Find all six-digit integers $\overline{a b c d e f}$:

where $a, b, c, d, e, f$ represent different digits, and $a \cdot e \neq 0$,
Let $\overline{a b c d}=x, \overline{e f}=y$.
According to the problem, we have
$$
3(100 x+y)=10000 y+x,
$$

which simplifies to $299 x=9997 y$.
Thus, we have
$$
\frac{x}{769}=\frac{y}{23}=k, k \in N .
$$

Therefore, $x=769 k, y=23 k$.
However, $a \neq 0$, and $y$ is a two-digit integer, so $k$ can only be 2, 3, or 4.

Thus, the six-digit numbers we are looking for are 153846, 229769, or 307692.