19. (6 points) Among three consecutive natural numbers, the smallest number is a multiple of 9, the middle number is a multiple of 8, and the largest number is a multiple of 7. The smallest sum of these three numbers is $\qquad$ .
【Analysis】According to the problem, these are three consecutive natural numbers. Since $7, 8, 9$ are also consecutive natural numbers, we first find the least common multiple (LCM) of $7, 8, 9$: $7 \times 8 \times 9=504$. Then, subtracting 9 results in a multiple of 9, subtracting 8 results in a multiple of 8, and subtracting 7 results in a multiple of 7, yielding $495, 496, 497$ as the numbers that meet the criteria.

【Solution】The least common multiple of $7, 8, 9$ is: $7 \times 8 \times 9=504$;
$$
\begin{array}{l}
504-7=497, \\
504-8=496,
\end{array}
$$
$$
\begin{array}{l}
504-9=495 ; \\
495+496+497=1488 .
\end{array}
$$

Therefore, the answer is: 1488.