, Example 6. Find a four-digit number, which when multiplied by 4 is exactly equal to its reverse.
Assuming the four-digit number $x y z t(x \neq 0$, $1 \neq 0, y, z$ are integers) exists, at this point
$$
\text { 4. } x y z t=t z y x \text {. }
$$

Since $4 \cdot x y z t \geqslant 4 x \cdot 1000$,
$\overline{t z y x}7$. Thus, $t=8$ or $t=9$.

But $t=9$ is impossible, because if $t=9$, then the left-hand side of (*) is 6, while the right-hand side is $2$, making (*) invalid. Therefore, only $t=8$.
Substituting $x=2, t=8$ into (*) yields:
$$
\begin{array}{l}
4(2000+100 y+10 z+8)=8000+100 z \\
+10 y+2 .
\end{array}
$$

Simplifying gives $13 y+1=2 z$.
Given $z \leqslant 9$ and (**), we have $13 y \leqslant 17$, so $y=0$ or $y=1$.

When $y=0$, by (**) we get $z=\frac{1}{2}$, which is not an integer, so $y \neq 0$. Therefore, only $y=1$.
When $y=1$, substituting into (**) gives $z=7$. Thus, the number sought is 2178.