AHSME 1991 Problem 26

An $n$-digit positive integer is cute if its $n$ digits are an arrangement of the set $\{1,2,...,n\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,...,n$. For example, $321$ is a cute $3$-digit integer because $1$ divides $3$, $2$ divides $32$, and $3$ divides $321$. How many cute $6$-digit integers are there? $\text{(A) } 0 \text{(B) } 1 \text{(C) } 2 \text{(D) } 3 \text{(E) } 4$