Question 15
For all positive integers $x,$ let $$
f(x) = \scriptsize\begin{cases} 1 & \text{if } x = 1, \\ \frac{x}{10} & \text{if } x \text{ is divisible by } 10, \\ x + 1 & \text{otherwise,} \end{cases}
$$ and define a sequence as follows: $x_1 = x$ and $x_{n+1} = f(x_n)$ for all positive integers $n.$ Let $d(x)$ be the smallest $n$ such that $x_n = 1.$ (For example, $d(100) = 3$ and $d(87) = 7.$) Let $m$ be the number of positive integers $x$ such that $d(x) = 20.$ Find the sum of the distinct prime factors of $m.$
f(x) = \scriptsize\begin{cases} 1 & \text{if } x = 1, \\ \frac{x}{10} & \text{if } x \text{ is divisible by } 10, \\ x + 1 & \text{otherwise,} \end{cases}
$$ and define a sequence as follows: $x_1 = x$ and $x_{n+1} = f(x_n)$ for all positive integers $n.$ Let $d(x)$ be the smallest $n$ such that $x_n = 1.$ (For example, $d(100) = 3$ and $d(87) = 7.$) Let $m$ be the number of positive integers $x$ such that $d(x) = 20.$ Find the sum of the distinct prime factors of $m.$