## Problem

Dear friends and visitors of the site.

Welcome to our first 2019 Problem:

Prove that 23 is a factor of 2^{11} - 1 (without direct computation).

## Answer Submission Is Not Available

Form is valid through January 2019

## Solution

Dear friends and visitors of the site!

Thank you for you letters concerning the January 2019 Problem.

The first to provide correct solutions were James and Olga.

We send them our sincerest Bravo!

Here is my solution:

2^{11} -1 = (2^{4})^{2 }x 2^{3} - 1 = 16^{2} x 8-1= (-7)^{2} x 8 - 1 (mod 23)

=49 x 8 - 1 (mod 23) = 3 x 8 – 1 (mod 23) = 0 (mod 23).

Proved Try to solve our February Problem!
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