NL-NWO 1997 Problem 1

For each positive integer $n$ we define $f (n)$ as the product of the sum of the digits of $n$ with $n$ itself. Examples: $f (19) = (1 + 9) \times 19 = 190$ , $f (97) = (9 + 7) \times 97 = 1552$ . Show that there is no number $n$ with $f (n) = 19091997$ .