543. Does there exist a six-digit number that increases by 6 times when the first three digits, without changing their order, are moved to the end of the number?
$\triangle$ Let the desired number be denoted as $\overline{a b c d e f}$. Then

$$
\overline{a b c d e f} \cdot 6=\overline{d e f a b c}
$$

Let

$$
\overline{a b c}=x, \quad \overline{d e f}=y
$$

The equation becomes

$$
(1000 x+y) \cdot 6=1000 y+x
$$

Simplify it:

$$
6000 x+6 y=1000 y+x, \quad 5999 x=994 y, \quad 857 x=142 y
$$

Since the numbers 857 and 142 are coprime, from the last equation we have $x=142, y=857$.

Therefore, the six-digit number is

$$
1000 x+y=142857
$$

Answer: It exists and is unique - 142857.