$$\gcd(\text {lcm}(x, y), \text {lcm} (x, z)) = \text {lcm}(x,\gcd(y, z))$$ where $x,y,Z$ are three integers. I came across this property on GCD but was not able to prove this.Can anyone suggest me a method.
I honestly have no idea how to approach this question. I have done easier questions eg. show $gcd(a,b)lcm(a,b)=ab$ by letting $a=Alcm(a,b)$ and $b=Blcm(a,b)$ and then using the fact that $A$ and $B$ are coprime. However this approach only works in two variables. Any help with the question in the title would be greatly appreciated, as well as the thought process behind the answer so that I can approach similar problems.
Error sending edittext in java server. See code How to resolve the "GPT detected. Please create a BIOS-Boot partition" error when using boot-repair? After installing Ubuntu, Windows didn't boot so I installed boot repair to fix this problem. But I am getting the message displayed below.  I tried to make a new partition with unformatted file system but still receiving the same message.  How do I fix it? Export multiple objects to .obj It is possible to export all object from a scene into different obj files (each object in a obj file separately)? Show every nonempty compact Hausdorff space is not the countable union of nowhere dense sets I know this proof is somewhat similar, or related to the Baire's Category Theorem but I can't seem to figure out how to do it.  The Baire Category theorem asserts that if X is a complete metric space or a locally compact Hausdorff space (my case), then the complement of a countable union of nowhere dense sets is always nonempty.  How can I use this on the proof? Determinant of a block lower triangular matrix I'm trying to prove the following: Let $A$ be a  $k\times k$ matrix, let $D$ have size $n\times n$, and $C$ have size $n\times k$. Then,  $$\det\left(\begin{array}{cc} A&amp;0\\ C&amp;D \end{array}\right) = \det(A)\det(D).$$  Can I just say that $AD - 0C = AD$, and I'm done? By $A\oplus B$ we denote the symmectric difference of two sets. The definition is $A\oplus B =(A\setminus B) \cup (B\setminus A)$. Now I hope to show that $A\oplus (B\oplus C) = (A\oplus B)\oplus C$. I remember that there's an elegant proof, but I forget its detail. (The first step is to show $A\oplus (A\oplus B) = B$, and maybe applying this formula, we can obtain associativity.) Migrated questions with a nonexistent author should still show the author's name in the questions list I just noticed . Its author doesn't have an account on this site, which makes it show up like this on the homepage:        It looks like a simple thing - but it gets in the way of the 'flow' of the page:        Couldn't we somehow get the author's name, so that migrated questions with authors that don't have an account here still show up, like this:        Much better. Please? What Is the Real Name of the #? I used to say "sharp sign" to refer to the # sign.   Today a friend told me that the correct term is number sign or hash sign or even just hash.  What is the difference between these options and what's the correct usage of the names for this sign? The  is a proposed propulsion mechanism for spacecraft in which some form of microwave device provides the propulsion. The consensus from the physics community, including many voices on this site, is that the proposed device would violate conservation of momentum.   Several experiments have been conducted, by three main groups: (i) , from Satellite Propulsion Research Ltd, (ii) a research group at the Northwestern Polytechnical University in Xi'an, China, and (iii) Eagleworks Laboratories at NASA's Lyndon B. Johnson Space Center in Houston, Texas. However, few of these experiments have been performed in vacuum (and none of the vacuum tests has been published in a peer-reviewed journal), which means the effect of air currents around the device cannot be ruled out.  This device has been the subject of many questions on this site, including          and their many closed duplicates.  Many of these questions ask what the proposed explanations are, but then veer off into territory of the form "assuming it is true, how does it work?", so they have been (rightfully) closed as counterfactual physics. I feel this sort of bundling does a lot of harm to the discussion, and there is some scope to ask what exactly is being proposed as an explanation. So:     What do the proponents of EmDrive claim is the theoretical justification of the device's function?   Please limit answers exclusively to references, and indicate whether they are peer-reviewed or not. Please limit answers to serious attempts at explanations, as opposed to vague claims about the 'quantum vacuum' - or indicate if an experimental paper does not have any such theoretical explanation or reference.  To be clear, this is not the place to discuss whether the explanations are correct or not. If you have specific questions about any of these sources, ask it separately. (However, if your question starts with "assuming that this works", be prepared for it to be closed.) If you feel there are not enough resources on this thread, you can  but please do not post another question asking people to look for more references. How to do one-class text classification? I have to deal with a text classification problem. A web crawler crawls webpages of a certain domain and for each webpage I want to find out whether it belongs to only one specific class or not. That is, if I call this class Positive, each crawled webpage belongs either to class Positive or to class Non-Positive.  I already have a large training set of webpages for class Positive. But how to create a training set for class Non-Positive which is as representative as possible? I mean, I could basically use each and everything for that class. Can I just collect some arbitrary pages that definitely do not belong to class Positive? I'm sure the performance of a text classification algorithm (I prefer to make use of a Naive Bayes algorithm) highly depends on which webpages I choose for class Non-Positive.  So what shall I do? Can somebody please give me an advice? Thank you very much! Total space of a finite rank locally free sheaf, Vakil's 17.1.4 & 17.1.G What's the difference between using > and >> in shell? So before I reformatted (to get rid of Windows), I simply used this command to open documents by putting it at the end of the ~/.bashrc file:  alias commandless= 'gnome-open ./Documents/the-linux-command-line.pdf; gnome-open ./Documents/linux.odt'   But now everytime I start the terminal, it says:  bash: alias: gnome-open ./Documents/the-linux-command-line.pdf; gnome-open ./Documents/linux.odt: not found   before even typing anything in, and when I run the command nothing happens, not even error messages. Cheap/free tool to view/convert Canon CR2 RAW format photo files on Windows? Prove that in a triangle $$r^2+r_1^2+r_2^2+r_3^2=16R^2-(a^2+b^2+c^2)$$ where the symbols have their usual meanings. I am looking for a smaller or elegant proof using trigonometry. A geometric proof would be super cool. I did prove it but it took me ages to do the calculation. Thanks. I have added the png transparent image with the setting and its not showing transparent black background is seen. I have the base color aluminum and i want to add that logo of png on to it. The image is black letter with transparent background why its showing in white letters and black background. May i know where im going wrong any suggestion or help. What's the easiest way to figure out if a window is opened modally or not?  CLARIFICATION:  I open a window calling   myWindow.ShowDialog();   I have a footer with an "OK" &amp; "Cancel" button that I only want to show if the window is opened modally. Now I realize I can set a property by doing this:  myWindow.IsModal = true; myWindow.ShowDialog();   But I want the window itself to make that determination. I want to check in the Loaded event of the window whether or not it is modal.  UPDATE  The IsModal property doesn't actually exist in a WPF window. It's a property that I have created. ShowDialog() blocks the current thread.   I'm guessing I can determine if the Window is opened via ShowDialog() by checking if the current thread is blocked. How would I go about doing that? Do effects with the duration of one minute end at the beginning or the end of your turn? The Rage class feature of the Barbarian allows you to rage for one minute given you either took damage or attacked every round. A round is defined as a 6-second interval. So the rage can last up to 10 rounds. Now my question is, does it end at the end of your 10th round or at the beginning of your 11th round? There are other effects like that, also concentration spells with the duration of 1 or 10 minutes. Sometimes those are activated before combat starts and end during combat. In general: Does an effect/concentration end at the end of your last turn during the timespan or at the beginning of your first turn after the timespan? In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading! Let $d=GCD(n^2+5,n^3-5n^2+6n)$ show that $d|630$