6. (5 points) There are four natural numbers, their sum is 243. If the first number is increased by 8, the second number is decreased by 8, the third number is multiplied by 8, and the fourth number is divided by 8, then the four resulting numbers are equal. Therefore, the product of the largest and smallest of the original four numbers is . $\qquad$
【Analysis】According to the problem, we can set the four numbers to be equal after the change as $x$. Then, by reverse deduction, the original four numbers can be: $x-8, x+8, x \div 8, x \times 8$. Based on their sum being 243, we can set up an equation to find the value of $x$, and then determine these four natural numbers, thus solving the problem.

【Solution】Let the four numbers be equal after the change as $x$,
Then the original four numbers are: $x-8, x+8, x \div 8, x \times 8$. According to the problem, we have:
$$
\begin{aligned}
x-8+x+8+x \div 8+x \times 8 & =243, \\
10 x+\frac{x}{8} & =243, \\
81 x & =1944, \\
x & =24,
\end{aligned}
$$

So the four natural numbers are:
$$
\begin{aligned}
24-8 & =16, \\
24+8 & =32, \\
24 \div 8 & =3, \\
24 \times 8 & =192,
\end{aligned}
$$

The product of the largest and smallest numbers among the original four numbers is: $3 \times 192=576$,
Answer: The product of the largest and smallest numbers among the original four numbers is 576.
Therefore, the answer is: 576.