CAPS 2009 Problem 2

For positive integers $a$ and $k$ , define the sequence $a_1,a_2,\ldots$ by $$a_1=a, \text{and} a_{n+1}=a_n+k\cdot\varrho(a_n) \text{for } n=1,2,\ldots$$ where $\varrho(m)$ denotes the product of the decimal digits of $m$ (for example, $\varrho(413)=12$ and $\varrho(308)=0$ ). Prove that there are positive integers $a$ and $k$ for which the sequence $a_1,a_2,\ldots$ contains exactly $2009$ different numbers.