Question 154
Given positive integers $a$ and $k$, the sequence $(a_n)_{n=1}^{\infty}$ is defined by
$a_1 = a$ and $a_{n+1} = a_n + k \cdot \varphi(a_n)$ for $n = 1, 2, \dots,$
where $\varphi(m)$ stands for the product of digits of $m$ in its decimal representation (e.g., $\varphi(413) = 12$, $\varphi(308) = 0$). Prove that there exist positive integers $a$ and $k$ such that the sequence $(a_n)_{n=1}^{\infty}$ contains exactly 2009 different numbers.
$a_1 = a$ and $a_{n+1} = a_n + k \cdot \varphi(a_n)$ for $n = 1, 2, \dots,$
where $\varphi(m)$ stands for the product of digits of $m$ in its decimal representation (e.g., $\varphi(413) = 12$, $\varphi(308) = 0$). Prove that there exist positive integers $a$ and $k$ such that the sequence $(a_n)_{n=1}^{\infty}$ contains exactly 2009 different numbers.