Prove that if $\{a,b,c\}$ forms a Pythagorean triple, then at least one of the three numbers must be divisible by $3$. Need some help with this one.
Show that if $x$, $y$, $z$ are integers such that $x^2 + y^2 = z^2$, then at least one of them is divisible by $2$, at least one is divisible by $3$, and at least one is divisible by $5$.  I know that $x,y,z$ are sides of a right-angled triangles. If I start putting values, then I can show, but have no clue about mathematical approach.  How do I solve it mathematically, using modular arithmetic?
When is the present perfect tense used instead of the past tense? When is the present perfect tense used instead of the past tense?    I know that the present perfect tense is used when some adverbs (e.g., never, ever) are present in the sentence; the same is true for sentences like the following one.     When you returned, I have been at home since 3:00 PM.   In which other cases should I use the present perfect? Do the following sentences require it?     I have walked downtown everyday for a year.        I have been at home since 3:00 PM. Cannot login to chat.meta.stackexchange.com; server error when trying to access chat homepage I just recreated my MSE account after having deleted it a few years ago. Now, I'm trying to log in to Meta Stack Exchange Chat, and whenever I try to do so, it just fails silently and I'm still logged out.  Additionally, in the same browser session that I'm supposedly logged out, I get a server error when accessing the chat homepage or the chat profile page of any user. I don't get any errors when trying to access any of these pages in a private session. I also was able to sign in to Stack Exchange Chat and Stack Overflow Chat without problems, since my accounts on SO and my parent site for SE chat haven't been deleted.  I think it might be because it's trying to sign me in to my old chat profile, which was not deleted, but that is associated to my old, deleted site account, which clash and thus cause it to fail. I recently bought a new laptop with following configuration   CPU: Intel i5 2450 RAM: 8 GB DDR3 GPU: NVIDIA GeForce GTX 520 1GB   I've setup a virtual machine using VirtualBox with Windows 7 as guest. I want to use my (host's) graphics card in it but the virtual machine is only showing 128 MB of video memory.  I am using Bumblebee and start the virtual machine using   $ optirun VBoxManage startvm "Windows 7"   Please suggest me how should I use my graphics card for 3D acceleration in virtual machine. Sauron's sight details? I am aware that the "Eye of Sauron" in Peter Jackson's movies was not Sauron's actual physical form.  What the answers at '' don't indicate is, despite this, what abilities of vision Sauron may have had.  Did he have the ability to see from afar? If so, what are his limitations or details of this power? Is it some kind of Elvish-sight like Legolas but stronger? Was it a magical power? Or perhaps with the help of the Palantir (I'm not sure of this last one because I dont remember if the Palantir could only see what other Palantíri were seeing or anything he wanted)? Make gedit embedded terminal colours the same as the default terminal I've just updated to 11.10 and now my embedded terminal in gedit does not match my default terminal. I hate looking at a white and black terminal. Does anyone know how to make the embedded terminal match the default terminal?  This is a screenshot:    I want my embedded terminal in gedit to match my default terminal. Disable all software installation How does LIGO remove the effects of environmental noise? Since LIGO is dealing with readings at nanometers, events such as vehicles driving nearby, and constant (but extremely minor) tremors of the earth can cause movement with the mirrors at nanometers.  Such tiny movements are extremely difficult to measure, but can easily cause errors when using such a small scale.  How does LIGO remove these factors, and take precautions to prevent false readings due to noise? How can I create a coffee cream effect with Photoshop? Today I would like to create a coffee cup with the cream floral on it as a text by using Photoshop. It looks like you order a cup of hot cappuccino and the bartender will draw a flower with cream on the surface of coffee but now I want to make it like a text to present a tittle.     I just need to do the cream text effect, no need to draw the whole cup of coffee since i can't use some images. So anyone can tell me how to do this cream effect on Photoshop.  Thanks in advance for any feedback! In this MWE, I want to obtain bold letter for the letter "H" (compare the two "H") :  \documentclass[10 pt]{beamer} \usetheme{default} \usepackage{lmodern} % load a font with all the characters \usepackage{fontspec} \usepackage{cmbright} \SetSymbolFont{largesymbols}{normal}{OMX}{iwona}{m}{n} \usepackage{amsmath}%loads amsmath that should go before unicode-math \usefonttheme{professionalfonts} \defaultfontfeatures{Mapping=tex-text}  % This seems to be important for mapping glyphs properly  \begin{document} \begin{frame} \[ \mbox{First bold H : }\|x\|_{\mathbf{H}},\; \mbox{Second bold H : }\mathbf{H}  \] \end{frame} \end{document}   but I don't know how doing it. THe "H" produced by this MWE is tiny but no bold effect. If I use      \SetSymbolFont{largesymbols}{normal}{OMX}{iwona}{m}{n}   I obtain bold tiny "H" but the font used still not the same. I hope somebody has already figured this one out. I just installed Geoserver 2.9 on a vanilla Ubuntu 16.04 distro. The Geoserver 2.8 method of enabling CORS with the shanbe.hezoun class does no longer work with Jetty 9.2.13.  There are mentions that CORS support is already packaged with Jetty 9.2.13 in the jetty-servlets.jar.  The Jetty lib which is compiled with Geoserver contains a jetty-servlet-9.2.13.v20150730.jar in geoserver/lib but not jetty-servlets.9.2.13.v20150730.jar. Are these supposed to be the same jar with a different name?  It should be possible to enable CORS either in geoserver/etc/webdefault.xml or in geoserver/webapps/geoserver/WEB-INF/web.xml.  My understanding is that the webdefault.xml is applied first and the web.xml thereafter.  I have tried following filter in both xml. I haven't got as far as adding a filter mapping. Adding the filter alone will cause the Geoserver/Jetty service to not start proper.  &lt;filter&gt;     &lt;filter-name&gt;cross-origin&lt;/filter-name&gt;     &lt;filter-class&gt;org.eclipse.jetty.servlets.CrossOriginFilter&lt;/filter-class&gt; &lt;/filter&gt; Get and Set Screen Resolution Phase shift of 180 degrees of transversal wave on reflection from denser medium Can anyone please provide an intuitive explanation of why phase shift of 180 degrees occurs in the Electric Field of a EM wave, when reflected from an optically denser medium?  I tried searching for it but everywhere the result is just used.The reason behind it is never specified. Why did Aragorn and Theoden ride out at the Hornburg? In the Battle of the Hornburg from The Two Towers, Aragorn and Theoden say "We must ride out." Why did they do this?  Was it just a matter of honor? Did they ride out on their horses in the middle of battle because they thought it was the end, as they were outnumbered? Was it because Aragorn knew Gandalf was coming so they were confident they were about to win? Converting GeoPDF to vector format like CAD or shapefile? I want to convert a GeoPDF that has vectors entity in it to a vector format and preserve the real coordinates of these vectors. The GeoPDF has been exported from ArcMap and has layers in it.  I have try GDAL, but it rasterize the PDF before exporting it. Which is not good for me. Also I don't want a solution that would convert this image back to vectors, too complicated for the data I have.  Is someone aware of any tool that can do that? The resulting format can be a CAD file or a shapefile or anything that is a vector format. How to calulate mosfet response time with load? I would like to calculate the response time of a PMOS circuit and of the circuit  Could you tell me how to do it?    I'm using this transitor.   I tried to use the rise time of the PMOS to find it can switch (and i don't know if that's the good way), but I don't now the results with the load. CL en RL. I assume CL at t=0ms is equal to 0V. I tried to use dv/dt, but I failed VIN1 = 28V. VCMD is a command  Thanks.  edit: I made a mistake. The Zener is also connected to the source. Sorry How do I hide a parent div if the child div doesn't contain an  tag? Why does this sort ignore the +/- character prefix? Expected number of games played in a game with 'win by two' rule. Calvin and Hobbes play a match consisting of a series of games, where Calvin has probability p of winning each game (independently). They play with a “win by two”” rule: the first player to win two games more than his opponent wins the match. What is the expected number of games played? My attempt: I use Markov chains. Let $S$ denote that starting state, $W$ and $L$ denote the states where Calvin and Hobbes win respectively. Then, I have $$\mu_S = 1+ p\mu_W+q\mu_L $$ $$\mu_W = 1+ q\mu_S $$ $$\mu_L = 1+ p\mu_S $$ where $\mu_j$ denote the expected number of games from state $j$. Solving the system, I get the solution $\mu_S = \frac{2}{1-2pq}$ which I think is correct. I however would like to know how to approach it with geometric distribution instead of Markov chains. I read one solution where the number of games is given as $1+\text{Geom}(p^2+q^2)$ but I do not understand it. My understanding is that for the game to stop, the required probability is $p^2 + q^2$ since this is the probability that either Calvin or Hobbes win two games in a row. How to use this information in a Geometric random variable? Thanks. How do I clone a generic list in C#? Let $A$ be a ring with $0 \neq 1 $, which has $2^n-1$ invertible elements and less non-invertible elements. Prove that $A$ is a field.