# Mock AIME 1 2006-2007 Problems/Problem 5

## Modified Problem

For a prime number , define the function as follows: If there exists , , such that

set . Otherwise, set . Compute the sum .

## Original Problem

Let be a prime and satisfy for all integers . is the greatest integer less than or equal to . If for fixed , there exists an integer such that:

then . If there is no such , then . If , find the sum: .

## Solution

The definition of is equivalent to the following: "If has a multiplicative inverse mod , is the member of the set such that . Otherwise, ."

Note that this really gives a well-defined function because that set includes exactly one member from each congruence class modulo , and each invertible element has inverses in only one such class.

From this point onwards, it's clear: as cycles through , also cycles through the same values in some order. We cover those values 11 times. Thus the answer is .