frac{q}{3}\right]+\left[\frac{q}{7}\right]-q\right)+q \right\rvert\, q=0,1, \cdots, 41\right\} \\
= & \{0,-6,-12,-14,-18,-20,-21, \\
& -24,-26,-27,-28,-30,-32,-33, \\
& -34,-35,-36,-38,-39,-40,-41, \\
& -44,-45,-46,-47,-49,-50,-51, \\
& -52,-53,-55,-57,-58,-59,-61, \\
& -64,-65,-67,-71,-73,-79,-85\} .
\end{aligned}
$$