1. (2005 Nordic Mathematical Contest) Find all positive integers $k$ such that the product of the digits of $k$ equals $\frac{25}{8} k - 211$.
1. The product of the digits of the number $k$ is an integer, so $8 \mid k, k=8 n, n \in \mathbf{N}^{*}$, then the product of the digits is $25 n-211$. Since $8 \mid k$, the last digit is even, so $25 n-211$ is even, and $n$ is odd. It is easy to prove: the product of the digits of the number $k$ is no greater than $k$, so $25 n-211 \leqslant 8 n$, so $n \leqslant 12$. On the other hand, $25 n-211 \geqslant 0$, so $n \geqslant 9$. Upon inspection, when $n=9,11$, $k=72,88$ satisfy the conditions.