Question 26
An $n$-digit positive integer is *cute* if its $n$ digits are an arrangement of the set $\{1,2,\ldots,n\}$ and its first $k$ digits form an integer that is divisible by $k,$ for $k=1,$ $2,$ $\ldots,$ $n.$ 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?