Greatest Common Divisor Integer
How can we prove that a binomial coefficient $n\choose m$is divisible by the ratio of $n$ and $\gcd(n,m)$?
Can I set up a Mac for a multi head / multi user environment? What are the colored/highlighted edges in edit mode? There are red and pink edges on my mesh.  What do these colored edges mean? How can I use them? What is the preferred way to remove spaces from a string in C++? I could loop through all the characters and build a new string, but is there a better way? What are PATH and other environment variables, and how can I set or use them? Questions about    setting environment variables the PATH   are very common here, and in most cases the answers are very similar to each other. In the future it would be nice to have a good Q/A for this.  So the question is: What are environment variables, like the executable PATH, and how can I change and use them on major operating systems?  A good answer would include a simple explanation of what environment variables and especially PATH mean to the OS, as well as simple guidelines on how to set and read them accordingly. Undo a particular commit in Git that's been pushed to remote repos Binary Bomb Phase 5 - lookup table translation -> string compare SQL server Read Only permission automatically for new databases How do I activate Dauntless Cathar's ability? has the following ability:     {1}{W}, Exile Dauntless Cathar from your graveyard: Put a 1/1 white Spirit creature token with flying onto the battlefield. Activate this ability only any time you could cast a sorcery.   Please clarify. If the card is IN my hand, how can I exile if FROM the graveyard if it is not there? Did the writers mean SEND it to graveyard, or actually exile it? Confused, the writing does not make sense. I've been reading up on STL containers in my book on C++, specifically the section on the STL and its containers. Now I do understand each and every one of them have their own specific properties, and I'm close to memorizing all of them... But what I do not yet grasp is in which scenario each of them is used.  What is the explanation? Example code is much prefered. Problem to launch java at Debian: "error while loading shared libraries: libjli.so" Are there any universal rules in appropriating “_an,” “_sh,” “_es,” “_ch” for the demonyms of countries? Suffixes indicating people and language of country vary by country:  _an: American, German, Italian, Belgian, Australian, Russian, Ukrainian, Korean,  Mexican, Brazilian, Chilean, Argentinean, Indonesian, Malaysian, Iranian, Egyptian, Somalian, Palestinian  _sh: English, British, Polish, Danish, Swedish, Turkish,    _ch: French, Dutch  _ese: Chinese, Taiwanese, Japanese, Vietnamese, Portuguese  _no: Filipino, Latino  _i: Israeli  _: Thai  Among the above, _an seems to outnumber other suffixes.  Are there any simple rule to govern appropriation of suffix turning a country name into adjective, indicating people and language of countries in the world in terms of say, region, race, or cultural background? Or is it just discretional? The limit to solve is $$\lim_{n\to\infty}\left(\frac{n^2+2}{2n^2+1}\right)^{n^2}$$ I tried to use L'Hôpital's rule but the derivatives are quite complex. I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$.  If $x\in V$, then since $V$ is a manifold, there is a local parametrization $\phi: U\to V$ where $U$ is open in $\mathbb{R}^k$. Without loss of generality, we can require $\phi(0)=x$. Then $T_x(V)$ is defined to be the image of $d\phi_0$ on $\mathbb{R}^k$.   An arbitrary element of $T_x(V)$ looks like $$ d\phi_0(v)=\lim_{t\to 0}\frac{\phi(0+tv)-\phi(0)}{t}=\lim_{t\to 0}\frac{\phi(tv)-x}{t} $$  but this doesn't seem very useful to show $T_x(V)=V$. What is the right approach? Installing Guest Additions causing problems I am trying to install Guest Additions in VirtualBox with Windows XP host and Ubuntu 12.04 guest. I followed the above steps:  Devices-&gt; Insert Guest Additions CD Image. After that I locate the folder /media/VBoxAdditions4.3.4_91027/ and run sudo ./VBoxLinuxAdditions.run   I want to be able to create shared folders and adjust the resolution of the screen. I achieved to do the first issue, but I couldn't do the second one. I didn't get any message, however it is not possible to change the resolution of the screen and I cannot switch to fullscreen. What am I doing wrong?  I found in the instal log the following message: unknown version of the X Window System. I've tried the solution of the proposed link but I got the following message:  Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies: virtualbox-guest-x11 : Depends: xorg-video-abi-11                     Depends: xserver-xorg-core (&gt;= 2:1.10.99.901) E: Unable to correct problems, you have held broken packages.   I tried also to install with the command sudo apt-get install virtualbox-guest-dkms but nothing is changing. During the installation of dkms I got the message: Error! Bad return status for module build on kernel 3.13.0-32-generic(i686). I have a php file which I will be using as exclusively as an include. Therefore I would like to throw an error instead of executing it when it's accessed directly by typing in the URL instead of being included.  Basically I need to do a check as follows in the php file:  if ( $REQUEST_URL == $URL_OF_CURRENT_PAGE ) die ("Direct access not premitted");   Is there an easy way to do this? I am installing an overhead automatic garage door opener. My only power source close to where I will be hanging the motor is an overhead light socket that is operated by a switch on the wall next to the house entry door.   I wanted to know if it is possible to rewire the light socket with a standard electrical plug outlet AND to remove the switch &amp; wire it to provide constant power to the new outlet. The overhead light is on my 15A garage GFI circuit &amp; the opener lists 120V/6A as its power requirements.  I live in the US. How can one prove that numbers of the form $4x^2+1$ can only be divided by primes of the form $4y+1$ (e.g. there is no $x$ for which $7$ divides $4x^2+1$)?   On a quick lookup, the statement is given as the first comment in  and of course, manual testing concurs with this How to terminate a script? I am aware of the die() command in PHP which exits a script early.  How can I do this in Python? People like myself are Stack Exchange trusted users with thousands of reputation points and hundreds of badges.  In other words, we've proven ourselves as people who are assets to Stack Exchange and are a key component in making it so profitable.  As such, I was surprised when I posted  on the  site, and then shortly afterwards tried to  on that same meta site, but was blocked with a message in red stating I had to wait 40 minutes before posting again.  Why is this limit present for trusted Stack Exchange contributors, and can it be changed? When should I use \input vs. \include? There are two different commands to incorporate another file into the source of some document, \input and \include. When should I use one or the other? What are the differences between them? Are there more things like them to be aware of?