As in the title, I'm trying to prove $-12^{482}$ has a multiplicative inverse modulo 175. Generally, I believe I could do this by showing that $gcd(-12^{482}, 175)=1$, (i.e that they're relatively prime) but I'm not sure about how to deal with the exponent without fully computing $-12^{482}$.
How do I go about proving this?  If $\gcd(a,b)=1$ and $\gcd(a,c)=1$, then $\gcd(a,bc)=1$. I'm very confused with gcd proofs.
Best way to reverse a string I've just had to write a string reverse function in C# 2.0 (i.e. LINQ not available) and came up with this:  public string Reverse(string text) {     char[] cArray = text.ToCharArray();     string reverse = String.Empty;     for (int i = cArray.Length - 1; i &gt; -1; i--)     {         reverse += cArray[i];     }     return reverse; }   Personally I'm not crazy about the function and am convinced that there's a better way to do it. Is there? Studying the equation $\ln(x)- 1+ x^2=0$ I have a large amount of duplicate records (25.000) in an even larger table (250.000 records). I've done the detection using the reporting function. The criteria I used to detect the duplicates were 3 fields (unique account identifier, unique user identifier and date). Based on a list of criteria, I would like to either merge or delete these records. I've got a handle on the logic associated with the merging and the deleting but what I'm having trouble with is defining the duplicate detection process.   I'm looking at either creating a custom object to take care of the detection or dumping the records into an SQL database and taking care of it offline. Any thoughts on which one of the two is the best approach?  Edit:  After reading the first comment, I think the more appropriate question would be "How do I approach the problem in SOQL".  The object I'm working with is the Event object and I'm really only looking at the standard fields, AccountId, OwnerId and ActivityDate. From what I've found online, there's no easy way to find records that have the same value for these three fields. I think it could be accomplished by nesting SOQL queries but I'd like to use a different approach if at all possible. Sorry if this is a stupid question, but I am mainly working with 5V PIC circuits and now want to integrate a component (nRF24l01+) that needs a 3.3V supply.  As I have a box full of ~7V wall-warts, I normally use a 7805 or similar LDO regulator to power the circuit. Now I want to also get 3.3V without using a separate external supply.  I'm looking for guidance on the "right" way to do this. I think the options are:   Use a voltage divider on the output of my 5V regulator to get 3.3V Use a 3.3V LDO regulator fed directly from the 7V wall-wart   Use the output of the 5V regulator as input to a 3.3V regulator   I can see that all of these would work (would they?) and that option 3 seems to waste least power, but I don't know what gotchas might arise by chaining two regulators like this.  What things do I need to consider when making this choice? Where did the world go? I was playing on my main world when my laptop ran out of power.  I found a charger and woke it up, but Minecraft had crashed.  Now, it doesn't appear in the world selection screen!  I have verified that it still exists in the Minecraft folder, and the directory contents look to be in order.  Does Minecraft save an internal flag somewhere when it crashes hard?  If so, how do I reset it? How should I unit test multithreaded code? Intro  I'm working on a lightweight package that allows you to put any content (typically LaTeX code) in an environment, then use that content multiple times, each time scanned separately, so you can use it both as verbatim output and interpreted as LaTeX.  Its main purpose is to typeset LaTeX code and its 'result' side-by-side; useful for documenting TeX packages without having to copy/paste code. But it has other purposes as well.  After experimenting with \scantokens until I gave up in tears, I arrived at the filecontents environment, which almost solved my problem singlehandedly!  The Problem  The problem is, when reintroducing filecontented code through \input, LaTeX adds an implicit newline at the end, which translates to whitespace that I can't seem to get rid of.  I've seen many semi-related questions floating around, with answers employing techniques like:   reading a file line-by-line, using \everyeof in some clever way, playing with \endlinechar or using packages such as catchfile   but none of them provides a clear answer to the following question:  The Question  How can I \input a file without getting the implicit newline?  Minimal Working Example  \documentclass[margin=1mm]{standalone}  \usepackage{filecontents}  \begin{filecontents*}{tmp.tex}     \LaTeX{}\end{filecontents*}  \begin{document}     \fbox{\input{tmp.tex}} \end{document}     Note the extra whitespace to the right of the box.  Note also that I'm definitely not adding any explicit whitespace. Even a manually produced file (so, not using filecontents) with no newlines at all exhibits the same problem. How to make a figure span on two columns in a scientific paper? how to install drivers for wifi adaptor Ralink 148f:7601 on ubuntu? Do local flows of left-invariant vector fields satisfy $\Phi_X^t\circ L_x=L_x\circ \Phi_X^t$? Given $G$ a lie group and $X$ a left-invariant vector field. Let $\Phi_X^t$ be the local flow of $X$. Why can we conclude that $\Phi_X^t \circ L_x=L_x \circ \Phi_X^t$? Thanks! Can you win a game of Splendor if you still have cards in reserve? Is there a way to have coloured hyperref hyperlinks in the PDF, but have them print "as if" they were black? I like having my hyperref hyperlinks a different colour, so it's obvious that they are links. But when I print a copy of the PDF, I don't want the links to be coloured, because you can't click them, so it doesn't matter and it's distracting.  Other than just having a separate "for printing" version of the PDF where I turn off the hyperlinks, is there a way to have the printed version come out without the coloured links? On all of the other SOFU sites a downvote means The question is unclear or not useful.  However, here on meta, a downvote has become a sign of I don't agree with the question, suggestion, feature request, bug, etc. being described in the question.   What does a downvote on Meta really mean? What other reasons do people downvote on Meta? Why is it okay for meta to redefine the meaning of the downvote?  The tooltips and instructions are all the same here on meta.   Note: I'm not saying I disagree with the practice, just curious. Every planar graph has a vertex of degree at most 5. I am trying to prove the following statement, any help!?     Prove that every planar graph has a vertex of degree at most 5. Evaluating $\int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x$ How does gravity escape a black hole? "Which" vs. "what" — what's the difference and when should you use one or the other? How do I locate all files that include specific word as part of name and end with specific extension? How to draw Venn diagrams (especially: complements) in LaTeX What I am up to is to write some exercises dealing with logical formulas for my students, like:     And the students should draw these formulas on . At the end of the lesson, I really would like to print the correct answer for them. I found a great resource on a forum thread at , which helped me a lot to make up some Venn diagrams with tikz, but have some problems with visualizing complements, like ~A.  A simple, modified version of the TeX file found on the forum linked above, can be seen below, which produces the following expression:    \documentclass{letter}   \usepackage{tikz}   \def\firstcircle{(90:1.75cm) circle (2.5cm)}   \def\secondcircle{(210:1.75cm) circle (2.5cm)}   \def\thirdcircle{(330:1.75cm) circle (2.5cm)}   \begin{document}     \begin{tikzpicture}       \begin{scope}     \clip \secondcircle;     \fill[cyan] \thirdcircle;       \end{scope}       \begin{scope}     \clip \firstcircle;     \fill[cyan] \thirdcircle;       \end{scope}       \draw \firstcircle node[text=black,above] {$A$};       \draw \secondcircle node [text=black,below left] {$B$};       \draw \thirdcircle node [text=black,below right] {$C$};     \end{tikzpicture}   \end{document}   Which looks like:    Could anyone please help me out plotting/defining some expressions dealing with complements? A nice example could be:    That should look like: (image from )    I do not insist on the red color :)  I would like to use the simplest possible solution, as I would like to mass generate the exercises with the help of . So any suggestion dealing with gnuplot, R or any other opensource packages is welcome. Thank you!    UPDATE (25/01/2011): added details based on answers.  Thank you @Leo Liu, you helped me a lot! I modified a bit the code you suggested to be able to color the area outside of the two circles also (in the H universe), but have no idea how to set a background to that polygon also. The code:  \begin{tikzpicture}[fill=gray] % left hand \scope \clip (-2,-2) rectangle (2,2)       (1,0) circle (1); \fill (0,0) circle (1); \endscope % right hand \scope \clip (-2,-2) rectangle (2,2)       (0,0) circle (1); \fill (1,0) circle (1); \endscope % outline \draw (0,0) circle (1) (0,1)  node [text=black,above] {$A$}       (1,0) circle (1) (1,1)  node [text=black,above] {$B$}       (-2,-2) rectangle (3,2) node [text=black,above] {$H$}; \end{tikzpicture}   And the image generated:    I will also look for even odd rule in the near future which does not make sense for me at the moment but looks really simple and promising! 'Some index files failed to download' while updating When running sudo apt-get update I get these following errors:  Err archive.ubuntu.com quantal InRelease Err archive.ubuntu.com quantal-updates InRelease Err archive.ubuntu.com quantal-backports InRelease Err archive.ubuntu.com quantal-security InRelease Err archive.ubuntu.com quantal Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-updates Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-backports Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-security Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Reading package lists... Done W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-updates/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-backports/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-security/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-updates/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-backports/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-security/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Some index files failed to download. They have been ignored, or old ones used instead.   I am using Ubuntu 12.04. This is the command I used after installing Ubuntu from window installer. I am sure my laptop is connected to internet. What can I do now?