731*. Can the number $n^{2}+7 n+8$ for any natural $n$ be represented as the product of two consecutive natural numbers?
$\triangle$ Let's apply the "bracketing" method (see the solution to problem 663 in § 14). The following inequality holds:

$$
(n+2)(n+3)<n^{2}+7 n+8<(n+3)(n+4)
$$

(check this!).

Since the products $(n+2)(n+3)$ and $(n+3)(n+4)$ are the closest to each other products of two consecutive natural numbers, and the number $n^{2}+7 n+8$ is enclosed between them, this number cannot be represented as the product of two consecutive natural numbers for any natural $n$.

Answer: cannot.