2004 AIME I Problems/Problem 15

Problem

For all positive integers $x,$ let

$f(x)=\begin{cases}1 & \rm{if \ x=1}\\ \frac x{10} & \rm{ if \ x \ is \ divisible \ by \ 10}\\ x+1 & \rm{otherwise}\end{cases}$

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 $m.$

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2004 AIME I Problems/Problem 15

Problem

Solution

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