Let $a,b,c$ be integers, no two of which are zero and $d=\gcd(a,b,c)$. Show that $d= \gcd(\gcd(a,b),c)$.
$\gcd(a,b,c)=\gcd(\gcd(a,b),c)\,$ [GCD Associative Law]
I'm developing a site using Drupal 8 beta-14. I've created a view block of different terms and now I want to display it using code. How can I display it programatically?  I used to do it in Drupal 7 using this code but I'm confused about Drupal 8.  $block = module_invoke('block', 'block_view', '4'); $text_block = render($block['content']); Guid vs INT - Which is better as a primary key? I've being reading around reasons to use or not Guid and int.  int is smaller, faster, easy to remember, keeps a chronological sequence. And as for Guid, the only advantage I found is that it is unique. In which case a Guid would be better than and int and why?  From what I've seen, int has no flaws except by the number limit, which in many cases are irrelevant.  Why exactly was Guid created? I actually think it has a purpose other than serving as primary key of a simple table. (Any example of a real application using Guid for something?)  ( Guid = UniqueIdentifier ) type on SQL Server The Half-Elven and their choices Earendil and Elwing were given the choice of kindreds after their journey to Valinor, This choice was extended to their sons (specifically sons, not daughters or grandchildren)  Elrond chose the race of the Eldar and his brother Elros chose to be counted amongst the Edain. Thus far Manwë's proclamation has been fulfilled.  There are some inconsistencies here:   Arwen was given the choice, but Elladan and Elrohir remained as Eldar in Imladris Tar-Vardamir was not given the choice The offspring of Imrazor and Mithrelas in Dol Amroth dont seem to have been given a choice   I can think of only one explanation, that the Mannish blood is dominant over the blood of the Eldar, thus, As Vardamir was the offspring of two mortals he did not fall under Manwë's proclamation despite being the son of a half elven.  However this does not seem to explain the other inconsistencies, Is it possible that Elladan and Elrohir and also the princes of Dol Amroth weren't considered important enough by Manwë or Eru to give them the choice.  Also as a side note Tolkien states there were only three unions between man and elf, but this seems to ignore Turin/Finduilas and Imrazor/Mithrelas  My question is, Why was Arwen given the choice and not the sons of Elros? Get material and color of material on a face of an object in Python I want to iterate over the faces of a mesh and get the diffuse color of the material that is assigned to the face. How can I get the material that is assigned to the face of the object and the diffuse color of that material? Do I have to use “do” in any “wh-” question? What will happen if a plane trys to take off whilst on a treadmill? Do I have to move swap partition to the right side? According to  I just expand my sda4 to include unallocated space, my question is after create 2 new partitions sda7, sda8, do I need to move the swap partition to the most right of the disk? (I really don't know this, I saw  which moving the swap partition to the right side)  Thanks. How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals? How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals? Key uses weak digest algorithm I'am getting this kind of message while updating my ubuntu 16.04 LTS:  W: http://dl.google.com/linux/chrome/deb/dists/stable/Release.gpg: Signature by key 4CCA1EAF950CEE4AB83976DCA040830F7FAC5991 uses weak digest algorithm (SHA1) W: http://dl.google.com/linux/chrome/deb/dists/stable/Release.gpg: Signature by key 3B068FB4789ABE4AEFA3BB491397BC53640DB551 uses weak digest algorithm (SHA1)   Can anyone help me? Please suggest me any particular unnecessary file that I can clean to back everything to normal condition(temporarily). (i.e. any log or archieve or anything ). My var/log has only 40MB and Home directory has 3GB of space(so I believe that's not a problem). Other than that what I can clean up to make space.   [user@host]$ df -h Filesystem            Size  Used Avail Use% Mounted on /dev/mapper/vg_inamivm-lv_root                        18G   17G     0 100% / tmpfs                 1.9G     0  1.9G   0% /dev/shm /dev/sda1             485M   71M  389M  16% /boot   I am in a debian machine.   UPDATE1:  output of  cd /; du -sxh *  6.1M    bin 61M     boot 156K    dev 22M        etc 3.3G    home 306M    lib 18M     lib64 16K     lost+found 4.0K    media 4.0K    mnt 408K    opt du: cannot access `proc/18605/task/18605/fd/4': No such file or directory du: cannot access `proc/18605/task/18605/fdinfo/4': No such file or directory du: cannot access `proc/18605/fd/4': No such file or directory du: cannot access `proc/18605/fdinfo/4': No such file or directory 0       proc 208K    root 9.7M    sbin 0       selinux 4.0K    srv 0       sys 8.0K    tmp 536M    usr 187M    var   Update2  Output of ls -la /  dr-xr-xr-x.  22 root root  4096 Aug  7 08:42 . dr-xr-xr-x.  22 root root  4096 Aug  7 08:42 .. -rw-r--r--.   1 root root     0 Aug  7 08:42 .autofsck dr-xr-xr-x.   2 root root  4096 Mar 28 16:53 bin dr-xr-xr-x.   5 root root  1024 Mar 28 16:54 boot drwxr-xr-x.  16 root root  3580 Sep  9 03:13 dev drwxr-xr-x.  69 root root  4096 Aug 23 09:19 etc drwxr-xr-x.   9 root root  4096 Jun 29 16:10 home dr-xr-xr-x.   8 root root  4096 Mar  7  2012 lib dr-xr-xr-x.   9 root root 12288 Mar 28 16:53 lib64 drwx------.   2 root root 16384 Mar  7  2012 lost+found drwxr-xr-x.   2 root root  4096 Sep 23  2011 media drwxr-xr-x.   2 root root  4096 Sep 23  2011 mnt drwxr-xr-x.   3 root root  4096 Mar  7  2012 opt dr-xr-xr-x. 355 root root     0 Aug  7 08:42 proc dr-xr-x---.   5 root root  4096 Aug 17 18:27 root dr-xr-xr-x.   2 root root  4096 May  2 09:13 sbin drwxr-xr-x.   7 root root     0 Aug  7 08:42 selinux drwxr-xr-x.   2 root root  4096 Sep 23  2011 srv drwxr-xr-x.  13 root root     0 Aug  7 08:42 sys drwxrwxrwt.   3 root root  4096 Sep 13 03:37 tmp drwxr-xr-x.  13 root root  4096 Mar 28 17:53 usr drwxr-xr-x.  18 root root  4096 Mar  7  2012 var I have, by default, 250MB of swap space in Ubuntu, and I want to expand it to a larger size. I need 800MB, which I think will be enough to open several applications without having to hit the current limit of swap. I hope someone can help me. Why are the United States often referred to as America? How to quit Skype on Android? What is a rocket engine thrusting against in space? While reviewing "First Posts" on Stack Overflow, I often come across new users posting comments as answers. It seems (at least to me) they are not doing this because they are unaware of how Stack Overflow works but because they can't comment on any question they are forced to put it as answer.  I have found duplicate questions for my request like:   ,  and    but the best explanation I have found is an  to .     I think the main reason this isn't supported is there's no way to   downvote comments, which would make them attractive to spammers. If   somebody comments on the top answer in a question the only way to get   rid of it is for six people to flag it. Comment flags don't show up on   10k tools, so it needs to be six people that happen to stumble across   the spam comment, or a moderator that checks the list manually. People   can comment every 15 seconds, so that can get out of hand pretty   quickly. Comments also aren't listed on the user's profile page, so   it's hard to see at a glance if a user has been spamming   So I am suggesting allowing new users (with less than 50 reputation points) to comment only 5 times every day or only 25 times times until they reach 50 reputation points or any other variation of it. In chat, the short code [metaso] expands to http://meta.stacexchange.com/ (no typo, the missing 'k' is the bug I'm reporting).  (And since this is too short a bug report, I'll note that I think that [metaso] should probably point to .) Given a convergent sequence $x_{n}$ and bounded sequence $y_{n}$  I need to prove that $\limsup (x_{n}+y_{n})=\lim x_{n}+\limsup y_{n}$, when $n$ tends to $\infty$.  I chose $z_{n}=x_{n}+y_{n}$, we know that $z_{n}$ is bounded as being sum of two bounded sequences, so from the Bolzano-Weierstrass Theorem, we know that there is a  subsequence of $z_{n}$, let's call it $z_{n_{k}}$, that converges to $\limsup (x_{n}+y_{n})$. $y_{n_{k}}$ is bounded as well, so there is a convergent subsequence $y_{n_{k_{j}}}$. All this gives me that  $\limsup (x_{n}+y_{n})\leq \lim x_{n}+\limsup y_{n}$,  What can I do for getting the other inequality?   Thank you and Good evening. Any ideas how to display a PDF file in a WPF Windows Application?     I am using the following code to run the browser but the Browser.Navigate method does not do anything!  WebBrowser browser = new WebBrowser(); browser.Navigate("http://www.google.com"); this.AddChild(browser); // this is the System.Windows.Window Finding the internal name and display name for a list column I am using SharePoint Server 2007 Enterprise with Windows Server 2008 Enterprise, and I am using publishing portal template. I heard for SharePoint list/list column, it has an internal name which could be used for programming (as identifier for list/list column), and also has a display name (which is showed for end user for list name or list column name).  How to find and set such internal name and display name for list/list column? CSS bug with printing site pages (now with 100% more suggested fix included!)