I am having trouble with this proof.     If $a,b$ are integers, then $(a+b)^3 \equiv a^3 + b^3 \pmod{3}$     What I am confused about is using the definition of modulo. So can $b^3 \pmod{3}$ be written as $3\, |\, [b^3 - (a+b)^3]?$ How does the $a^3$ term come into play with writing the modulo definition.   Also I apologize for the messiness of the math equations, I am new to math exchange and haven't gotten the TeX concepts down yet.
a, b are integers. p is prime. I want to prove: $(a+b)^{p} \equiv a^p + b^p \pmod p$  I know about Fermat's little theorem, but I still can't get it I know this is valid: $(a+b)^{p} \equiv a+b \pmod p$ but from there I don't know what to do.  Also I thought about $(a+b)^{p} = \sum_{k=0}^{p}\binom{p}{k}a^{k}b^{p-k}=\binom{p}{0}b^{p}+\sum_{k=1}^{p-1} \binom{p}{k} a^{k}b^{p-k}+\binom{p}{p}a^{p}=b^{p}+\sum_{k=1}^{p-1}\binom{p}{k}a^{k}b^{p-k}+a^{p}$ Any ideas?  Thanks!
The extension of smooth function If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an arbitrary domain $V \supseteq \bar U$, is there a smooth function $\tilde f$ on $V$ extending $f$? Prove that any two consecutive terms of the Fibonacci sequence are relatively prime Prove that any two consecutive terms of the Fibonacci sequence are relatively prime.   My attempt:    We have $f_1 = 1, f_2 = 1, f_3 = 2, \dots$, so obviously $\gcd(f_1, f_2) = 1$. Suppose that $\gcd(f_n, f_{n+1}) = 1$; we will show that $\gcd(f_{n+1}, f_{n+2}) = 1$. Consider $\gcd(f_{n+1}, f_{n+2}) = \gcd(f_{n+1}, f_{n+1} + f_n)$ because $f_{n+2} = f_{n+1} + f_n.$ Then $\gcd(f_{n+1}, f_{n+1} + f_n) = \gcd(f_{n+1}, f_{n}) = 1$ (gcd property).  Hence, $\gcd(f_n, f_{n+1}) = 1$ for all $n &gt; 0$.  Am I on the right track? Any feedback would be greatly appreciated.  Thanks, "Die from cancer" vs. "die of cancer" Is there a difference between those expressions: "Die from cancer" or "Die of cancer"? Are they both correct? Extruding a circular shape from a non-circular object I need to extrude a circular shape from a surface that's basically just a grid of quads. What's the best practice to do this without screwing up the topology too much?  I basically want to have a circular face I can extrude, in order to have a cylinder like protuberance on the object, something like this: I am currently using 19.04 and I got "No new release found for 19.10" My Logitech G5 Laser mouse has 9 buttons but it's SetPoint software only allows 5 of them to be used as "mouse buttons" and the remaining 4 can only be assigned special windows functions (Forward, Back, Cruise Up, etc) or Keystroke Assignments.  Many of the games I play will take up to 10 "MOUSE BUTTON" inputs for in-game actions but wont recognize the remaining 4 buttons even when assigned keystroke assignments.  Is there a way to make these last 4 buttons "Generic Buttons" as well as buttons 3,4,and 5? (1 is left click and 2 is right click).  My other question is if the above is not possible, maybe I can use AutoHotkey to remap keystrokes as assigned by my SetPoint software to mouse button commands?  If so how? Ubuntu touch - remote access to device in our company we developed an android app installed on nexus 7 tablets and on tablet we release to several customers. This app stays in foreground active 24/7 but we have problems in monitoring  the devices due to the restrictive access policies of unrooted Android devices.   We can't for example reboot the device remotely.  We are evaluating the possibility of installing Ubuntu touch to replace android on this tablets. Does Ubuntu touch offer more freedom for the remote access of the device? May I do remotely everything I could do normally with the tablet?   Moreover, are the apps developed with ubuntu sdk stable enough to run 24/7?  Thanks Reordering polygon shapefile permanently? Can I do it with reflection or something like that? i'm surprised there is so little documentation on this issue. But i'm trying to duck the audio on all the mix buses in Unity so that you can clearly hear the VO of the game.  Now it works great except that I have some reverb zones in the game and any sound that plays in these reverb zones, the actual reverb itself is NOT DUCKED.  So the VO is accompanied with a huge wall of wishy washy muddy noise.  Anyone deal with this before? How can we deal with this? Old usernames stick to Community Wiki posts On a whim the other day, I changed my meta name from Lord Torgamus to Popular Demand.  I just noticed that the old name has "stuck" to the author tag of ; everywhere else on meta, my new name displays as expected.  Is this a feature?  Is it a bug?  Mostly I'm just curious, as I can't think of a strong reason in favor of keeping this behavior or a strong reason for changing it.  A question somewhat like this has been raised before as , but the previous poster implied that the issue had something to do with migration, which my situation shows is not the case. I have an idea or subject that should have a Q&amp;A, and want to propose a new site. How can I do this?  Related: I know why questions, answers and some other aspects of the site are color-coded and what they mean but I can't find why badges are color-coded. I'll include a snapshot to show what I mean. Additional horizontal space when the last line of an environment is full I have created an environment env which just adds a \medskip and some text (say "Blablabla") after the text inside \begin{env} ... \end{env}. The problem is that if the last line is full, then I get an additional horizontal space between the text and "Blablabla".  Here is an MWE:  \documentclass{article} \newenvironment{env}{ ENVIRONMENT:\par }{ \medskip \par\noindent Blablabla }  \begin{document}  \section*{First case: the last line is not full}  \begin{env} Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nulla malesuada porta tortor non vestibulum. Vivamus vel tincidunt purus. Curabitur quam arcu, suscipit sed diam sed, ornare efficitur felis. Duis at arcu iaculis, viverra ligula sit amet, hendrerit eros. Proin finibus venenatis sollicitudin. Nunc ullamcorper sem ut lacinia posuere. Integer pulvinar nulla lorem, ut pulvinar ipsum egestas at. Duis quam ipsum, posuere ac pulvinar vitae, auctor non leo. \end{env}  \section*{Seconde case: the last line is full}  \begin{env} Lorem ipsum dolor sit amet, consectetur adipiscing elit. Quisque suscipit vitae neque eu gravida. Quisque et enim quam. Integer elementum diam a ex laoreet, ac egestas lacus volutpat. Nunc ullamcorper non urna tincidunt ultrices. Etiam eu ultricies diam. Proin mollis, nunc sit amet volutpat varius, ipsum tellus bibendum nulla, in eleifend lacus felis pellentesque nisl. Phasellus laoreet, libero quis auctor congue, nulla nulla aliquam metus, in hendrerit arcu justo quis tortor. Nunc nec nisi euismod, consectetur tellus nec, suscipit libero.  \end{env}  \end{document} Is floating point math broken? Consider the following code:   0.1 + 0.2 == 0.3  -&gt;  false   0.1 + 0.2         -&gt;  0.30000000000000004   Why do these inaccuracies happen? Java generics type erasure: when and what happens? How to trim whitespace from a Bash variable? Am not doing anything fancy, just trying to update a field on Contact in JS of LWC but getting this exception.  Uncaught TypeError: 'set' on proxy: trap returned falsish for property 'Name' throws at mydomain/auraFW/javascript/mhontaYdOya4Y_lBu7v9yg/aura_prod.js:2:27687  HTML Code:   &lt;template&gt;      &lt;template if:true={wiredContact}&gt;          {wiredContact.Name}          &lt;lightning-input value={wiredContact.Name} onchange={updateName}&gt;&lt;/lightning-input&gt;     &lt;/template&gt;  &lt;/template&gt;   JS:   import { LightningElement ,wire,track,api } from 'lwc'; import myContact from "@salesforce/apex/ContactController.fetchContact";  export default class Myrefreshapextest extends LightningElement {    @track wiredContact;   @wire (myContact)         fetchedContact({error, data}){             if(data){                 console.log(JSON.stringify(data));                 this.wiredContact = data;             }else if (error){                 console.log(error);             }     }      updateName (event){         console.log(JSON.stringify(event.detail.value));         console.log(JSON.stringify(this.wiredContact));         this.wiredContact.Name = event.detail.value;     }  }   Apex:  public class ContactController {      @AuraEnabled(cacheable=true)     public static Contact fetchContact(){         return [SELECT Id,Name FROM COntact LIMIT  1];     } }   On top of my head, am not doing anything wrong, anyone has idea what's wrong with my code?  When I print console.log(JSON.stringify(this.wiredContact)); I get old values so am pretty sure it exists.  I tried with @track and @api, but same response. Can anyone shed some light? Undefined reference to vtable Is there an easy way to delete an element from an array using PHP, such that foreach ($array) no longer includes that element?  I thought that setting it to null would do it, but apparently it does not work.