1991 AHSME 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?

\textbf{(A) } 0\qquad \textbf{(B) } 1\qquad \textbf{(C) } 2\qquad \textbf{(D) } 3\qquad \textbf{(E) } 4

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.