Find the smallest value of $a\in\mathbb{N}$ such that $7^{21}\cdot 2^{50}+a$ divisible by $9$. The only thing I know is that an integer is divisible by $9$ is the sum of its digits is divisible by $9$.
I've read about Fermat's little theorem and generally how congruence works. But I can't figure out how to work out these two:   $13^{100} \bmod 7$ $7^{100} \bmod 13$   I've also heard of this formula:  $$a \equiv b\pmod n \Rightarrow a^k \equiv b^k \pmod n $$  But I don't see how exactly to use that here, because from $13^1 \bmod 7$ I get 6, and $13^2 \bmod 7$ is 1. I'm unclear as to which one to raise to the kth power here (I'm assuming k = 100?)  Any hints or pointers in the right direction would be great.
The LaTeX Wikibook :     Please understand: you do not have to use floating tables. It only   looks more professional. If you want to place your tables where they   lie in your source code, do not use table at all! This is a very   common misunderstanding among newcomers.   LaTeX works with an intricate , but I often prefer immediate placement with the H placement specifier (enabled by the float package).  My understanding is that captions and table/figure numbering require the table/figure environments. Is this really so, or are there ways around that? What other reasons are there for using table/figure if we know that we don't want a particular object to float? Answer draft is lost if you click to sign up When writing an answer on a site where you don't have an account or are not logged in, below the answer entry textbox are options to create an account or continue as guest by providing a nickname and email. I had written an answer of several paragraphs on a network site, clicked &quot;Sign up with Google&quot; (resulting in association to my existing network-wide profile) but after doing so instead of showing my answer to be posted, only the first five or six words were retained.  I know that I had no account on this site previously because I just received the association bonus... I cannot tell whether the bug also affects totally new users or those with accounts but who are not logged in. This is a very unwelcoming user experience.  Please fix it. If the intended workflow is to click &quot;Post Answer&quot; first and associate the account afterwards, then the sign up options should not be shown until the answer is posted, and certainly not above the Post Answer button. This problem also occurs if you already have an account on the site, and click 'sign up with google' (which allows you to successfully log in). How can I reduce the gap between the subfloat and its caption? I tried to use   \captionsetup[subfigure]{aboveskip=0pt}    but it doesnt work for me. Below is the snippet i have and the output  \begin{figure} \captionsetup[subfigure]{labelformat=empty} \captionsetup[subfigure]{aboveskip=0pt}     \centering     \subfloat[Ground Truth]{{\includegraphics[width=\textwidth]{ood-ms/ood-1-gt.png} }}     \\ \vspace{-1\baselineskip}     \subfloat[Recurrent Neural Network]{{\includegraphics[width=\textwidth]{ood-ms/ood-1-rnn.png} }}     \\ \vspace{-1\baselineskip}     \subfloat[]{{\includegraphics[width=\textwidth]{ood-ms/ood-1-vrnn.png} }}     \\ \vspace{-0.5\baselineskip} \noindent\rule{14cm}{0.4pt}     \subfloat[]{{\includegraphics[width=\textwidth]{ood-ms/ood-3-gt.png} }}     \\ \vspace{-1\baselineskip}     \subfloat[]{{\includegraphics[width=\textwidth]{ood-ms/ood-3-rnn.png} }}     \\ \vspace{-1\baselineskip}     \subfloat[]{{\includegraphics[width=\textwidth]{ood-ms/ood-3-vrnn.png} }}     \\ \vspace{-0.5\baselineskip} \noindent\rule{14cm}{0.4pt}         \subfloat[]{{\includegraphics[width=\textwidth]{ood-ms/ood-1-gt.png} }}     \\ \vspace{-1\baselineskip}     \subfloat[]{{\includegraphics[width=\textwidth]{ood-ms/ood-1-rnn.png} }}     \\ \vspace{-1\baselineskip}     \subfloat[]{{\includegraphics[width=\textwidth]{ood-ms/ood-1-vrnn.png} }}     \\ \vspace{-0.5\baselineskip} \noindent\rule{14cm}{0.4pt}     \subfloat[]{{\includegraphics[width=\textwidth]{ood-ms/ood-3-gt.png} }}     \\ \vspace{-1\baselineskip}     \subfloat[]{{\includegraphics[width=\textwidth]{ood-ms/ood-3-rnn.png} }}     \\ \vspace{-1\baselineskip}     \subfloat[]{{\includegraphics[width=\textwidth]{ood-ms/ood-3-vrnn.png} }}     \\ \vspace{-1\baselineskip}     \caption{}     \label{fig:ms} \end{figure}   Here is the output What is the best way to find the users home directory in Java? How does "warp drive" not violate Special Relativity causality constraints? I'm talking about this nonsense:   Now, I'm aware that there are problems with the practicalities (or possibilities) of methods involving exotic matter with negative mass, and that kind of thing has been addressed here:   But in this case my question is different and more specific. Sweeping all the other problems of constructing such a beast under the rug, wouldn't this violate causality?  SR says that the simultaneity and relative ordering of events is dependent on the relative motion of the observer, but that no matter how it works out, if event A causes event B, no observer will see B first. One of the many problems with FTL travel (or signals) is that if any information is sent faster than light, there will exist an inertial frame in which the signal arrives before it is sent, a blatant violation of causality. (I think we can all agree, can we not, that the threshold should be extremely high before breaking assumptions that we're in a causal universe?)  So how is any scheme to travel faster than light, even if it may locally satisfy the equations of GR, not a blatant violation causality according to SR?  Is my understanding of GR incomplete and there's something in it that obviates the causality implications of SR? Or am I correct that FTL claims should be dismissed with extreme prejudice on this basis alone? (Pending a serious replicable experiment showing causality to be breakable.) Get image pixels of prescribed color Prevent back button after logout Laplace transform of integral constant GPU compute grayed out Rename profile folder on Windows 7/8 Can't login with Google on recently released iPhones and iPads I just switched from an iPhone 7 to an iPhone XR (both are on latest iOS version available).  When I try to login into the Stack Exchange app with my Google account it says:     Request failed   Could not perform this request with your account. Please try logging   back in. If this persist, let us know on meta.   this happens consistently.  I have 2FA enabled on my Google account if it matters.  Is this a known bug? Best practice for Rails App to run a long task in the background? I have a Rails application that unfortunately after a request to a controller, has to do some crunching that takes awhile.  What are the best practices in Rails for providing feedback or progress on a long running task or request?  These controller methods usually last 60+ seconds.    I'm not concerned with the client side... I was planning on having an Ajax request every second or so and displaying a progress indicator.  I'm just not sure on the Rails best practice, do I create an additional controller?  Is there something clever I can do?  I want answers to focus on the server side using Rails only.  Thanks in advance for your help.  Edit:  If it matters, the http request are for PDFs.  I then have Rails in conjunction with Ruport generate these PDFs.  The problem is, these PDFs are very large and contain a lot of data.  Does it still make sense to use a background task?  Let's assume an average PDF takes about one minute to two minutes, will this make my Rails application unresponsive to any other server request during this time?  Edit 2:  Ok, after further investigation, it seems my Rails application is indeed unresponsive to any other HTTP requests after a request comes in for a large PDF.  So, I guess the question now becomes: What is the best threading/background mechanism to use?  It must be stable and maintained.  I'm very surprised Rails doesn't have something like this built in.  Edit 3:  I have read this page: .  I would love to read about various experiences with these tools.  Edit 4:  I'm using Passenger Phusion "modrails", if it matters.  Edit 5:  I'm using Windows Vista 64 bit for my development machine; however, my production machine is Ubuntu 8.04 LTS.  Should I consider switching to Linux for my development machine?  Will the solutions presented work on both? Option clash for package xcolor I'm writing a report, and have a rather large table in my appendix. I'd like to emphasize some specific values by coloring some single cells. I'm trying to do it with xcolor. But receiving the error:  ! LaTeX Error: Option clash for package xcolor. See the LaTeX manual or LaTeX Companion for explanation. Type H &lt;return&gt; for immediate help. ... l.17 The package xcolor has already been loaded with options: [] There has now been an attempt to load it with options [table]   My preamble looks as follows  \documentclass[a4paper,oneside,article]{memoir} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage[danish]{babel} \usepackage{lmodern} \usepackage{graphicx}     \graphicspath{{Figures/}} \usepackage{fullpage} \usepackage{modroman} \usepackage{float} \usepackage{caption} \usepackage[numbers]{natbib} \usepackage{url} \usepackage{pgfplots} \usepackage{mathtools} \usepackage[table]{xcolor}   And take out of my table  \begin{table}  \centering   \begin{tabular}{|c|c|c|}        \hline    \cellcolor{green!25}D &amp; x &amp; D/$L_0$ \\        \hline    65 &amp; 2600 &amp; 0,417\\        \hline    60 &amp; 2400 &amp; 0,385\\   \end{tabular}  \caption{Bestemmelse af bølgehøjde}  \label{tab:Q1} \end{table}   Read some other questions asking more or less the same. And the answer to them seems to be a "collision" between packages. I have no idea how to find the problem. So my question is:   Where is the "collision"? How on earth would I have found it without help ? User and associated Post I would like to either prevent or handle a StackOverflowException that I am getting from a call to the XslCompiledTransform.Transform method within an Xsl Editor I am writing. The problem seems to be that the user can write an Xsl script that is infinitely recursive, and it just blows up on the call to the Transform method.  (That is, the problem is not just the typical programmatic error, which is usually the cause of such an exception.)  Is there a way to detect and/or limit how many recursions are allowed?  Or any other ideas to keep this code from just blowing up on me? Export/Deploy Drupal site/page to static? (no cache) I wonder if there is an existing  module that can help in the subject. I am not looking for a caching module (like Boost or Varnish), but a way to export my whole site (or only certain pages) into static.  The idea is that I want my web server will be a very simple and light and will only serve static files (html/css/js) - no database on production, no php engine will be installed on server. All the Drupal engine will be on another server, export will be done each time adding/editing/changing a node or view, and a new set of html files will be created in a directory structure.  These files will later on be copied to the production site that will serve always and only static files, no .htaccess that will check requests if file exist etc.   Is there a module for that, or a workaround that this can be achieved with Drupal?  Thanks I am using nginx/0.7.68, running on CentOS, with the following configuration:  server {     listen       80;     server_name ***;     index index.html index.htm index.php default.html default.htm default.php;      location / {             root   /***;             proxy_pass   http://***:8888;             index  index.html index.htm;     }     # where *** is my variables   The proxy_pass is to a DNS record whose IP changes frequently. Nginx caches the outdated IP address, resulting in a request to the wrong IP address.  How can I stop nginx from caching the IP address, when it is outdated? Attributive or Possessive noun: the Dell Company's staff. or the Dell Company staff? In the following is it better to use a possessive noun with an apostrophe or an attributive noun without an apostrophe?     The following list details the assumptions that have been made in   conjunction with the Dell Company’s staff.   Or      The following list details the assumptions that have been made in   conjunction with the Dell Company staff.   Assume Company is the name of a company, such as Dell. How can I start automatically a command on Ubuntu 18.04? I have a small command like this  xinput set-prop "XP-PEN G430S Pen Pen (0)" --type=float "Coordinate Transformation Matrix" 1.155756, 0.000000, 0.000000, 0.000000, 1.529, 0.000000, 0.000000, 0.000000, 1.000000   to use my XP-PEN tablet, I have to start it automatically every time that I start my PC. How can I do that? I searched around, and currently there are two methods suggested; installing an enhancement for Windows Subsystem for Linux and installing an XServer.  I want to know which method is the most hassle-free (easy to install AND to use), and which one is less memory-heavy.  I just want Synaptic and CMake. Why couldn't that be a builtin feature?