The product of the digits of $k$ is an integer so $8 \mid k, k=8n$, and the product of the digits is $25n-211$. Now since $8 \mid k$ the rightmost digit is even, so $25n-211$ is even; then $25n$ is odd, so $n$ is odd. Also the product of the digits of $k$ is at most $k$ (trivial to show) so $25n-211 \leq 8n, 17n \leq 211, n \leq 12$. On the other hand, it is a non-negative number so $25n-211 \geq 0, 25n \geq 211, n \geq 9$. Now we just have to check $n=9, 11$ because it is odd, that is, $k=72, k=88$. Inspection shows both work.

your answer is correct.