I've read  but I don't know how to use the same method to prove my example. Induction seems very long and unnecessary. I tried to come up with something using:  $n^7 \equiv n \mod{7}$ and $n^{11} \equiv n \mod{11}$  but I am not sure that's useful. I am now assuming I have to show that $n^{31} - n$ is a multiple of 7 and 11 and thus a multiple of 77. How do I arrive in this conclusion?
I'm asked to show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$. I've tried to tackle this using Fermat's Little Theorem and Euler's Theorem, but I can't even get started. My main problem seems to be the "for every $a \in \mathbb{N}$" part, because if it restricted $\gcd(a, 1001) = 1$ I would be able to factor out 1001 and try to show that it's factors divide $a^{61} - a$. I'm kinda lost here. Any help is appreciated.
Why does "unisex" mean both sexes? There are some places where men and women are segregated — for example public toilets and public swimming pool changing areas.  By this I mean, for example, that there are "Men's toilets" and "Ladies' toilets".  However, in some more cosmopolitan swimming baths, and other places, there are such things as unisex changing areas, or unisex toilets.  From my understanding uni- implies one, as in universe, unilateral or unicycle.  However when it comes to unisex, this means both sexes, so applies to two things.  How does uni- mean one in unisex? Does it mean one? Can the 'bin size' in a histogram be thought of as a regularity constraint? I know from the  that it is not possible to introduce direct abbreviations like \newcommand{\bal}{\begin{align}} % doesn't work \newcommand{\eal}{\end{align}}   % doesn't work  or \newcommand{\bal}{\align}    % doesn't work \newcommand{\eal}{\endalign} % doesn't work  What is puzzling me is that \newenvironment{foo}{\align}{\endalign}  does work, at least if one is well-behaved enough to diligently type \begin{foo}...\end{foo}, since \foo...\endfoo won't work either. I'd really appreciate some help to shed some light on what is going on behind the scenes :-) If parsing the output of ls is dangerous because it can break on some funky characters (spaces, \n, ... ), what's the best way to know the number of files in a directory?  I usualy rely on find to avoid this parsing, but similarly, find mydir | wc -l will break for the same reasons.  I'm working on Solaris right now, but I'm looking for a answer as portable across different unices and different shells as possible. A Product function for matrix products How to force Wine into acting like 32-bit Windows on 64-bit Ubuntu? In Euclidean geometry texts, it is always mentioned that point is undefined, it can only be described. Consider the following definition: "A point is a mathematical object with no shape and size." I do not understand what is the problem with this definition. Please give the detailed reasons. Thanks in advance! In Episode 1: The Phantom Menace, Anakin's mother, Shmi, said that he does not have a father, although it seemed to me that she doubted that.  Is there a canonical answer as to who Anakin's father is?  If so, who? I wanted to change my own password (as an admin) so I did the following command:  sudo passwd   and then it prompted for a password but when I tryed to relog, my password was not changed. I'm now wondering what password did I changed since it did prompted me to enter a password but I'm not sure what password did it changed exactly.  I know that if I do sudo passwd &lt;MY_USER&gt; it works, I'm just wondering without a user what happens.  Thank you Reputation misaligned in new topbar In the sites dropdown, the reputation scores are misaligned:    This seems to happen only on Firefox (56.0), both on macOS (10.13) and Windows (7, 10) (thanks @Tom and @MonicaCellio). I've tested a couple of other browsers which do not have this problem. I've already cleared my cache. How do I resolve this installation error? I am getting the following error while installing Drush with composer require drush/drush.     Using version ^8.1 for drush/drush   ./composer.json has been updated   Loading composer repositories with package information   Updating dependencies (including require-dev)   Your requirements could not be resolved to an installable set of packages.            Problem 1   - drush/drush 8.1.0 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.1 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.2 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.3 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.4 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.5 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.6 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.7 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.8 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - drush/drush 8.1.9 conflicts with drupal/drupal [No version set (parsed as 1.0.0)].   - Conclusion: don't install drush/drush 8.1.12   - Conclusion: don't install drush/drush 8.1.11   - Conclusion: don't install drush/drush 8.1.10   - Conclusion: don't install drush/drush 8.x-dev   - Installation request for drupal/drupal [No version set (parsed as 1.0.0)] -> satisfiable by drupal/drupal [No version set (parsed as 1.0.0)].   - Conclusion: remove phpdocumentor/reflection-docblock 3.1.1   - Installation request for drush/drush ^8.1 -> satisfiable by drush/drush [8.1.0, 8.1.1, 8.1.10, 8.1.11, 8.1.12, 8.1.2, 8.1.3,   8.1.4, 8.1.5, 8.1.6, 8.1.7, 8.1.8, 8.1.9, 8.8.x-dev, 8.x-dev].   - Conclusion: don't install phpdocumentor/reflection-docblock 3.1.1   - drush/drush 8.8.x-dev requires phpdocumentor/reflection-docblock ^2.0 -> satisfiable by phpdocumentor/reflection-docblock [2.0.0,   2.0.0a1, 2.0.0a2, 2.0.0a3, 2.0.1, 2.0.2, 2.0.3, 2.0.4, 2.0.5].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0a1, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0a2, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.0a3, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.1, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.2, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.3, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.4, 3.1.1].   - Can only install one of: phpdocumentor/reflection-docblock [2.0.5, 3.1.1].   - Installation request for phpdocumentor/reflection-docblock (locked at 3.1.1) -> satisfiable by phpdocumentor/reflection-docblock [3.1.1].      Installation failed, reverting ./composer.json to its original   content. Review Queues and Ignored Tags Steps to Recreate:   Have enough rep to use some of the review queues. Ignore a popular tag. Go to your profile-> prefs-> check 'hide ignored tags'. Open a review queue.   You will get questions from your ignored tag(s) in the review queue.  Personally, I ignored a tag on Movies.SE (and Scifi.SE) for a film that is out elsewhere in the world, but not locally.  It's a simple way to avoid being exposed to spoilers.  I then load up the First Posts review queue on Movies.SE, and I get an answer to a question about the end of the film.  Can the review queues be altered to skip posts on your ignored tags when you have the 'hide ignored tags' option enabled? Array versus List: When to use which? How do I go about proving this?  If $\gcd(a,b)=1$ and $\gcd(a,c)=1$, then $\gcd(a,bc)=1$. I'm very confused with gcd proofs. Riemann and Darboux Integral of a product of two functions Why doesn't this division work in Python? How to interpret regression coefficients in logistic regression? Inspired by this Hagon Von Eitzen's answer( ) I started investigating how I could express natural numbers as differences of squares. Using the method explained in the solution, I investigated natural numbers up to $60$ manually. I observed the following:  There is no way to express numbers of the form $4k+2$ as a difference of two squares.  2.There is one and only one way to express prime numbers (except $2$) as a difference of two squares.  The smallest number that can be expressed as a difference of two squares in two different ways is $15$ and the smallest number that can be expressed as a difference of two squares in three different ways is $45$.  As, $15= 8^2-7^2=4^2-1^2$. $45=23^2-22^2=9^2-6^2=7^2-2^2$. I could prove point $2$ easily but could not prove point $1$. Also I am interested in knowing how I could find the smallest natural number that could be expressed as a difference of two squares in four different ways without manual calculation. Strange artifacts along window borders after waking computer from sleep mode NB: Nvidia has released an official fix; please see ,  and/or .  Upon waking up Ubuntu 16.04 from Suspend, the following graphical glitches appear around all windows, context menus, and similar:      This is currently known to affect the following NVIDIA cards running the proprietary drivers:  NVIDIA GeForce 610M, driver version 375.39, 340.102 NVIDIA GT 630M, driver version 375.39 NVIDIA GTX 650, driver version 375.39 NVIDIA GT 740M, driver version 378.13 NVIDIA GTX 745, unknown driver version NVIDIA GTX 850M, driver version 375.39, 378.13 NVIDIA GT 940M, driver version 375.39 NVIDIA GTX 950M, unknown driver version NVIDIA GTX 960M, driver version 378.13 NVIDIA GTX 970, driver version 375.39 NVIDIA GTX 970M, driver version 375.39 NVIDIA GTX 980 TI, unknown driver version NVIDIA GTX 1060, driver version 375.26, 375.39, 378.09 NVIDIA GTX 1070, driver version 378.09  The above above list has been sourced from reports from other users here on AU,  thread, and  thread. NB: This is be no means an exclusive list. It probably effects nearly all Nvidia cards.  Furthermore, this issue has been replicated on a clean install where the only change was a software update and the installation of the latest NVIDIA drivers.  What is going on, and how can it be resolved? "Can’t help but" vs. "can help but"