Find all numbers that are their own multiplicative inverse in $mod$ $p$ where $p$ is prime.   I recall that when $p$ is prime, all integers from 1 to the modulus minus 1, so all numbers from $1$ to $p-1$, have multiplicative inverses in mod $p$. So, the numbers that are their own multiplicative inverse would be $1$ and $1-p$. Can someone please explain why?
I like to be thorough, but if you feel confident you can skip the first paragraph.  Review: A ring is a set $R$ endowed with two operations of + and $\cdot$ such that $(G,+)$ is an additive abelian group, multiplication is associative, $R$ contains the multiplicative identity (denoted with 1), and the distributive law holds.  If multiplication is also commutative, we say $R$ is a commutative ring. A ring that has no zero divisors (non-zero elements whose product is zero) is called an integral domain, or just a domain.  We want to show that for a domain, the equation $x^2 = 1$ has at most 2 solutions in $R$ (one of which is the trivial solution 1).    Here's what I did:   For simplicity let $1,a,b$ and $c$ be distinct non-zero elements in $R$.  Assume $a^2 = 1$.  We want to show that letting $b^2 = 1$ as well will lead to a contradiction.  So suppose $b^2 = 1$, then it follows that $a^2b^2 = (ab)^2 = 1$, so $ab$ is a solution as well, but is it a new solution?  If $ab = 1$, then $abb = 1b \Rightarrow a = b$ which is a contradiction.  If $ab = a$, then $aab = aa \Rightarrow b = 1$ which is also a contradiction.  Similarly, $ab = b$ won't work either.  So it must be that $ab = c$.  So by "admitting" $b$ as a solution, we're forced to admit $c$ as well.  So far we have $a^2 = b^2 = c^2 = 1$ and $ab = c$.  We can procede as before as say that $(abc)^2 = 1$, so $abc$ is a solution, but once again we should check if it is a new solution.  From $ab = c$, we get $a = cb$ and $b = ac$, so $abc = (cb)(ac)(ab) = (abc)^2 = 1$.  So $abc$ is not a new solution; it's just one.  At this point I'm stuck.  I've shown that it is in fact possible to have a ring with 4 distinct elements, namely $1,a,b$ and $c$ such that each satisfies the equation $x^2 = 1$ and $abc = 1$.  What am I missing?
how many terms will there be once you collect terms with equal monomials? What is the sum of all the coefficients? If you expand $(x_1+x_2+\cdots+x_k)^n$, how many terms will there be once you collect terms with equal monomials? What is the sum of all the coefficients?  I'm kind of lost here. This came up with other questions on the topic of binomial coefficients, and i have figured those out. I have spent almost an hour trying to figure where to start and haven't made any progress. How can I format a date as dd/mm/yyyy or mm/dd/yy ?   Like in VB format("dd/mm/yy",now)  How can I do this in C#? What happens with the appearance of the folder icons in dark mode of macOS Mojave Finder? In High Sierra I used some custom icons for my folders in the Finder; but with the new update to Mojave and the dark mode, the predefined icons also become darker. However the custom icons don't.  Inspecting an element you find a dark preview, but when you extract the icon this is the clear, the same as there was in High Sierra. E.g.:   Previously visualization of the developer predefined icon. It's dark:  But if you extract that icon, it's clear:    There are no dark icons in the system library, at least not in the same directory as always:  /System/Library/CoreServices/CoreTypes.bundle/Contents/Resources   Are they being actively obscured by the operative system without there being icons with that real appearance? In this case, any idea to apply this method to the custom icons without modifying it into a new coloration? How do I find out which program and process ID accesses a given IP address in Windows? Is it possible to catch, which program is accessing a specific IP address?   I found my computer is flooding and wish to check addresses one by one.  May be it is possible to set up some audit for this? Find an equation in spherical coordinates for the surface represented by the rectangular equation The rectangular equation is $$x^2+y^2-8z^2=0$$ $$x^2+y^2=8z^2$$ Know in the relationship between rectangular and spherical coords. we can manipulate our given to fit the form: $$x^2+y^2+z^2=9z^2$$ $$\rho=x^2+y^2+z^2, \space z=\rho\cos(\phi)$$ $$\rho^2=9\rho^2\cos^2(\phi)$$ $$1=9\cos(\phi)$$ $$\frac{1}{3}=\cos(\phi)$$ $$\arccos(\frac{1}{3})=1.23 \space rads$$ And so the equation in spherical coords. is $\phi=1.23$ I know my math is correct but I have the wrong answer so I'm not sure where I went wrong. Handling of special LaTeX characters in text Should covariates that are not statistically significant be 'kept in' when creating a model? I have several covariates in my calculation for a model, and not all of them are statistically significant.  Should I remove those that are not?  This question discusses the phenomenon, but does not answer my question:   There is nothing in the answer to that question that suggests that non-significant covariates be taken out, though, so right now I am inclined to believe that they should stay in.  Before even reading that answer, I was thinking the same since a covariate can still explain some of the variance (and thus help the model) without necessarily explaining an amount beyond some threshold (the significance threshold, which I see as not applicable to covariates).  There is another question somewhere on CV for which the answer seems to imply that covariates should be kept in regardless of significance, but it is not clear on that.  (I want to link to that question, but I was not able to track it down again just now.)  So...   Should covariates that do not show as statistically significant be kept in the calculation for the model?  (I have edited this question to clarify that covariates are never in the model output by the calculation anyway.)  To add complication, what if the covariates are statistically significant for some subsets of the data (subsets which have to be processed separately).  I would default to keeping such a covariate, otherwise either different models would have to be used or you would have a statistically significant covariate missing in one of the cases.  If you also have an answer for this split case, though, please mention it. Creating Digital Terrain Model from Digital Elevation Model using ArcGIS for Desktop? I have a DEM and I want to create a DTM from that DEM. Is it possible to do this in ArcGIS? Add a way for moderators to cast a normal, non binding close/open vote I think moderators should have the ability to cast a normal, non binding close and open vote like they were a normal user (while of course retaining their ability to cast a binding vote where necessary). This can be used in &quot;grey areas&quot; where a moderator can choose to give his or her opinion, but not make a decision alone; a good example of this is . This also helps in areas not so grey (questions that everyone would agree should be closed) since the OP has the time to see close (1) and possibly edit his or her question. The moderator can then optionally come back after a few minutes and close it for real if it's not improved (or maybe during that time other 4 people voted to close it). This would also fit perfectly in :  Since we’re about to add community moderators on both Stack Overflow and Server Fault, I need to document what it is, exactly, we expect moderators to do. The short answer is, as little as possible! \end{abstract} conflicts with arabtext package I am using elsarticle  Everything was find before I used \usepackage{arabtext}.   When I compiled to PDF, I got the following error message.    And if I type "s", everything became fine again but I don't want to type an "s" every time I compile PDF preview.   Please anyone advise.  Thank you very much in advance for your help.  Here is the code  \documentclass[12pt, authoryear]{elsarticle}    % use "amsart" instead of "article" for AMSLaTeX format \usepackage[colorlinks=true,linkcolor=black, citecolor=blue, urlcolor=blue]{hyperref} \usepackage{geometry}                       % See geometry.pdf to learn the layout options. There are lots. \geometry{a4paper}                          % ... or a4paper or a5paper or ...  %\geometry{landscape}                       % Activate for rotated page geometry %\usepackage[parfill]{parskip}          % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx}               % Use pdf, png, jpg, or epsÂ§ with pdflatex; use eps in DVI mode                             % TeX will automatically convert eps --&gt; pdf in pdflatex         \usepackage{amssymb}  \usepackage{amsmath} \usepackage{tipa} \usepackage{arabtex} %SetFonts  %SetFonts   \title{Translating from English to Bahasa Indonesia: Which procedures should be taught to EFL students preparing to be a translator?} \author[mkl]{Mohammad Kholid}  \author[fai]{Faisal Mustafa\corref{cor1}} \ead{faisal.mustafa@unsyiah.ac.id}  \cortext[cor1]{Corresponding author}  \address[mkl]{Syiah Kuala University, Banda Aceh, Indonesia} \address[fai]{Syiah Kuala University, Banda Aceh, Indonesia} \date{}                         % Activate to display a given date or no date  \begin{document} %\section{} %\subsection{}  \begin{frontmatter}  \begin{abstract}   Abstract content is here.  \end{abstract}  \begin{keyword}  translation procedures \sep English to Bahasa Indonesia \sep teaching translation \end{keyword} \end{frontmatter} Right now, I have everything in a partition (except EFI system partition). How can I split the partition into two, one for / and one for /home?   How would you use parted to do that step by step?  How would you use gparted to do that step by step?  How would you use LVM to do that step by step?   One thing I am not sure of is that I can't just shrink the partition and create a second partition on the space that is freed by shrinking, because that doesn't move /home to the new partition.  Thanks.    $ sudo parted -l [sudo] password for t:  Model: ATA TOSHIBA MQ01ABF0 (scsi) Disk /dev/sda: 500GB Sector size (logical/physical): 512B/512B Partition Table: gpt Disk Flags:   Number  Start   End    Size   File system  Name                  Flags  1      1049kB  538MB  537MB  fat32        EFI System Partition  boot, esp  2      538MB   500GB  500GB  ext4  $ sudo df -h [sudo] password for t:  Filesystem      Size  Used Avail Use% Mounted on udev            1.7G     0  1.7G   0% /dev tmpfs           340M  1.4M  338M   1% /run /dev/sda2       457G  6.7G  428G   2% / tmpfs           1.7G   27M  1.7G   2% /dev/shm tmpfs           5.0M  4.0K  5.0M   1% /run/lock tmpfs           1.7G     0  1.7G   0% /sys/fs/cgroup /dev/sda1       511M  6.1M  505M   2% /boot/efi tmpfs           340M   20K  340M   1% /run/user/1000 tmpfs           340M     0  340M   0% /run/user/106 What makes a Feynman diagram real or virtual? Simple question: as the title says, what makes a real Feynman diagram real, and what makes a virtual diagram virtual? Or in other words, how do I tell whether any given diagram is real or virtual? I've never gotten a really satisfying explanation of this. I would imagine it has something to do with virtual particles, but all internal propagators are virtual particles and I know for a fact that having internal lines doesn't make a diagram virtual. Use the values of \title, \author and \date on a custom title page I am trying to create a title page of my own.  Is there a way to make use of the \title, \author and \date commands inside the title page?  For instance:  ... \title{Something Cool} \author{Cool Dude} \date{\today}  \begin{document} \begin{titlepage}  The title is \title It was written by \author on \date  \end{titlepage} \end{document}   Basically I just want to know if there is a way to leverage the information stored in \title, \author and \date inside the title page. In Cocoa KVO, why doesn't a change on a NSMutableArray proxy notify observers? How to cluster time series? I have a question about cluster analysis. There are 3000 companies, which have to be clustered according to their power usage over 5 years. Each company has values for every hour during 5 years. I would like to find out if some companies have the same pattern in usage power over the time period. The results should be used for daily prediction of power usage. If you have some ideas how to cluster time series in SPSS, please share with me. Following a global preference for , as of now, the preferences for enabling keyboard shortcuts and disabling the fixed topbar are network-wide* as well.  If you had either checked off on any site in the network, that setting has been preserved and applied to the whole network for your account already. Future changes will also apply to all* Q&amp;A sites.  Find them in their usual spot as previously on the  pages.    * With the exception of , which will support the fixed topbar after the . Best practices for making a WordPress site "movable"? I'm planning to buy a domain and install a WordPress site on it at a web-hosting company.  But first I'll need to iron-out some bugs...  I got a working LAMP-stack at my home-computer, so I'm thinking about first installing my site locally - off the Internet.  I'll be mostly tweaking and making "test-posts" to check that everything works, but I'll probably also make some posts I'd like to keep when I move the site and take it online.  When I'm done chasing bugs; I guess I'll just package the public_html|www directory-tree and upload it to the web-hosting company, and also export the database from my home-computer to the web-hosting.    What I'm wondering if there are some "best practices", some do-s and don't-s, some major pits to take note of?    For example, how can ensure the smooth transition of domain-name in posts and links as such, from localhost to whatever-i-end-up-buying.com (I know from previous attempt of running WordPress at my home computer, that this didn't always work quite well...). Does this email mean that I have been granted the visa? I received the following email this morning. Does it mean that my visa has been granted?  UK Visas &amp; Immigration has now assessed your UK visa application and made a decision. Your documents and the decision will be sent back to either the UK Visa Application Centre where you applied, where we will contact you by email over the next few days to collect them, or if you are using the courier return service, will be sent directly back to the address provided. Please note that TLScontact does not know the outcome of the assessment and has played no role in the decision-making process. Kind regards, TLScontact UK VIsas &amp; Immigration leaflet geojson coordinate problem I just started to play with leflet/geojson a little. But my coordinates are not rendered properly and I have no clue what is going on.  My coordinates are: 52.23943, 4.97599. They work correct with the setView function.  var map = L.map('leaflet_map').setView([52.23943, 4.97599], 15);   But using a geojasonFeature they are, hmmm, 'projected', somewhere east of Somalia.  var geojsonFeature = {             "type": "Feature",             "properties": {             "name": "Coors Field",             "amenity": "Baseball Stadium",             "popupContent": "This is where the Rockies play!"             },             "geometry": {                 "type": "Point",                 "coordinates": [52.23943, 4.97599]             }         };         var myLayer = L.geoJson().addTo(map);         myLayer.addData(geojsonFeature).bindPopup("I am a gjson point.");   Anyone who knows what is happening here?  EDIT  Out of pure curiosity I changed the coordinates around [4.976143930893815,52.23925499011473] and the point appears at its correct location. A known bug!? OpenLayers and OSGB 1936 (CRS 27700) I've been working with loads of shapefiles projected according to OSGB 1936 (CRS 27700). The project CRS is set to EPSG:27700 and "On the fly reprojection" to EPSG:27700 is set (I keep this as a default as I work with OSGB 1936 95% of my time).  I decided to download OSM layers from the OpenLayers plugin. When I load this, the EPSG in the bottom right of the screen goes from EPSG:27700 to EPSG:3857   Question 1: should this be happening anyway as it would suggest that the whole project has been re-projected to EPSG:3857?   As soon as the OSM layer is loaded it's way out of alignment but a quick pan seems to bring it altogether quiet nicely.   What I have noted is that (Question 2), if I try to re-specify the CRS to 27700, then the shapefiles "move" by a few tens of meters with respect to the OSM. Does this suggest that QGIS is doing a second re-projection (from what to what?)?   Also (Question 3), does it mean that if I import another shapefile that would be originally a dataset digitised with EPSG:27700 it would be re-projected to EPSG:3857 because the CRS in the bottom right of the screen was set to EPSG:3857 after the OSM import. This is all a bit confusing and logic would suggest that the import of a non-ESPG:27700 (project default CRS) should not go and change the EPSG setting in the bottom right of the screen. Anita Graser's book says on page 34 "Your project CRS will automatically be changed to Pseudo Mercator when you load a background map using the OpenLayers plugin option".   Does that mean there is no way of keeping ESPG:27700 active in the project when importing basemaps from OpenLayers? If so, what alternatives are there?  What I am trying to bottom out is to understand the correct way of importing basemaps (usually Google, Bing or OSM through the OpenLayers plugin) into my projects (always EPSG:27700) without losing alignment and getting layers to correctly re-project. I'm trying to support greater use of QGIS in my company but don't want to lose credibility by promoting the use of a tool that is misaligning datasets. Don't get me wrong. QGIS is a great tool and I am very fond of it but importing layers with different CRS's is quite common these days and getting this right from the start is half the job done.  I did quite an extensive Google search before posting and didn't come up with a recent, clear cut answer to my questions above. There is a bit of content out there on problems with older versions of QGIS, PROJ4, etc. but nothing recent for QGIS 2.0 so worth an update.