Prove Divisibility test for 11  "If you repeatedly subtract the ones digit and get 0, the number is divisible by 11"  Example:   11825 -> 1182 - 5 = 1177  1177 -> 117 - 7 = 110  110 -> 11 - 0 = 11  11 -> 1-1 = 0  Therefore 11825 is divisible by 11. Note 11825 = 1075*11  I was thinking that we let x = $a_ka_{k-1}.....a_1a_0$ where the following $a_i$'s are the digits.
We know that $1331$ is divisible by $11$. As per the $11$ divisibility test, we can say $1331$ is divisible by $11$. However we cannot get any quotient. If we subtract each unit digit in the following way, we can see the quotient when $1331$ is divided by $11$.  $1331 \implies 133 -1= 132$ $132 \implies 13 - 2= 11$ $11 \implies 1-1 = 0$, which is divisible by $11$.  Also the quotient is, arrange, all the subtracted  unit digits (in bold and italic) from bottom to top, we get $121$. Which is quotient. I want to know how this method is working? Please write a proof.
I was reading through the Wikipedia article on  and came across the fact that the most distant Quasar is 29 Billion Light years. This is what the  exactly says      The highest redshift quasar known (as of June 2011[update]) is ULAS-J1120+0641, with a redshift of 7.085, which corresponds to a proper distance of approximately 29 billion light-years from Earth.   Now I come to understand that the  singularity is believed to be around 13.8 Billion years ago.  So how is this possible? Does the presence of such a quasar negate the Big Bang Theory?   I'm not a student of Physics and was reading this out of (whimsical) curiosity. Is there something I'm missing here or the "proper distance" mentioned in the fact is a concept that explains this?  Edit: My Bad! Here's how..  A simple google search led me to  which says the farthest quasar found is 12.9 billion LYs and not 29 billion. So in the end we have just proven that wikipedia needs more moderation. I came across this example when trying to mount a usb device inside a openvz container and I have never seen the construct in the second line before. Can you explain what it signifies?  #!/bin/bash . /etc/vz/vz.conf How do I get a book graphic and description from the Amazon Book API? sends an ISBN number to Amazon and gets back a small bit of XML including author, title, and publisher.  However, I also want to get small, medium and large graphic and book descriptions of the title.  Problem: I can find no REST URL examples/documention that work, either at Google or when logged into my "AWS Account" at Amazon Associates.  I find a lot of examples from 2003-2005 but they are all out-of-date and give errors, it seems that Amazon's cloud web services have obfuscated their simple REST API documentation for their books.  Can anyone point me to some documentation on how I can get detailed information about books at Amazon via REST/XML?   Here's what I have tried so . Can I use I2C over 2m cables? Got a new iPhone - where am I supposed to receive the verification code? I am trying to put in a switch and a light it controls in a storage building and  is how I wired it.   Click for larger view  When I checked the wiring it registered 120V when the switch was on, and right around 50V when the switch was off.  Why would power still be getting to the light (I'm guessing there is a problem with the switch) and why would it only be 50V?  Oh, and where everything branches off at the beginning I have a junction box there where all of the wires are joined there. Reading shapefile using C#? How can you find and replace text in a file using the Windows command-line environment? I am writing a batch file script using Windows command-line environment and want to change each occurrence of some text in a file (ex. "FOO") with another (ex. "BAR"). What is the simplest way to do that? Any built in functions? Which Harry Potter works are considered canon? How can I benchmark my HDD? I've seen commands to benchmark one's HDD such as this using dd:  $ time sh -c "dd if=/dev/zero of=ddfile bs=8k count=250000 &amp;&amp; sync"   Are there better methods to do so than this? Was Elrond, in Tolkien's opinion, more inherently powerful than Galadriel? I always thought it was the other way around, but then I read this (Letter 246):     "Of the others only Gandalf might be expected to master him – being an emissary of the Powers and a creature of the same order, an immortal spirit taking a visible physical form. In the 'Mirror of Galadriel', 1381, it appears that Galadriel conceived of herself as capable of wielding the Ring and supplanting the Dark Lord. If so, so also were the other guardians of the Three, especially Elrond. But this is another matter. It was part of the essential deceit of the Ring to fill minds with imaginations of supreme power."   This "especially Elrond" confuses me. I don´t think that Elrond is inherently more powerful than Galadriel, who is described as the greatest of the Noldor, except for Fëanor (and possibly even his equal).  A quote from Unfinished Tales says:     "...In this he (Tolkien) emphasized the commanding stature of   Galadriel already in Valinor, the equal if unlike endowments of   Fëanor; and it is said here that so far from joining in Fëanor's   revolt she was in every way opposed to him."   Furthermore,      "A queen she was of the woodland Elves, the wife of Celeborn of   Doriath, yet she herself was of the Noldor and remember the Day before   days in Valinor, and she was the mightiest and fairest (so even   fairer than Arwen I would guess) of all the Elves that remained in   Middle-earth."   (The Silmarillion, of the Rings of Power)  And again...     "Galadriel was the greatest of the Noldor, except Fëanor maybe, though   she was wiser than he, and her wisdom increased with the long years."   (the Shibboleth of Fëanor)  Restating it:     "These two kinsfolk (Fëanor and Galadriel), the greatest of the   Eldar in Valinor, were unfriends for ever." "Who together with the   greatest of all the Eldar, Luthien Tinuviel, daughter of Elu Thingol,   are the chief matter of the legends and histories of the Elves."   (Unfinished Tales)  The latter means that Luthien, Fëanor and Galadriel are the chief matter of the history and legends. It seems to me that Tolkien thought of those three as the creme de la creme of the elven crop.  Reading all that I have serious problems believing that Elrond could be superior to her. As great as Elrond is, descending from Melian, he is no equal of Fëanor, who is said to be the mightiest of all elves, and if Galadriel is his equal, then there is no way that Elrond would be mightier than her.  Maybe I just read this "especially Elrond" wrong; it could as well mean that especially Elrond was deceived into thinking that he could master the ring, because that was the deceit of the ring. And Tolkien said explicitly that only Gandalf could possibly master it.  So, what do you think? I haven't been able to do this exercise:     Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse   image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X \subseteq B$.   Since $X \subseteq B$, all the elements in $X$ belong to the codomain of $f$.  Since $f$ is surjective, it means that all elements in the codomain $B$ have some preimage in $A$. Given that $X \subseteq B$, all elements in $X$ must also have a preimage in $A$.  Have $\triangle = f^{-1}(X)$, $\triangle$ is now a set containing the preimages of the elements in $X$. Because of this, $\triangle \subseteq A$.  If we evaluate $f(\triangle)$, we...... nope, I don't know what I'm doing now.    What do you think? Review Queues and Ignored Tags Steps to Recreate:   Have enough rep to use some of the review queues. Ignore a popular tag. Go to your profile-> prefs-> check 'hide ignored tags'. Open a review queue.   You will get questions from your ignored tag(s) in the review queue.  Personally, I ignored a tag on Movies.SE (and Scifi.SE) for a film that is out elsewhere in the world, but not locally.  It's a simple way to avoid being exposed to spoilers.  I then load up the First Posts review queue on Movies.SE, and I get an answer to a question about the end of the film.  Can the review queues be altered to skip posts on your ignored tags when you have the 'hide ignored tags' option enabled? subsurf splits the mirrored model into two objects I tried to subsurf this model that I mirrored but when I use subsurf it splits the model into two objects.  Before Subsurf:   After Subsurf:   Any help would be appreciated. Input file in chapter title I cannot figure out how to import a file in a chapter title:  \chapter{\input{"Introduction.txt"}} \chapter{\texorpdfstring{\input{"Introduction.txt"}}{Introduction}}   Both of these fail with      Use of \@chapter doesn't match its definition.   Any help will be greatly appreciated. find exec '{}' not available after > Exec allows us to either pass all arguments at once with {} + or to pass them one by one with {} \;  Now let's say I want to rename all jpeg, no problem doing this:  find . \( -name '*.jpg' -o -name '*.jpeg' \) -exec mv '{}' '{}'.new \;   But if I need to redirect output, '{}' isn't accessible after redirection.  find . \( -name '*.jpg' -o -name '*.jpeg' \) -exec cjpeg -quality 80 '{}' &gt; optimized_'{}' \;   This wouldn't work. I'd have to use a for loop, storing find's output into a variable before using it. Let's admit it, it's cumbersome.  for f in `find . \( -name '*.jpg' -o -name '*.jpeg' \)`; do cjpeg -quality 80 $f &gt; optimized_$f; done;   So is there a better way? Different colors for different types of blocks in beamer In a beamer presentation, I’d like to set up different colors for normal blocks, definition blocks, theorem blocks, proof blocks, example blocks…  Ideally I’d set those up using \usebeamercolor commands. I can do it for example blocks, and for all other kinds of blocks at once, as illustrated in the sample below (compiled using XeLaTeX). But I can’t find how to do it for specific, non-example, types of blocks.  I suppose “” points towards some solution, but I’d prefer (somehow) inheriting from the existing special block environment definitions rather than define new environments from scratch.  \documentclass{beamer} \usepackage{fontspec}  \useinnertheme[shadow=true]{rounded}  \setbeamercolor{block title}{use=structure,fg=structure.fg,bg=structure.fg!20!bg} \setbeamercolor{block body}{parent=normal text,use=block title,bg=block title.bg!50!bg}  \setbeamercolor{block title example}{use=example text,fg=example text.fg,bg=example text.fg!20!bg} \setbeamercolor{block body example}{parent=normal text,use=block title example,bg=block title example.bg!50!bg}  \begin{document} \begin{frame}     \begin{block}{My block}         A block.     \end{block}     \begin{definition}[My definition]         A definition.     \end{definition}     \begin{theorem}[My theorem]         A theorem.     \end{theorem}     \begin{proof}[My proof]         A proof.     \end{proof}     \begin{example}[My example]         An example.     \end{example} \end{frame} \end{document} How to efficiently count the number of keys/properties of an object in JavaScript? What's the fastest way to count the number of keys/properties of an object?  It it possible to do this without iterating over the object?  i.e. without doing  var count = 0; for (k in myobj) if (myobj.hasOwnProperty(k)) count++;   (Firefox did provide a magic __count__ property, but this was removed somewhere around version 4.) Getting key with maximum value in dictionary? I'm having problems understanding the following situation. Suppose two 1-tonne cars are going with the same orientations but opposite senses, each 50 km/h with respect to the road. Then the total energy is   $$\begin{eqnarray}E=E_1+E_2&amp;=&amp;\frac{1\mathrm t\times(50\mathrm{km}/\mathrm h)^2}2+\frac{1\mathrm t\times(50\mathrm{km}/\mathrm h)^2}2\\&amp;=&amp;1\mathrm t\times(50\mathrm{km}/\mathrm h)^2\\&amp;=&amp;2500\frac{\mathrm t\times\mathrm{km}^2}{\mathrm h^2}.\end{eqnarray}$$  Now if we look at it from the point of view of one of the cars, then the total energy is   $$\begin{eqnarray}E=E_1+E_2&amp;=&amp;\frac{1\mathrm t\times(0\mathrm{km}/\mathrm h)^2}2+\frac{1\mathrm t\times(100\mathrm{km}/\mathrm h)^2}2\\&amp;=&amp;\frac{1\mathrm t\times(100\mathrm{km}/\mathrm h)^2}2\\&amp;=&amp;5000\frac{\mathrm t\times\mathrm{km}^2}{\mathrm h^2}.\end{eqnarray}.$$  I know that kinetic energy is supposed to change when I change the frame of reference. But I understand that then there must be some other kind of energy to make up for it so that the energy in the system stays unchanged. But I don't see any other kind of energy here. I only see two total energies of the same system that seem to be different. Could you explain this to me?  Please note that while I don't understand any physics, I do understand college level mathematics, so if necessary please use it. (I doubt anything more than high school maths should be needed here, but I want to say this just in case.)