Show that $(n-1)^2\vert n^k-1$ if and only if $n-1\vert k$, for naturals $n,k$.  I couldn't go much further, the only thing that i could conclude is that $n-1\vert n^k-1$, since it is true that $a-b\vert a^k-b^k$ for all natural $k$. Now i can't proceed from here.  Any hints on this question?
I need help!  I need to prove that for any $2 \le n,k$ positive integers $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$  Thanks!
CHECK constraint in MySQL is not working How do I programmatically get the field type of a specific field? How to make a ISO file out of a folder containing Windows setup files I copied the Windows 8 CD to my HDD. I want to convert that folder to an ISO and use that to create a bootable USB.  Please tell me how I can make an ISO image out of that folder containing files.  This is my folder which I want to convert to a ISO file: How to change shortcut key for switching between virtual desktops in windows 10? In linux I use a shortcut to switch between 4 workspaces: win+1/2/3/4.   I would like to have something like this in Windows 10.  UPDATE SOLUTION 2017, WINDOWS 10x64   Update your windows version to most recent through this  . You may also need  Download and unzip this  (0.9.1) works for me. Change content of file settings.ini to    [KeyboardShortcuts] Switch=LWin Move=LAlt, Shift, Ctrl MoveAndSwitch=LAlt, Shift Previous=Left Next=Right PlusTen=     Run virtual-desktop-enhancer.exe as administrator. How to save an activity state using save instance state? I've been working on the Android SDK platform, and it is a little unclear how to save an application's state. So given this minor re-tooling of the 'Hello, Android' example:  package com.android.hello;  import android.app.Activity; import android.os.Bundle; import android.widget.TextView;  public class HelloAndroid extends Activity {    private TextView mTextView = null;    /** Called when the activity is first created. */   @Override   public void onCreate(Bundle savedInstanceState) {     super.onCreate(savedInstanceState);      mTextView = new TextView(this);      if (savedInstanceState == null) {        mTextView.setText("Welcome to HelloAndroid!");     } else {        mTextView.setText("Welcome back.");     }      setContentView(mTextView);   } }   I thought it would be enough for the simplest case, but it always responds with the first message, no matter how I navigate away from the app.  I'm sure the solution is as simple as overriding onPause or something like that, but I've been poking away in the documentation for 30 minutes or so and haven't found anything obvious. How to remove duplicate entries in dash? What is the easiest way to update/upgrade an offline Ubuntu machine with the help of a Windows machine that has internet access?  This question also applies to installing new packages offline and not only updates/upgrades.  The Linux system shall remain completely cut off from the Windows machine, meaning there should be no ethernet connection or the likes between the two computers.  Update:  Please have patience, since I haven't updated this since two weeks. There are some complications getting apt-offline to work properly on Lubuntu and all other methods seem to be too old and deprecated:  AptOff - last update 2006 AptMedium - last update 2007 Keryx - last update 2011, looks like this project was discontinued  I hope I will be able to somehow update the offline machine, finally.  While I was still not able to update/upgrade the offline system due to authentication errors, I accepted an answer, because it's taking too long. What is $n!$ when $n=0$? How do spell lists change if the party levels up without taking a long rest? I'm playing in a D&amp;D 5e campaign with a bard and a wizard.  Our most recent session ended while we were in the middle of exploring a dungeon. The DM informed us that we leveled up from 4 to 5.  The wizard and bard were excited to get level 3 spell slots for the first time. They began discussing the level 3 spells they were looking forward to casting at the beginning of the next session. I asked, "Don't you guys need to have a long rest before you can cast new spells, though?"  The bard argued that he doesn't need a long rest, because he learns spells immediately upon leveling up and doesn't need to prepare them.  The wizard argued his "number of spells available" count increased immediately upon leveling up, and it wouldn't make much sense if his actual number of prepared spells stayed the same.  Ultimately, the DM decided that it would be more fun if everyone got access to their new toys at the same time. He permitted the wizard to add a level 3 spell to his prepared spell list.  Who is right? Do bards (and other casters that do not prepare spells) get access to new spells more immediately than wizards (and other casters that prepare spells)?  I looked at , but "Can a mage use his newly gained 2nd-level spell slot?" seems like a separate matter from "Can a mage use a newly gained spell?". Why wasn't Restricted Section of the library better protected? Convert stored procedure result to model object Live refresh shows "modified" on tagged questions even when unmodified How find this $x^3-5x+10=2^y$ let $x,y$ is positive integer,and such $$x^3-5x+10=2^y$$ find all $x,y$.  since $$x=1\Longrightarrow 1^3-5+10=6$$ can't $$x=2,2^3-5\cdot 2+10=8=2^3$$ so $x=2,y=3$  $$x=3,LHS=27-15+10=22$$ $$x=4,LHS=64-20+10=54$$ $$x=5,LHS=125-25+10=110$$ $$x=6,LHS=216-30+10=236$$ $$\cdots$$  I find $$(x,y)=(2,3)$$  I only find $x\le 7$ this solution.  maybe this have other solution.and This problem is from Mathematical olympiad problems Thank you The  (click on your number of helpful flags in your profile) is cool, however some posts have a gray background, some don't. What does it mean?  example: Why is 4 not an instance of Number? Just curious:    4 instanceof Number => false new Number(4) instanceof Number => true?   Why is this? Same with strings:    'some string' instanceof String returns false new String('some string') instanceof String => true String('some string') instanceof String also returns false ('some string').toString instanceof String also returns false   For object, array or function types the instanceof operator works as expected. I just don't know how to understand this.  [new insights]  Object.prototype.is = function() {         var test = arguments.length ? [].slice.call(arguments) : null            ,self = this.constructor;         return test ? !!(test.filter(function(a){return a === self}).length)                : (this.constructor.name ||                   (String(self).match ( /^function\s*([^\s(]+)/im )                     || [0,'ANONYMOUS_CONSTRUCTOR']) [1] ); } // usage var Newclass = function(){};  // anonymous Constructor function var Some = function Some(){}; // named Constructor function (5).is();                     //=&gt; Number 'hello world'.is();           //=&gt; String (new Newclass()).is();        //=&gt; ANONYMOUS_CONSTRUCTOR (new Some()).is();            //=&gt; Some /[a-z]/.is();                 //=&gt; RegExp '5'.is(Number);               //=&gt; false '5'.is(String);               //=&gt; true There is a remote Windows server on a private network which I can connect to via Remote Desktop Connection.  I would like to be able to make TCP/IP connections from my computer to other computers on that server's network.  Remote Desktop Connection makes it possible to share printers, drives, and other local resources through the connection.  Is there any way to "tunnel" a TCP/IP connection via RDC?  I'd like something similar to the port-forwarding provided by SSH.  I don't see any way to do this via RDC, but I'm hoping the capability is there and I just don't know about it. When should I use mmap for file access? POSIX environments provide at least two ways of accessing files.  There's the standard system calls open(), read(), write(), and friends, but there's also the option of using mmap() to map the file into virtual memory.  When is it preferable to use one over the other?  What're their individual advantages that merit including two interfaces? I am trying to solve the following exercise.      Assume that $m$ and $n$ are co-prime positive integers, and that $k$ is a field. Prove that the ideal generated by $x^{m} - {y^n}$ is a prime ideal in $k[x,y]$.   I tried the following approach : Consider the ring homomorphism ${\varphi}:k[x,y]\rightarrow{k[t]}$ defined by $\varphi(x) = t^n$, and $\varphi(y) = t^m$, and extend $\varphi$ to $k[x,y]$ to ensure that it is a ring homomorphism. Then the ideal $(x^m - y^n)$ is clearly contained in the kernel of $\varphi$. It remains to prove the reverse set inclusion. i.e. to prove that this ideal is exactly equal to the kernel of $\varphi$. I don't know how to go about this! Rice with worm and possibly their eggs I bought a bag of brown rice from a store. After having opened and used for a while, I found there were moths flying in the house. At the same time, I found worms in the rice. I suspect that the moths are from the worms(rice-size, white body, dark head). I guess but am not sure if they are called rice moths. There were also many little sand-like things in the rice bag, are they the eggs of the rice moths?  Will eating this infested rice pose a health risk? How to add custom objects to the "add object" menu?