I'm stuck on this basic problem,  Show that $39$ divides $17^{48}-5^{24}$  I am familiar with the basic arithmetic properties of modular. But I have absolutely no idea how I'm supposed to apply them to solve this type of question. Some advice on the pattern I'm supposed to be looking for would be very helpful.
I've read about Fermat's little theorem and generally how congruence works. But I can't figure out how to work out these two:   $13^{100} \bmod 7$ $7^{100} \bmod 13$   I've also heard of this formula:  $$a \equiv b\pmod n \Rightarrow a^k \equiv b^k \pmod n $$  But I don't see how exactly to use that here, because from $13^1 \bmod 7$ I get 6, and $13^2 \bmod 7$ is 1. I'm unclear as to which one to raise to the kth power here (I'm assuming k = 100?)  Any hints or pointers in the right direction would be great.
How do I help my third-grader focus on homework? My child just entered third grade, and the homework load is much higher this year, with a lot of writing and math drills. It is all work that she can do easily, but it's causing the whole family a lot of pain, because she's have a really hard time just sitting down and doing it — asking many questions about things she clearly actually knows, and getting distracted by the smallest thing going on around her. (Setting up a special quiet area isn't helping, because the distractions could be her pencil or a sudden burst of word association based on something in the lesson.)  So, a page of writing and a couple of pages of math which could be done in fifteen minutes total is taking an hour and a half. She can easily explain the reasoning behind the math problems, but when it comes to writing down what she just said, it's like pulling teeth. She'll slowly write a word, sounding it out in slow motion as she writes each word. And she'll say that she doesn't understand a question, but when she actually gets her mind pointed at it, she can write a great answer. And she's clearly picking up the concepts in class; I have no doubt that she actually understands, which is part of our frustration.  My wife feels like she's just wanting attention, but I think it's more about our daughter's own ability to focus. We both feel that sitting down and working painfully through each question isn't really helping her. That works, but my wife feels like it's just rewarding the behavior, and I feel like it's not really helping her learn any better focus on her own, which, you know, she'll need in actual life. And, we're happy to spend child with our kid, this pretty much the opposite of quality time.  Neither rewards or consequences seem to have any effect, and nor do charts. All of those things basically just seem to add another layer of complication, distraction, and misery. What else can we do? I'm looking for a word that can describe a person who does nice things for other people (e.g holding the door open, carrying someone's things) but only for self gain; this person only does nice things to make themselves look good in the eyes of others, but doesn't actually care about the well-being of the people they're helping. Why doesn't the definition of a category require an explicit notion of morphisms equality? I am learning basics of the category theory (CT).  I do understand that CT is a modern powerful framework to describe various branches of mathematics in a unified way.  I do admit that category's definition tells a lot about the whole thing and focuses on the most important properties, shadowing unnecessary details. However, the more I go forward, the more it feels like it should explicitly require a notion of morphisms equality (like it does with morphism composition). It feels so because equality or inequality of morphisms arises everywhere, since the very beginning: any commutative diagram eventually reduces to an equality of two different paths and many other things involve equality as well. Even composition is defined grounding upon an (implicit) equality!  I'll try to give an example of my concern. Consider $\mathbb{SET}$. Its objects are sets and morphisms are functions. Now, let's recall that functions are sets as well - to be more precise, a function $f : A \mapsto B$ is a subset of a $A \times B$ obeying certain properties, which I omit. Thus, one could argue that $f$ is both a morphism between objects $A, B \in Obj(\mathbb{SET})$ and an object defined as a set of pairs $\{ (a_1 \in A, b_1 \in B), ..., (a_n \in A, b_n \in B) \}$ and those are equal because both are essentially exactly the same thing.  I hope my concern is clear and I need some kind of "conceptual insight" from knowledgeable people to move forward. Please give me the right perspective. How can I make my browser lie about my operating system? I'm using Chromium (Version 30.0.1599.114 Ubuntu 12.10) under Linux. I'm trying to access  that claims that it doesn't work under Linux. Of course, this is most likely crazy since the web is platform-independent (unless they're using some kind of evil plug-in).  I've used a user agent switcher to supply a Windows user agent, but to no avail. Upon viewing the source, it appears that the site is using Javascript to determine the operating system. I'm looking for a way to get my browser to report in Javascript that it is Windows or MacOS.  Do you know how to do it? What does the x mean on MCB? Any clue? Or is it just there for better visual? Check all dependencies of a Record Type in the Organisation before deactivating it How to change the href for a hyperlink using jQuery How can you change the href for a hyperlink using jQuery? i have a huuuuge problem.  Created a snapshot and image from my instance...instance has 1000GB of disk.  Now i need to reduce it but everytime i try to create an image or a snapshot from that, i get the same error "Disk need to be at least 1000GB".  There is a way to reduce disk size directly on instance? Some workaround? How do I call functions loaded from a Static Resource JavaScript file? I'd like to put some shared Javascript to Resource file to reuse it between several components. This Javascript is general purpose set of utility functions. However, functions from my resource file are not available from the Components. Static resource (named TestScript):  function helloWorld() {      return 42; }   Component:  &lt;ltng:require scripts="{!$Resource.TestScript}" &lt;ui:button label="Do job" press="{!c.doJob}"/&gt;   Controller:  doJob: function(cmp, event, helper) {      var i = helloWorld();      // failed with Uncaught ReferenceError: helloWorld is not defined }   I don't like to use  &lt;ltng:require scripts="{!$Resource.***resourceName***}" afterScriptsLoaded="{!c.afterScriptsLoaded}" /&gt;   because my static resource is a set of functions and I'd like just use them as functions later in my controllers. How can I achieve that (if possible)? If not, what's the suggested solution? I'm trying to install a service using InstallUtil.exe but invoked through Process.Start. Here's the code:  ProcessStartInfo startInfo = new ProcessStartInfo (m_strInstallUtil, strExePath); System.Diagnostics.Process.Start (startInfo);   where m_strInstallUtil is the fully qualified path and exe to "InstallUtil.exe" and strExePath is the fully qualified path/name to my service.  Running the command line syntax from an elevated command prompt works; running from my app (using the above code) does not. I assume I'm dealing with some process elevation issue, so how would I run my process in an elevated state? Do I need to look at ShellExecute for this?  This is all on Windows Vista. I am running the process in the VS2008 debugger elevated to admin privilege.  I also tried setting startInfo.Verb = "runas"; but it didn't seem to solve the problem. Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian. Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.   This question is from group theory in Abstract Algebra and no matter how many times my lecturer teaches it for some reason I can't seem to crack it.   (Please note that $e$ in the question is the group's identity.)  Here's my attempt though...  First I understand Abelian means that if $g_1$ and $g_2$ are elements of a group $G$ then they are Abelian if $g_1g_2=g_2g_1$...  So, I begin by trying to play around with the elements of the group based on their definition...  $$(g_2g_1)^r=e$$  $$(g_2g_1g_2g_2^{-1})^r=e$$ $$(g_2g_1g_2g_2^{-1}g_2g_1g_2g_2^{-1}...g_2g_1g_2g_2^{-1})=e$$  I assume that the $g_2^{-1}$'s and the $g_2$'s cancel out so that we end up with something like,  $$g_2(g_1g_2)^rg_2^{-1}=e$$ $$g_2^{-1}g_2(g_1g_2)^r=g_2^{-1}g_2$$  Then ultimately...  $$g_1g_2=e$$  I figure this is the answer. But I'm not totally sure. I always feel like I do too much in the pursuit of an answer when there's a simpler way.  Reference: Fraleigh p. 49 Question 4.38 in A First Course in Abstract Algebra. How to create a color of the same brightness and saturation? Let's consider three colors: Red, Green and Blue. For blue I created a specific shade, something like #F2F5F7. If you see this color, it has a certain level of shade, brightness and saturation value of blue.  I want to create the same equivalent color but for Green and Red, how to get that? PHP regular expressions: No ending delimiter '^' found in How can I make a more realistic glass shader? Torus object stretched in array curve modifier I have a chain object. And i have a curve it follows. I got it to work for following curve, but chain object is not circles but both parts are more like stretched circles (ovals). Object i made with 2 toruses one rotated 90 degrees and then joined them together. I tried single torus too. But same issue. on a curve it just looks stretched oval. What can be the issue?   Thanks I'm trying to figure out how to assign higher priority to one PPA over another PPA. I had a look on bunch of the answers on this forum but none of them solves my problem.  We have two PPA sources on some of our servers which we stick into separate files in /etc/apt/sources.list.d . Some of these repos provide same packages (in terms of names) but different versions and possibly different binaries. Is there a way how we can PRIORITISE one repo over another ?  I read apt_preferences tutorial but it could not be more obscure and didn't find the answers for my question. Here is the situation. WE have the following sources:  $ ls -l /etc/apt/sources.list.d  total 12 -rw-r--r-- 1 root root 66 Jan  2 16:50 nginx-source.list -rw-r--r-- 1 root root 84 Jan  2 16:49 ruby-ng-experimental-source.list   where nginx-source.list content is:  deb     http://ppa.launchpad.net/nginx/stable/ubuntu precise main   and ruby-ng-experimental-source.list content is:  deb     http://ppa.launchpad.net/brightbox/ruby-ng-experimental/ubuntu precise main   Both of these sources provide nginx-full package of different versions etc. When I list the priorities ruby-ng-experimental-source.list is on top of output hence nginx-full is installed from that repo:  $ apt-cache policy nginx-full                                  nginx-full:   Installed: 1:1.2.3-1~38~precise1   Candidate: 1:1.2.6-1~43~precise1   Version table:      1:1.2.6-1~43~precise1 0         500 http://ppa.launchpad.net/brightbox/ruby-ng-experimental/ubuntu/ precise/main amd64 Packages  *** 1:1.2.3-1~38~precise1 0         100 /var/lib/dpkg/status      1.4.1-1ppa0~precise 0         500 http://ppa.launchpad.net/nginx/stable/ubuntu/ precise/main amd64 Packages      1.1.19-1ubuntu0.1 0         500 http://mirror.rackspace.co.uk/ubuntu/ precise-updates/universe amd64 Packages      1.1.19-1 0         500 http://mirror.rackspace.co.uk/ubuntu/ precise/universe amd64 Packages   How do I prioritize repo listed in nginx-source.list over the one which is on top of the list now ? I tried something like this:  $ cat /etc/apt/preferences Package: nginx-full Pin: origin http://ppa.launchpad.net/nginx/stable/ubuntu Pin-Priority: 1000  Package: nginx-full Pin: origin http://ppa.launchpad.net/brightbox/ruby-ng-experimental/ubuntu Pin-Priority: 100   But that has changed priorities of both sources to the same number - or something like that ? So this has NOT changed the priorities and nginx-full is still being installed form the brightbox PPA  $ apt-cache policy nginx-full   nginx-full:   Installed: 1:1.2.3-1~38~precise1   Candidate: 1:1.2.6-1~43~precise1   Package pin: (not found)   Version table:      1:1.2.6-1~43~precise1 1000         500 http://ppa.launchpad.net/brightbox/ruby-ng-experimental/ubuntu/ precise/main amd64 Packages  *** 1:1.2.3-1~38~precise1 1000         100 /var/lib/dpkg/status      1.4.1-1ppa0~precise 1000         500 http://ppa.launchpad.net/nginx/stable/ubuntu/ precise/main amd64 Packages      1.1.19-1ubuntu0.1 1000         500 http://mirror.rackspace.co.uk/ubuntu/ precise-updates/universe amd64 Packages      1.1.19-1 1000         500 http://mirror.rackspace.co.uk/ubuntu/ precise/universe amd64 Packages   Anyone has any idea how to achieve what I'm trying to do ? How to animate noise texture "seed"? It's possible to animate a change noise texture by animating the texture coordinates with a mapping node or by animating the texture space, however this only works on one axis at a time (on the other axes the texture will appear to be sliding across the object's surface).  For example, the top face has the desired effect but the other faces appear to be moving/sliding upwards:    Even with all three axes location and rotation animated it still appears to be sliding rather than statically changing.    Is there some way to make each face appear like the top face in the first gif but still be seamlessly blended with the other faces? This should work for other shapes as well.  In other words, I'm looking for a way to animate the "seed" of the texture. Creating a .desktop file for a new application Is smell a bad way of determining whether meat is still good? In the spirit of a  I would like to know if there's a basis for understanding why there  that have non-zero electric charge but zero spin?  Can such a quantum theory be written down and self-consistent? Do the current symmetries of our present QFTs not allow such a particle?