Where $(,)$ is the greater common divisor and $[,]$ the smallest common multiple.  In general how do you solve this kind of problems?
These two exercises I encountered recently seem to develop some type of connection between GCD and LCM I can't quite figure out. Exercise 1:  Find all the numbers $x$ and $y$ such that: $a) \ GCD(x,y)=15, \ LCM(x,y)=150$ $b) \ GCD(x,y)=120 \ LCM(x,y)=1320$ $c) \ GCD(x,y)=100 \ LCM(x,y)=990$  Exercise 2:   Find all the numbers $m,n$ such that $GCD(m,n)=pq , \ LCM(m,n)=p^2qs$  where $p,q,s$ are prime  The first thing that is known to me is that $GCD(x,y) \cdot LCM(x,y)= x \cdot y$ Also $LCM(x,y)$ is at most $x \cdot y$ while $GCD(x,y)$ is at most $\max \{x,y\}$. Last thing is that $GCD(x,y)|LCM(x,y)$. Using all this I tried to solve the first exercise: $a)$ First two obvious pairs are $x=15, y=150$ and $y=15, x=150$. Now neither of the numbers can be bigger than $150$ or smaller than $15$. So we are looking for numbers in the range $15-150$ that satisfy $x \cdot y = 15 \cdot 150$ Another such pair is $(x,y)=(30,75), \ (x,y)=(75,30)$. Similarly for $b)$ we find that the only possible values are permutations of the set {$120,1320$} and in $c)$ since $100$ does not divide $990$ no such numbers exist. Now exercise 2 is what made me think there is actually another connection I'm not quite aware of since now it's about arbitrary prime numbers and the previous method doesn't work anymore. My intuition is that it has something to do with $GCD$ or $LCM$ of the $GCD(x,y), \ LCM(x,y)$
Why Didn't People Think That Voldemort Died After His Attack on Baby Harry? After the Avada Kedavra curse rebounded from Harry onto Voldemort, Voldemort disappeared. I'm curious as to why so many key players believed Voldemort was still "alive" somewhere, his powers decimated, but, under the right circumstances, able to return to full power. It seems the witches and wizards in chapter one of Philosopher's Stone, The Boy Who Lived, were happy to believe that Voldemort was gone forever. Why didn't everyone believe Voldemort had died, rather than just disappeared? Was it the lack of a body? Or were there other reasons?  Here's a couple of quotes from Philosopher's Stone if anyone wants to review them. Otherwise, just disregard.     ‘A fine thing it would be if, on the very day You- Know-Who seems to have disappeared at last, the Muggles found out about us all. I suppose he really has gone, Dumbledore?’   ‘It certainly seems so,’ said Dumbledore. ‘We have much to be thankful for.’   Philosopher's Stone - page 13 - Bloomsbury - chapter 1, The Boy Who Lived   ‘But what happened to Vol– sorry – I mean, You-Know-Who?’   ‘Good question, Harry. Disappeared. Vanished. Same night he tried ter kill you. Makes yeh even more famous. That’s the biggest myst’ry, see ... he was gettin’ more an’ more powerful – why’d he go?   ‘Some say he died. Codswallop, in my opinion. Dunno if he had enough human left in him to die. Some say he’s still out there, bidin’ his time, like, but I don’ believe it. People who was on his side came back ter ours. Some of ’em came outta kinda trances. Don’ reckon they could’ve done if he was comin’ back.   ‘Most of us reckon he’s still out there somewhere but lost his powers. Too weak to carry on. ’Cause somethin’ about you finished him, Harry. There was somethin’ goin’ on that night he   hadn’t counted on – I dunno what it was, no one does – but somethin’ about you stumped him, all right.’   Philosopher's Stone - pages 46-47 - Bloomsbury - chapter 4, The Keeper of the Keys How to change css property using javascript Is Kant's "noumenal self" argument on freedom flawed? My PC has a build-in VT1828S chip. The connection between PC and monitor (which has built-in speakers) is through an HDMI cable.  Audio preferences shows 2 devices for audio output: analog stereo and RS880 Audio Device [Radeon HD 4200] digital stereo (HDMI).  When I plug a headphone in the rear jack and I choose the former device, everything works. When I choose the latter device, instead, no sound is heard, and video playback is too fast. It is as if video frames want to synchronise with the audio timing, but the audio timing information is wrong.  HDMI audio worked perfectly with Ubuntu 11.04, but stopped after 11.10 upgrade. Encode multi-class response variable In a classification problem when the response variable has multi-class, e.g., "sunny","rainy","cloudy", how should we encode it? I know that for predictors like this, usually we do , and if a predictors have too many classes, then we might just use the .   When this multi-class issue occurs in the response variable, I guess we can just use LabelEncode() rather use using One Hot encoding right? Because if we use One Hot encoding, then we will have 2 variables as the response variable, and the machine learning algorithm in sklearn usually expects the response variable not to be a vector right? (I mean it expects a long 1D vector with length equal to the number of observations but not a 2D matrix). But on the other hand, if we just map "sunny", "rainy","cloudy" to {1,2,3} or {0,1,2}, or whatever 3 numbers, that will create a less than or greater than relationship among "sunny", "rainy","cloudy", which is not inherited in the original problem. How to set systemd for automatically force fsck disks after crash (hard poweroff)?  When I used sysvinit (in Arch Linux) as /sbin/init I used the hack: in the rc.local I create /forcefsck file; in the rc.local.shutdown I remove it. At boot-up rc.sysinit enable force fsck if /forcefsck exists.  How to do same in systemd? Maybe it has built-in facilities for automatic fsck after crush? I bought a rtl8811au chipest based usb wifi for my work stationd that supports dual band. But when I plugged in the usb my ubuntu doesn't recognized it. So I had to find the driver and compiled it 'cause the one inside the usb package was not compatible with the last kernel version.  My experience with some drivers get from github platform was unsatisfied.  All the drivers I tested are an issue in my user experience. It seems that, with these drivers, my wifi usb stops working randomly, specially when the browser is open and it loads some javascript.   In that context the ping command works normally and all the other connections such as p2p like torrent, or app like telegram.  When that issue shows up, I have some options:   Waiting to work again with do nothing Unplug and replug the usb dongle Restart network service or turn off and turn on wifi   With all these options I have to wait and it is so heartbreaking !  So I think that the problem is relative to the driver 'cause I get it from github and it can contain some error.  I tried drivers like:         and the problem still persists on all these drivers.  Do you know some driver that works properly ? The formal solution of the time-dependent Schrödinger equation Consider the time-dependent  (or some equation in Schrödinger form) written down as $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{ H}~ \Psi . $$ Usually, one likes to write that it has a formal solution of the form $$ \tag 2 \Psi (t) ~=~ \exp\left[-i \int \limits_{0}^{t} \hat{ H}(t^{\prime}) ~\mathrm dt^{\prime}\right]\Psi (0). $$ However, this form for the solution of $(1)$ is actually built by the method of successive approximations which actually returns a solution of the form $$ \tag 3 \Psi (t) ~=~ \hat{\mathrm T} \exp\left[-i \int \limits_{0}^{t} \hat{H}(t^{\prime})~\mathrm  dt^{\prime}\right]\Psi (0), \qquad t&gt;0, $$ where $\hat{\mathrm T}$ is the time-ordering operator.  It seems that $(3)$ doesn't coincide with $(2)$, but formally $(2)$ seems to be perfectly fine: it satisfies $(1)$ and the initial conditions. So where is the mistake? 2 GROUP BY WITH DISTINCT , SUM , COUNT : PHP MYQL Ref :   I got answer (thanks coytech ) but i need one more column in its :  as per :  id | mid | pid | owgh | nwgh | 1    3      12    1.5    0.6 2    3      12    1.5    0.3 3    3      14    0.6    0.4 4    3      15    1.2    1.1 5    4      16    1.5    1.0 6    4      17    2.4    1.2  7    3      19    3.0    1.4   i got answer      Select mid ,          COUNT(distinct pid) as cpid  ,          SUM(nwgh) as totalnwgh  from test  GROUP BY mid   sqlfiddle :   mid  cpid       totalnwgh 3      4          3.8 4      2          2.2    But above i need one more column that's as below : totowgh  mid cpid        totalnwgh  totowgh 3      4          3.8        6.3 (DISTINCT value as per pid column) 4      2          2.2        3.9   where totowgh = 6.3 come by DISTINCT value of owgh column  as per distinct pid column  mid = 3 has count 5 but distinct pid  = 4 for mid=3 same way "distinct" owgh = 6.3 for mid=3 and distinct pid.  As pid=12 , column owgh value 1.5 is twice ...hence its is count 1 time hence,  1.5 + 0.6 + 1.2 + 3 = 6.3 ( please not this is as per DISTINCT value of pid )  Please note : i need distinct owgh value as per distinct pid or group by pid .. because if i replace value of owgh 0.6 with 1.5 then it will be 5.7 instead of 7.2 but value of owgh 0.6 belong to pid = 14 and not pid = 12 hence totalcount of owgh change ...but i need is 7.2  SEE WHAT I MEANS   : Will reformatting my hard drive fix bad sectors on it? Are plants creatures or objects? This has occurred to me due to conflict between Fabricate and Blight. Fabricate says it can not target creatures and lists trees as example targets. Blight on the other hand can only target creatures and lists mundane normal plants as example targets.     Which of these conflicting spell descriptions is correct?    Note: It also occurred to me that plants might also be whichever one is convenient, unlikely but possible. I was wondering if it's possible to run an Ubuntu virtual machine using VirtualBox in background, I mean, without any window open.  The idea is to connect via SSH to the Linux host, that would be running in background.  Has anyone ever done something like this? Is it possible? Are there 'stack' sites for Mac or Windows? There is Ubuntu and Unix/Linux site but is there special 'stack' site for Mac and Windows?  For context, the question I am looking to ask: I accidentally connected a Windows NTFS formatted HDD to a Mac. And the Mac (may be) quick-formatted it. I think that the information is there, but the FAT table is gone. How I can recover the information. get list item by id using COM sharepoint 2013 I need to get item from list sharepoint by Id I use this code and it give me all the items of the list :  &lt;script&gt;  function parseURLParams(url) {         var queryStart = url.indexOf("?") + 1,             queryEnd = url.indexOf("#") + 1 || url.length + 1,             query = url.slice(queryStart, queryEnd - 1),             pairs = query.replace(/\+/g, " ").split("&amp;"),             parms = {}, i, n, v, nv;          if (query === url || query === "") {             return;         }          for (i = 0; i &lt; pairs.length; i++) {             nv = pairs[i].split("=");             n = decodeURIComponent(nv[0]);             v = decodeURIComponent(nv[1]);              if (!parms.hasOwnProperty(n)) {                 parms[n] = [];             }              parms[n].push(nv.length === 2 ? v : null);         }         return parms;     }  $( window ).load(function() {   var urlString = document.URL;  urlParams = parseURLParams(urlString);  ExecuteOrDelayUntilScriptLoaded(retrieveActualite, "sp.js");          var siteUrl = '/sites/securite';      function retrieveActualite() {              var clientContext = new SP.ClientContext(siteUrl);             var oList = clientContext.get_web().get_lists().getByTitle('Actualite');             var camlQuery = new SP.CamlQuery();              camlQuery.set_viewXml('&lt;Query&gt;&lt;Where&gt;&lt;Eq&gt;&lt;FieldRef Name=”ID” LookupId="TRUE"/&gt;&lt;Value Type="Text"&gt;' + urlParams.IdArt[0] + '&lt;/Value&gt;&lt;/Eq&gt;&lt;/Where&gt;&lt;/Query&gt;');             this.collListItem = oList.getItems(camlQuery);             clientContext.load(collListItem);             clientContext.executeQueryAsync(onQuerySucceededActualite, onQueryFailed);         }          function onQuerySucceededActualite(sender, args) {              var listItemInfo = '';              var listItemEnumerator = collListItem.getEnumerator();             while (listItemEnumerator.moveNext()) {                 var oListItem = listItemEnumerator.get_current();                   var desc=oListItem.get_item('Description');                 $('#idArticle').append('&lt;label class="titleArticle"&gt;'+oListItem.get_item('Title')+'&lt;/label&gt; &lt;br&gt;&lt;br&gt;&lt;p class="ContenuArticle" &gt;'+desc+'&lt;/p&gt;');                          }         }          function onQueryFailed(sender, args) {              alert('Request failed. ' + args.get_message() + '\n' + args.get_stackTrace());         }       });    &lt;/script&gt;   so how can I get only the considered item ? Modelling a spiral around a sphere I'm trying to figure out the best way to model the sphere-spiral here:    Is it possible to make this shape directly? Or do you have to make it as a cylinder and wrap it to a sphere? So I just got this Nikon D3100 Camera.  And I am trying to take a picture using autofocus with the kit lens(18-55mm).    The system is in Auto mode, AF-A, single frame and the lens A-M switch is on "A",  VR is "ON".  When I press shutter half way down, focus confirmation light in lower left corner of viewfinder neither lit nor blinks(hear clicking). When I press the shutter all the way down, the shutter seems locked (hear clicking, but shutter not released).    The manual says the shutter will lock itself if it can't get focus. But the place is well lit and have distinct color and I can easily focus using my old Canon.    The camera works fine in manual focusing, but I can't get it to take picture under auto-focusing.  I wonder is there something wrong with the setting that I don't know or is that I got a lemon?  Thanks Blurry Text On Model After Render How can I disable my internet connection from terminal? I don't want my users to change the system wallpaper. How do I the lock the wallpaper down? Update  Users are changing wallpaper by right clicking the picture and choosing set as wallpaper. How to restrict this behaviour?  Is it possible to remove Backgrounds Tab from Appearance Preferences? If yes how? When replacing non-grounding type receptacles is a GFCI receptacle required, or can any GFCI device be used? Under the NEC (it was 210(7)(d)(3), but appears to have moved around some), it is permissible to replace an ungrounded receptacle with a GFCI receptacle, then wire further ungrounded or non-grounded (i.e. a NEMA 5 without a ground wire hooked up) receptacles on the circuit to the load side of the GFCI, all in lieu of running a ground wire in order to replace the existing ungrounded receptacles with grounded receptacles.  However: does this permission extend to replacing the branch circuit overcurrent protection device with a GFCI-type circuit breaker (or a CAFCI breaker that provides ground fault protection for personnel)?  Or can only GFCI receptacles be used for this retrofit? The NEC's verbiage does not make mention of breaker-type GFCIs...