How many solutions are there to the equation $na\equiv _m0$ for $0\leq a <m$?
Modular arithmetic: division, fractions, solving linear congruences
Maximal order of an element in a symmetric group If we let $S_n$ denote the symmetric group on $n$ letters, then any element in $S_n$ can be written as the product of disjoint cycles, and for $k$ disjoint cycles, $\sigma_1,\sigma_2,\ldots,\sigma_k$, we have that $|\sigma_1\sigma_2\ldots\sigma_k|=\operatorname{lcm}(\sigma_1,\sigma_2,\ldots,\sigma_k)$.  So to find the maximum order of an element in $S_n$, we need to maximize $\operatorname{lcm}(\sigma_1,\sigma_2,\ldots,\sigma_k)$ given that $\sum_{i=1}^k{|\sigma_i|}=n$. So my question:      How can we determine $|\sigma_1|,|\sigma_2|,\ldots,|\sigma_k|$ such that $\sum_{i=1}^k{|\sigma_i|}=n$ and $\operatorname{lcm}(\sigma_1,\sigma_2,\ldots,\sigma_k)$ is at a maximum?   Example  For $S_{10}$ we have that the maximal order of an element consists of 3 cycles of length 2,3, and 5 (or so I think) resulting in an element order of $\operatorname{lcm}(2,3,5)=30$.  I'm certain that the all of the magnitudes will have to be relatively prime to achieve the greatest lcm, but other than this, I don't know how to proceed. Any thoughts or references? Thanks so much. Prove that $\displaystyle \sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$.   I tried using the partial fraction decomposition $a_j = \frac{1}{2j} - \frac{1}{j+1} + \frac{1}{2(j+2)}$, but I don't see how that helps. Debugging scripts, what is the difference between -x to set -euxo pipefail? The main way I know to debug scripts is adding -x to the shabang (#!/bin/bash -x).  I recently came across a new way, adding set -euxo pipefail right under the shabang, as in:  #!/bin/bash set -euxo pipefail   What is the main difference between the two ways of debugging? Are there times you would prefer one above the other?  As a freshman, , I couldn't extract such conclusion. What do I need to install LED strip lighting? I've been looking to install LED strips, either RGB or white for lighting and ambiance. It wouldn't be the main lighting of the room, so no necessity for high power LEDs.  The perimeter of the room is a little less than 25 meters. Looking around in online stores, I saw that the reels come in maximum lengths of 5 meters, so I would need five of those.  What I'm not sure is how to connect the strips together.  I was looking at , or  with .  If I choose the white LEDs, what power supply, controller, and connector would you recommend to be able to daisy chain the strips?  If I choose the RGB LEDs, what kind of data repeaters would work to be able to control all the strips at once? Is it possible to add images to comments? I was wondering if it is possible to post a comment with an image.  Recently, during a question about data-visualization in Cross Validated, there was a controversy about using one type of chart for another.   I had posted a question about pie-charts, after reading some references I decided to build an example and clarify the point of using waffle-chart vs pie-chart. In a way, my example was not an answer it was more properly a comment on waffle-charts. But it seems to be impossible to post a comment with an image (Maybe, there is a way but I can't figure out how).  I don't know if it is relevant to post images in comments in other SO sites. So, if there is no way to post images in comments... why not making it possible? And if there is a way to post an image with a comment, how I can do it? I've had a very strange Finer error caused by recent macOS High Sierra betas that affected Macs running macOS 10.13 as well as macOS 10.12 Sierra. This error causes an entire Finder window to show visual artifacts when opening certain folders on either an APFS or HFS+ volume.  After moving the cursor or the affected Finder window, the window may look something like this;    Attempting to use Disk Utility to repair the affected volume does not resolve this issue, nor does restarting the affected Mac (even in Safe Mode).  Without formatting this disk, how can I prevent Finder from exhibiting this graphical error when opening particular folders? Wording of "possible/exact duplicates" when closing the question When marking question for "duplicate", the reason in dialog window is "exact duplicate", yet when the process is completed there's a comment added that says "possible duplicate". Maybe those two should be evened out to either "possible duplicate" or "exact duplicate"?  Edit: Take  for instance, it says "Possible" on top but "exact" at the bottom    and the top remark was added by the system only after the question was closed, i.e. after it was concluded that it was an exact duplicate rather than a possible one.  So the question is: can the text on top be changed to "Exact Duplicate" please? When I use the MouseUp event, I can get it to fire with a mouse right-click. But MouseLeftButtonUp won't fire with either click!  &lt;Button MouseLeftButtonUp="btnNewConfig_MouseUp"  Name="btnNewConfig"&gt;     &lt;StackPanel Orientation="Horizontal"&gt;         &lt;Image Source="Icons\new.ico" Height="24" Width="24" Margin="5"/&gt;         &lt;TextBlock VerticalAlignment="Center"&gt;New&lt;/TextBlock&gt;     &lt;/StackPanel&gt; &lt;/Button&gt;   I know this is most likely something simple. Thanks for the help! Can't boot Windows 7 after installing Ubuntu I installed Ubuntu yesterday alongside Windows 7 and whenever I start my computer I don't have the option to boot Windows. When the computer starts it goes straight to Ubuntu without giving me an OS selection menu. When it goes to Ubuntu it gives me Ubuntu repair options and Ubuntu normal start option. I still have all of the Windows 7 files on my hard drive, but Windows 7 has no way to boot. In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?  Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.  Or, if you could redirect me to a place that explains how to do it, please do so.  My dad said there isn't, but I just had to make sure.  Thanks. Is there any parameter (in terms of physical quantities such as mass, length, charge...) which can be used to decide when to treat a system quantum mechanically and not classically? Find lost device without Gmail account configured and without SIM card I recently misplaced a phone that did not have gmail account configured and did not have a SIM card in it either. Is there any way I can track it ?   All of the tracking apps either require sending an SMS (needs SIM card) or OTA install(needs Gmail account).   I do have the IMEI number of the phone. Differences between LuaTeX, ConTeXt and XeTeX Schrödinger Equation for Two Electrons? I'm having trouble finding a simple answer to this question (maybe because there isn't one), but I'm just confused about how the Schrödinger Equation would look for two electrons. I understand that it would exist in 6 dimensional configuration space, but how does the potential look? It's confusing that the potential would be different for each electron depending on where the other one is. Is there a way to refresh Notepad? Say, for example, that there's one process that's writing to a file. While the process is running, I opened the file in notepad. The process keeps  writing to the file.   Other than closing and reopening the file, is there any way for me to "refresh" the data that notepad is showing? I have two areas of my house which are cantilevered. One is a bay area which extends about 2'. The other is the bottom of a fireplace box (gas fireplace). I have placed two layers of R-13 (I think) insulation in this area, but inside the house it's like a heat sink where all of the heat is sucked out ... very cold to the touch in the winter time. Obviously there must be a better way to insulate this area.  My question is, what is the best way to insulate a cantilevered area so as to allow the inside area of the house to not be so cold in the winter time? it seems my bash-completion got corrupted. Every time I try to use command substitution, for example when I type  ls $(ls Deskto&lt;Tab&gt;   ..result is..  ls $(ls Desktobash unexpected eof while looking for matching ")".   I tried removing my .bashrc file and cp /etc/skel/bashrc ~/ to get a new one. That works until I add something into my new bashrc. Even when I do echo '#" &gt;&gt; ./.bashrc it breaks again. Removing my bash_aliases did not help either.  Also sudo apt-get install --reinstall bash-completion did not do the trick.  However there are some threads about similar problem, nothing works.  ANY hint will be greatly appreciated! :) Multiple workspaces on Ubuntu 18.04.1 LTS and later with GNOME Shell Can you explain the "Axiom of choice" in simple terms? As I'm sure many of you do, I read the  webcomic regularly. The  involves a joke about the Axiom of Choice, which I didn't get.     I went to Wikipedia to see what the Axiom of Choice is, but as often happens with things like this, the Wikipedia entry is not in plain, simple, understandable language. Can someone give me a nice simple explanation of what this axiom is, and perhaps explain the XKCD joke as well? How to download all required Ubuntu drivers Is there a package of Ubuntu drivers as driver pack solution? I tried to search for it, but I did not find it. If there is, how can I easily get it?