Suppose $a$, $b$, and $c$ are integers. Prove that if $a \mid  b$ and $a  \mid  c$, then $a \mid  (b +c)$
Let $a$, $b$, and $c$ be integers, where a $\ne$ 0. Then $$ $$ (i) if $a$ | $b$ and $a$ | $c$, then $a$ | ($b+c$) $$ $$ (ii) if $a$ | $b$ then $a$|$bc$ for all integers $c$; $$ $$ (iii) if $a$ |$b$ and $b$|$c$, then $a$|$c$.  Prove that if $a$|$b$ and $b$|$c$ then $a$|$c$ using a column proof that has steps in the first column and the reason for the step in the second column.  My book is really vague. I'm not really sure what to do..  First I was thinking something like this: $a\mid b\Rightarrow b=as$ and  $b\mid c\Rightarrow c=bt$ and $a\mid c\Rightarrow c = au$ $$ $$ $b + c + c = b + 2c$  $$ $$ $=as + 2(bt+au)$  Yea then I just got lost.
Additional Boxes - Criteria, Longevity, Manipulability This feature has existed for a while (I'm not sure since when exactly), but I'm mainly interested in the Gen VI implementation (hence the tag).  Usually, at the start of a game, you have some number of Boxes available. In B/W, those were 8, in X/Y I think it was 15. Under certain conditions though, you receive additional Boxes that just appear at the end of your current Boxes list. Which brings me to Part 1 of my question:     1) What are the exact conditions for additional Boxes to appear and do you always get 8 additional Boxes? (Using Pokemon Bank reveals that in X/Y, the maximum number of Boxes is 31, which I have experienced to be unlocked in 2 steps of 8 Boxes each.)   Since I'm a person who really likes to order and organize stuff, my Boxes have names and are used to categorize my Pokemon. This results in some Boxes ending up empty for a while most of the time. So here's to     2) Do additional Boxes you receive stay forever or do they disappear if the conditions for them are no longer met?      3) If they disappear, is there a way (such as renaming or putting any Pokemon in the last Box) to make them stay no matter what, even if you released all your other Pokemon? How do I remove directories that begin with a dot after my server was infected? My server had been infected and the attacker spread a bunch of .5randomletters folders around my file system, with malicious code inside of them. I want to search for all the folders that begin with a dot, review them and then remove them.  How can I do that? How do I know what symbols/characters are available in a font package? I usually use xelatex, but I'm exploring pdflatex now. When loading specific fonts in xelatex, I can always know what characters the font has available, since I can inspect the font file in any font viewing program, or I can find out the hard way by just entering the unicode and see if the character is displayed in the output. But how does this work with a font package?  Say, for instance, that I would like to use phonetic symbols in the  font. How do I know whether the font has phonetic symbols (I assume it does), and how do I know what command I need to type in order to make them appear? My only resource is , but this only tells me what characters/commands there are available in LaTeX2e by default (depending on the font encoding), and what symbols I can find in specific packages. Phonetic symbols, for instance, are said to be in the tipa package. Does that mean I am locked to tipa if I want to use phonetic symbols with pdflatex?  Here's a pointless MWE:  \documentclass{article} \usepackage{gentium} \begin{document} Please give me some phonetic symbols here? \end{document} Kill some processes by .exe file name Where can I find stock or custom ROMs for my Android device? Why does connecting a high-current device to my digital circuit cause weird behavior? What is postselection in quantum computing? A quantum computer can efficiently solve problems lying in the complexity class . I have seen a claim the one can (potentially, because we don't know whether BQP is a proper subset or equal to PP) increase the efficiency of a quantum computer by applying postselection and that the class of efficiently solvable problems becomes now postBQP = .  What does postselection mean here? I know  that uses $x={\sqrt{2}}^{\sqrt{2}}$ to prove that there must exist an irrational to an irrational power that evaluates to a rational. But I don't know if $x$ itself is known to be irrational or rational.  Could someone enlighten me? Thanks! Why does this not work with an imaginary exponent? Can we get a better wording for protected questions? For context, see .  In essence:     "This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation (not including the ) on this site."   Suggested addition to existing message is in bold. PSTricks error "Undefined control sequence  \c@lor@to@ps" I have a problem with this file tex:  \begin{document} \begin{figure} \begin{center} \begin{pspicture}(0,0)(1,3.8) \psline{-&lt;}(0,0)(0,.2) \psline{-}(0,.2)(0,.4) \psline{*-&gt;}(0,.4)(0,.9) \psline{-*}(0,.9)(0,1.4) \psline{-&lt;}(0,1.4)(0,2.4) \psline{-}(0,2.4)(0,3.4) \psline{*-}(0,3.4)(0,3.6) \psline{&gt;-}(0,3.6)(0,3.8) \rput(1,.4){$y=0$}%testo \rput(1,1.4){$y=1$} \rput(1,3.4){$y=3$} \end{pspicture} \end{center} \caption{} \end{figure} \end{document}   but I don't understand where is the error. Background on color perception:  Due to the nature of the human eye and visual processing by the brain, there's an optical illusion that the same color will look different depending on the background.   It is known as the , illustrated below — center rectangles are identical:   Source:   An illustration  makes the effect of the background's color even more obvious:     Source:   (Note how the two upper-most colors in the second image appear the same, but are actually completely different.)    What about web design?  My question pertains to web design, specifically in this case, the color of link text on different backgrounds.  For a given color of text, how can one find the complementary color to make it appear the same on a certain background?    Let's take a look at an example for this question:  Consider my example below, with yellow text #FFF000 on the following backgrounds:    white #FFFFFF, black #000000, and grey #555555.          As expected, doesn't look the same at all. ()  How can we make it look the same on all three? Clearly, the text on the white background needs to be a darker yellow, and the one on the grey needs hue/tone adjustments. (The surrounding text will also affect it — but that's likely harder to account for.)  Can this perfect color somehow be calculated, based on the Hex, RGB or another system? That's where the crux of this question lies, as it would be much more efficient than manually approximating. Who are the clients of my shared iTunes library? Is it possible to see who is currently connected and listening to a shared iTunes library? Is it possible to see which songs they listen to and how many/often?  It's not really important; I'm just curious about it. Would be nice to know which colleagues love my music… "Outside" or "outside of"? Console flashes on launching, then closes (ms-windows) ASP.NET MVC Relative Paths In my applications, I often have to use relative paths. For example, when I reference JQuery, I usually do so like this:  &lt;script type="text/javascript" src="../Scripts/jquery-1.2.6.js"&gt;&lt;/script&gt;   Now that I'm making the transition to MVC, I need to account for the different paths a page might have, relative to the root. This was of course an issue with URL rewriting in the past, but I managed to work around it by using consistent paths.  I'm aware that the standard solution is to use absolute paths such as:  &lt;script type="text/javascript" src="/Scripts/jquery-1.2.6.js"&gt;&lt;/script&gt;   but this will not work for me as during the development cycle, I have to deploy to a test machine on which the app will run in a virtual directory. Root relative paths don't work when the root changes. Also, for maintenance reasons, I cannot simply change out all the paths for the duration of deploying the test - that would be a nightmare in itself.  So what's the best solution?  Edit:  Since this question is still receiving views and answers, I thought it might be prudent to update it to note that as of Razor V2, support for root-relative urls is baked in, so you can use   &lt;img src="~/Content/MyImage.jpg"&gt;   without any server-side syntax, and the view engine automatically replaces ~/ with whatever the current site root is. Sheafs and closed immersion I know it's possible to match a word and then reverse the matches using other tools (e.g. grep -v). However, is it possible to match lines that do not contain a specific word, e.g. hede, using a regular expression?   Input:  hoho hihi haha hede   Code:  grep "&lt;Regex for 'doesn't contain hede'&gt;" input   Desired output:  hoho hihi haha How can you find and replace text in a file using the Windows command-line environment? Short story about interstellar travel gone wrong Trying to find a story I read a couple of years ago in an anthology, not sure when the story dates from but likely to be recent - the last decade or so.   The story has passengers travelling between stars via a series of jumps. The calibration of the drive is affected by a special metal, any of which has to be stored in a shielded compartment (I think the jumps were measured in 'degrees'). One of the passengers unwittingly takes some of this metal on board, which throws off the calculations for the jumps, and they jump into an alternative universe where there are no signs of any humans ever existing. They start to try to reverse their steps to get back to the known universe, but the reverse calculation seems impossible.  That's as much as I read of the story before I lost the book. After many dreams (feverish nightmares!) re-living this story, I need to know how it ends. I know it's not much to go on. The story name, author, or the anthology it appears in would be fantastic.