For natural numbers $a$ and $b$, show that $a \Bbb Z + b \Bbb Z = \gcd(a, b)\Bbb Z $
Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that $m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}$
Is it permissible to use somebody else's words (without quotations) to describe a concept once their work is cited? Quite often one comes across a really succinct and precise explanation of a concept. I'm wondering if it is permissible to use this explanation verbatim, if the source work is cited in your paper (next to the explanation)? What's in question here is at most 3 sentences. Stack Overflow as a reference in a professional paper/presentation Suppose I'm doing an academic paper, journal article, or other professional presentation.  Would you say that it would be possible to use Stack Overflow as a reference?    Of course I'm pretty sure that the answer here would be "yes."   So here's the follow up: When, if ever, would it be a good idea to cite a Stack Overflow question as a reference in an professional paper/presentation of some kind? How to run firefox and firefox ESR simultaneously The main reason I want to use firefox ESR is to use a proprietary database tool for work that only allows oracle-java based web logins for linux (or a windows only thin client). I also want to use the full version of the outlook web application which only works in firefox. I don't want to run the NPAPI plugins for normal web browsing and email.   I have the repo version of firefox installed and firefox ESR installed in /opt/firefox and linked to /usr/local/firefox-esr  When I run firefox any attempt to run firefox-esr either brings up a window without the NPAPI plugins.   If I try running with the --new-instance or --no-remote options I receive the following error:  Firefox is already running, but is not responding. To open a new window, you must first close the existing Firefox process, or restart your system.   Regardless which one I start first, I can only open that one, and get the above error if I attempt to sidestep.  What makes this an especially annoying problem is that I often have to respond to emails that I need to access the database to deal with. So either I close and open many windows. Did spacetime start with the Big bang? Did spacetime start with the Big Bang?  I mean, was there any presence of this spacetime we are experiencing now before big bang? And could there be a presence/existence of any other space-time before the big bang? Make cd automatically ls I find that I often do the following:  %&gt; cd bla/bla %&gt; ls   I would like it that whenever I cd into a directory it automatically does an ls.  I fiddled with my .bashrc for a while, but couldn't figure out how to make it happen. The storage on my iPhone is nearly full. Most of the space is used by apps. I'd like to see which apps consume the most space on my device.  Is there a way to get a list of the apps with their corresponding used space? By "used space I mean the installation size (bonus for added user data). Which is stronger, a rope without knots or a rope with knots? Suppose I have two identical ropes, one of which I manually tie some knots in. If I use them to hang clothes, which one is more durable?      Personally I think that rope with knots will be more durable, but I can't come up with a satisfactory reason. Why $SU(3)$ and not $U(3)$? Is there a good reason not to pick $U(3)$ as the colour group? Is there any experiment or intrinsic reason that would ruled out $U(3)$ as colour group instead? Wifi not working (Intel) on HP Spectre x360 13 Does the order of declaration matter in Java/C#? In C++ I can't use a method if I declare it after the calling method.    Does this order matter in other languages like Java or C#? What's the difference between a shunt resistor and a four-terminal resistor? I want to know if the four-terminal resistor and the shunt resistor are the same.   Is there another use for the four-terminal resistor except measuring current? I am drawing some graphs with tangents. I calculated the tangent should be centered at x=57/27 which is approximately x=2.11. I'm currently using the pos and sloped keys for nodes to draw tangents, but I'm not happy about the result; the tangents don't really align.   I think this is because the pos key calculates the actual length of the graph while the graphs don't have the same length.  I'm sure I can somehow let LaTeX do the calculations for me so as to align the slopes vertically at an x value of my choice. But how?  PS: The below document requires a working installation of gnuplot to compile.    \documentclass{standalone}   \usepackage{pgfplots} \pgfplotsset{compat=1.10} \usetikzlibrary{arrows, decorations.markings,intersections}  \makeatletter \tikzset{     nomorepostaction/.code=\makeatletter\let\tikz@postactions\pgfutil@empty,      my axis/.style={         postaction={             decoration={                 markings,                 mark=at position 1 with {                     \arrow[ultra thick]{latex}                 }             },             decorate,             nomorepostaction         },         thin,         -,         every path/.append style=my axis     } } \makeatother  \definecolor{headtitle}{RGB}{167,63,63}  \begin{document} \begin{tikzpicture}   \begin{axis}[%       axis line style={my axis},       xlabel = $U_A$,       ylabel = $S$,       xmin = 0, xmax = 3.25,       ymin = 0,       xtick={1,1.5},       xticklabels={$U_{B,\mathrm{initial}}$,$U_{A,\mathrm{initial}}$},       xticklabel style={             inner sep=0pt,             anchor=north east,             rotate=35         },       ytick=\empty,       height=.66\linewidth, width=\linewidth-4.5\tabcolsep,       axis lines*=left,       every axis y label/.style={at=(current axis.above origin),anchor=east},       every axis x label/.style={at=(current axis.right of origin),anchor=west}]%     \addplot+[mark={},draw=headtitle] gnuplot[raw gnuplot] {%       set samples 51;       plot [0:3] log(3*(3-x)+1);       } node[sloped,above,pos=.63333,       minimum height=1cm,minimum width=1cm] (SB) {} node[pos=0.6,pin=45:{\color{black}{$S_B$}}] {};      \addplot+[mark={},draw=headtitle] gnuplot[raw gnuplot] {%       set samples 51;       plot [0:3] log(9*x^2+6*x+1);       } node[sloped,above,pos=.83333,       minimum height=1cm,minimum width=1cm] (SA) {} node[pos=0.25,pin=-15:{\color{black}{$S_A$}}] {};       \addplot+[mark={},draw=headtitle] gnuplot[raw gnuplot] {%       set samples 51;       plot [0:3] log(9*x^2+6*x+1)+log(3*(3-x)+1);       } node[sloped,above,pos=.775,       minimum height=1cm,minimum width=1cm] (St) {} node[pos=0.525,pin=165:{\color{black}{$S_\mathrm{total}$}}] {};      \path (SB.south east) edge[headtitle,very thick] node {} (SB.south west);      \path (SA.south east) edge[headtitle,very thick] node {} (SA.south west);      \path (St.south east) edge[headtitle,very thick] node {} (St.south west);   \end{axis}% \end{tikzpicture}% \end{document} I have a machine running a couple of vagrant VM. The problem I have is that sometimes I forget to shutdown those VM before I shutdown or reboot my machine. Because of that my machine get stuck with this message: waiting for vboxnet0 to become free  I searched about solutions and I found this page :    I tried what they for shutdown, but it doesn't work.  I wrote an sh file for that command:  #!/bin/bash  cd ~/workspace/git/mediaservice vagrant halt   any suggestions? How can I understand and prove the "sum and difference formulas" in trigonometry? The "" formulas often come in handy, but it's not immediately obvious that they would be true.  \begin{align} \sin(\alpha \pm \beta) &amp;= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \cos(\alpha \pm \beta) &amp;= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \end{align}  So what I want to know is,    How can I prove that these formulas are correct? More importantly, how can I understand these formulas intuitively?   Ideally, I'm looking for answers that make no reference to Calculus, or to , although such answers are still encouraged, for completeness. Kill process after it's been allowed to run for some time I want to limit the time a grep process command is allowed to run or be alive.  For example. I want to perform the following:  grep -qsRw -m1 "parameter" /var   But before running the grep command I want to limit how long the grep process is to live, say no longer than 30 seconds.  How do I do this?  And if it can, how do I return or reset to have no time limit afterwards. I have this problem I'm working on. Hints are much appreciated (I don't want complete proof):  In a normed vector space, if $ x_n \longrightarrow x $ then  $ z_n = \frac{x_1 + \dots +x_n}{n} \longrightarrow x $   I've been trying adding and subtracting inside the norm... but I don't seem to get anywhere.  Thanks! It seems i'm not gonna find an answer for this question which does not use LHopital's rule. How do you prove limit of (lnx)/x approaches 0 as x moves toward infinity WITHOUT using LHopital's rule? Substitute fresh garlic instead of garlic powder? I have a recipe that calls for garlic powder, but I only have fresh garlic on hand.   What ratio should I use to substitute? how do you make commands not show up on the chat I am making a server, and I put a comparitor clock down to constantly have clear weather. How do I silence the commands? I am running the latest version of X3 Terran Conflict (3.1) and after about 30 min - 1 hour it's memory usage goes well above 2 GB.  I only have 4 GB of ram, running Windows 7 64 bit.  As a result of X3 using more ram than is available, my hard drive starts paging and makes the game and other applications become unusable.  I have to restart the game.  Is anyone aware of this issue or how to fix it and is there any way in Windows or through some tool, to limit how much memory is given to the game?  Is this a bug in the game? Can this even be fixed?  I'm using some mods but I doubt any of them are causing this as they are simple mods like adding a tractor beam etc.