# Mock AIME II 2012 Problems/Problem 3

## Problem

The of a number is defined as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the of is (). Find the of .

## Solution

The sum of the digits when we get this down to a one digit number is the same thing as the number modulo 9. This is because let’s say we start with a number . When we take the sum of the digits modulo 9, this is the same thing as taking the number modulo 9. However, the sum of the digits modulo 9 is the same thing as the sum of the sum of the digits modulo 9, and this pattern repeats.

We are left with finding . Note that , therefore we get . By Euler’s Totient Theorem, or . We now need to find , which is equal to . Note that , and , so this pattern repeats in the exponents. Since , we get . From this, we get , so we need to find . Since , we get an answer of .