What is the name and what are the properties of the smallest integer divisible by all integers $1\ldots n$?  Clearly, $n!$ would do the trick, but it is generally not the smallest such number.  So far, I only came up with a recursive definition, i.e.  $f(n)=\frac{n\,f(n-1)}{\mathrm{gcd}(n,f(n-1)}$ starting with $f(1)=1$.  This nicely gives e.g. the first 12 such numbers: 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720.  Someone must have come across this before... Thanks.
What's the smallest positive integer which is divisible by all integers $1, 2, \dots, n$?  Is there a simple way to represent the answer?  Call it $f(n)$ here.   Clearly factorial ($n!$) satisfies the condition of divisibility.  But once you get to $f(4)$ (which is $12$), factorial ($24$) is too large, because it includes as factors both $4$ and $2$, which are redundant.  Multiplying all the prime numbers up to $n$ gives you another estimate, which is this time too low, because you need to include primes multiple times when they're repeated in the factorization of the inputs.  So the answer is somewhere between factorial, and the product of primes.  Is there a simple answer?
Question about the order of the product of two elements in an abelian group My reference for learning abstract algebra says that in an abelian group $G$ the product of two elements (say $a$ with order $n$ and $b$ with order $m$) has order $mn$ if $gcd(n,m) = 1$.   What I don't understand is where the $gcd(m,n)=1$ comes into use in proving this fact? For example, $ a^{mn} = (a^n)^m = (e)^m = e$ (identity element). A similar argument for $b^{mn}$ together with the fact that $G$ is abelian is enough to show that $(ab)^{mn} = e$. From here, how do I conclude for an integer $r &lt; mn$ that $(ab)^r \neq e$  Also, are there any other conclusion that can be drawn if $gcd(n,m) \neq 1$ ?  Thanks for any feedback in advance ! How do I install the Intel Graphics driver in my system? Can someone help me out and explain or point me in the right direction on how to check video drivers and see if my video card running okay?  I had 10.04 installed on my Thinkpad r61 with Compiz Manager and life was great, until the machine took water damage. I bought an ASUS (X54H) since and am trying out 12.04, but the desktop just doesn't look right.  I always struggled with video driver installation. There are no proprietary drivers available in the hardware manager.  When I run lspci | grep VGA:  00:02.0 VGA compatible controller: Intel Corporation 2nd Generation Core Processor Family Integrated Graphics Controller (rev 09)   I will greatly appreciate your help. I want to use Linux more, but like I said video drivers appear to be my biggest concern. I have also tried 12.04 on my desktop PC, but again failed to configure video card, so switched back to Windows 7. Can a character use the Misty Step spell to escape from the Iron Bands of Bilarro? Can a character escape the  (DMG p. 177, listed in the SRD as the ) with the  spell? To clarify, I am not asking if the bands prevent teleportation. I am asking whether the bands are teleported with the character as he teleports using misty step. Please stop bounties from turning off community moderation Possible Duplicate:                tl;dr: Please get rid of "This question has an open bounty and cannot be closed"  Questions with bounties on them can't be closed:    This is by design, and the recommended course of action is to flag for moderator attention, at which point they'll decide whether to refund the bounty and close the question or not.   But why should a bounty disable regular community moderation?  We should be able to cast close votes on bountied questions as usual, and if a question gets enough, the bounty's refunded and the question closed.  An obvious benefit would be reduced burden on moderators, but I doubt that this is really a significant part of their workload.  More important is allowing the entire community's wisdom to be brought to bear.  Consider this question:      Having a simple MFC application with web browser controller that navigates to some site and then to another one on and on.      Using a powerfull machine (16 cores and lots of memory) OS is Server 2008 R2. I can see when launching my app over 30 instances I'm getting the OS UI (each window including my app, mspaint, taskmgr etc..) become very sluggish and slow.      I have monitored the machine resources (CPU, memory and GDI objects) but it seems that all of them are not even close to be in some level of stress.      What is the problematic resource that influence the machine in such way?      10x, Guy   It doesn't look like a good question: somewhat poorly written, vague, not a ton of information, etc.  I would vote to close this question.  However, I don't do much work with MFC and maybe there's something I'm missing.  (I can readily imagine a similarly bad looking question with the extremely concrete answer: .)  I'm comfortable casting a vote to close, not closing unilaterally.  Flagging for a moderator just passes the buck to them, when the correct action is letting the community weigh in. How to automatically generate a stacktrace when my program crashes Keeping the time stamp when syncing with Ubuntu One? I would like to keep the original time stamp of a file when synchronizing with Ubuntu One.  Now Ubuntu One is messing with my time stamps, and I no longer know when I last worked on the file.  Ubuntu One also seem to mess up time stamps on the original source host.  This really is a show stopper, and we are just about to implement Ubuntu One across the many users in the organization. Script for display wrapper class Account list of records? Hi i am trying to write script for display list of records from wrapper class.   @remoteaction     public static wrappercls getaccounts(){         string account = 'select id,name,accountnumber from account';         listacc = database.query(account);         count = listacc.size();         system.debug('#####'+count);         wrappercls wra = new wrappercls(count,listacc);         return wra;     }      public class wrappercls{         public integer count11;         public list&lt;account&gt; account;          public wrappercls(integer count11,list&lt;account&gt; account){             this.count11 = count11;             this.account = account;         }     }    This class returns list of account records and record count parameter.Need to display these records in page.Is it possible if possible Please help me on this..  Script  &lt;script&gt;   function getallaccounts(){     exremote.getaccounts(         function(result,event){             if(event.status){              }             else {              }        },{escape:true}     ); } &lt;/script&gt;   Thanks in Advance... tabular input by columns (i.e. transpose a table) I would like to be able to transpose a tabular(x) table like in the following example. It is not necessary to be completely automated, but I don't want to change the data order, which is a painful and stupid. By "not completely automated" I mean that e.g. the headers can be done seperately etc.  Let's see the MWE:  \documentclass{article}  \usepackage{booktabs} \def\Midrule{\midrule[\heavyrulewidth]}  \begin{document}  \centering  \begin{tabular}{*6c}\toprule \bfseries Name &amp; Alice &amp; Bob &amp; Chuck &amp; Dave &amp; Eve\\\Midrule \bfseries Sex &amp; Female &amp; Male &amp; Male &amp; Male &amp; Female\\\midrule \bfseries Age &amp; 18 &amp; 19 &amp; 20 &amp; 21 &amp; 22\\\bottomrule \end{tabular}  \bigskip  Which I want to look like  \bigskip  \begin{tabular}{*3c}\toprule \bfseries Name &amp; \bfseries Sex &amp; \bfseries Age\\\Midrule Alice &amp; Female &amp; 18\\\midrule Bob &amp; Male &amp; 19\\\midrule Chuck &amp; Male &amp; 20\\\midrule Dave &amp; Male &amp; 21\\\midrule Eve &amp; Female &amp; 22\\\bottomrule \end{tabular}   \end{document}     Note: I know that , but I would like to get it without expl3 (mostly because people would might use my code quite often have an old LaTeX installed and I cannot be convincing them to change this). Why does "kill -9 0" end my console session? I just wonder why kill -9 0 ends my tty console session?   What exactly is PID 0, it is not listed on a ps aux? Backup of clients' directory to Windows Server 2008 We have a machine running Windows Server 2008, and we need to take backup of folder from a desktop on LAN. This folder is not shared.   Is there any tool or software to do this, or can a scheduled task be made which should automatically take the backup of that folder date-wise &amp; make a copy on server? Ie can we do it through Schedule Tasks option ?  Desktop systems have Windows Vista/Windows 7. How do I change the extension of multiple files? I have a directory with numerous files. Part of the files have the 755 permissions and the other part have 644 permissions. I'd like to convert the files with 755 permissions to 644. I have tried the following line by running it from the directory itself:  find . -perm 755 -exec chmod 644 {}\;   However as a result, the permission changed only for the directory itself and after changing it back I found out that the files permissions remained unchanged. Do I miss something? queries with db_select don't respect integers There are many examples of this, and I'd like to give a few:   A person who puts a lot of effort to help the community and earns reputation points. But that reputation is the motivation behind helping the community, not because they really want to. A student approaches a professor, and helps them in their teaching and doing research in order to have a good LOR (letter of recommendation). To a SEO, the content of the website must be good (well, this one is a little tricky, but you get the spirit.)   I'm not saying that the reputation/LOR/high-ranked is not deserved. It is. I just want to focus more on the selfish reason(s) behind that.  Any kind of answer can be accepted: single word, phrase, proverb or idiom. I updated a virtual machine running Ubuntu server 15.04 to the just released new stable 15.10.  There were no apparent problems, but after reboot the machine lost its network connection. lspci showed the correct card and lsmod proved that the driver had been loaded.  Hints about editing or removing /etc/udev/rules.d/70-persistent-net.rules did not succeed (still no network, no new file), the famous /lib/udev/write_net_rules did not even exist.  I don't have much clue regarding udev and systemd, so it took me some time to find in /sys/class/net that the network interface name changed. It became ens32, which was renamed from eth0. After I changed /etc/network/interfaces network communication was enabled again.  But - why does this happen at all? And if it happens, shouldn't the interfaces file be updated automatically?  Now that I deleted the virtual network card and created another one, eth0 becomes eno16777728, which is not so, uhm, nice.  Should this behaviour (at least that of the upgrade procedure) be reported as a bug?  Furthermore, is /etc/udev/rules.d/70-persistent-net.rules obsolete now? And (although only partly related) what about /etc/udev/rules.d/70-persistent-cd.rules?)  Edit - additional question (after accepted answer) The dependency of /etc/network/interfaces from the actual interface name bothered me. I tried a new installation of *ubuntu in another virtual machine. To my surprise, /etc/network/interfaces contained a reference to the loopback device only. However, with my original VM this change prevented the network to come up. So what else do I need to change?  Edit Well, from another test I found that a new Ubuntu Server has the network interface card in /etc/network/interfaces, while XUbuntu has not. Maybe it is using network-manager or something else. "NG NS" on ticket after airport security Old-style/Antique typesetting in LaTeX/TeX Getting the class name of an instance? Granted there are already quite a few badges for community service on this site, I thought, how about some more!  The theme is merging enthusiast/fanatic with suffrage/vox populi. Something like this (names and numbers open to suggestions)  Enthusiastic Voter - Voted 20 times per day consecutively for 30 days ????? - Voted 30 times per day consecutively for 50 days Fanatic Vox - Use the maximum number of 40 votes consecutively for 100 days My significant other and I finish our master degrees at the same time and thus we are applying for PhD programmes at the same time. One major factor of accepting a potential offer would be if we could study at the same university, so I feel that mentioning a two-body problem would be advantageous for both, us and the admission committee. But where do I mention this? Is a line in the statement of purpose suitable here?     If you admit me, please also consider admitting my significant other Jane Roe.   Some additional information: We both do research in computer science. While I do research in natural language processing my partner does research in computer vision.