How to prove Euler's theorem using Lagrange's theorem?      If $a$ and $n$ are relatively prime then $a^{\phi(n)}\equiv 1 \pmod n$   Wikipedia says that it can done and that $\phi(n)$ is the order of the multiplicative group of integers modulo $n$. But I'm not sure how to proceed with the proof. Any ideas/hints?
From A Classical Introduction to Modern Number Theory by Ireland and Rosen, page 33:     Corollary 1 (Euler's Theorem). If $(a,m) = 1$, then $a^{\phi(m)} \equiv 1\,(m)$.      Proof. The units in $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. If $(a,m) = 1$, $\bar{a}$ is a unit. Thus $\bar{a}^{\phi(m)} = \bar{1}$ or $a^{\phi(m)} \equiv 1\,(m)$.   If I'm interpreting this correctly, this proof implicitly uses the fact(?) that if $G$ is a group and $x\in G$, then $x^{|G|}=1$, where $1$ is the identity element of $G$.  If this is indeed true, can someone explain why? (I have had no prior exposure to group theory.)
I need to put some formal statistics on the document itself somewhere in the middle, like "This document contains XXX figures and YYY tables (and may be even ZZZ formulas and PPP theorems)". Is there a common way to output the final value of some counter in the middle of the document?  Edited after answering: I have asked a more specific question before, "How to output the number of bibliographic entries referenced in the document?", which is now deleted because the answer to this one covers also this case.  For the reference, some useful counters, which can be used with totcount:   page - page count enumiv - bibliography entries count figure - figures count table - tables count I recently got a computer from school. It came with Windows 8.1 installed. I then installed another copy of Windows 8.1 (so it would use the built-in Windows Pro license) and upgraded it to Windows 10. I then installed Ubuntu alongside both Windows 8 and 10. Note that I made a separate 255MB ext2 partition for /boot. Now, when I boot the computer, I first see a GRUB prompt (I already un-hid the grub prompt 'cuz I like it that way) with Windows 8 and Ubuntu options. However, there is no Windows 10 option. If I select Ubuntu, the computer boots into Ubuntu. However, if I select the Windows 8 option, I then get sent to the Windows bootloader, where it then gives me options to boot either Windows 8 or Windows 10. (However, by this point, the computer has already loaded the kernel and all. Apparently Microsoft misunderstood the concept of a bootloader and decided that that meant &quot;boot practically the entire system before actually showing the bootloader&quot;. Typical Microsoft.) How can I disable the Windows bootloader and add a Windows 10 option to GRUB? EDIT: Someone has answered telling me to disable the Windows bootloader. However, I still do not know how to add Windows to the GRUB OS list. Can anyone help with this? EDIT 2: After removing Windows 10 from the Windows bootloader with EasyBCD, it seemed that, after a few reboots, it would continually unhide itself and Windows 10 would reappear. However, I then later deleted Windows 10 from the list of OSes in msconfig, and it seems to have stayed deleted. However, I STILL have not managed to get Windows 10 to appear in GRUB. It's as if GRUB simply doesn't detect it. Also, I'll put this in the question so that people see it - I have tried using boot-repair, however, it has NOT helped any. I'm trying to find the name of a book I had as a kid back in the 80's/90's that was a collection of science fiction short stories and in particular one story in it.  The book: I remember the cover being mostly orange with the face of an alien taking up most of the cover. Scaly looking 'grey' style alien with hooded green eyes. They may have also been a 60's/70's style space ship in the background with a man and woman.  The story: Was set after all human life had ended in a great war and the story followed one of the many autonomous bombers still flying the skies carrying out their missions repeatedly bombing already leveled cities. At least that's how I remember it.  Any help/clues you can give would be much appreciated as I'd love to share these stories with my kid.  Some other stories it had, if any of these can be identified it might help me find the book:   Two guys talking in a room and one give the other a wooden carving of an apple and starts telling him the history of the object and how it's used to hypnotise people who hold it as they can't resist stroking it and it puts them in a catatonic state. A bunch of robots/machines wandering the post-apocalyptic country side, free of their duties to mankind. Their adventure comes to an end when they run into a hermit living in the country and their programming kicks in and they immediately become his servants. A story about first contact with an alien race, the aliens come to earth and land at the coordinates of an air-force base but upon arrival there is confusion as the humans cannot see the aliens and the aliens cannot see the humans. The aliens complain that they've been tricked and that their ship is sinking. The humans think this is strange until they realise the aliens are incredibly small and have landed in a puddle on the airfield. URLs are linked despite being inside a code span How can I highlight text before a character with listing? I can't find .bash_profile in Ubuntu 14.04 in my /home/user directory. I used the ls -a command to see the .bash_profile, but there isn't such a file. Here is a sentence excerpted from ,     Teen suicide is a growing health concern. It is the third-leading cause of death for young people ages 15 to 24, surpassed only by homicide and accidents, according to the U.S. Center for Disease Control and Prevention.   I've heard people use both "people aged from X to Y" and "people ages from X to Y" quite often, but I've always been believing that only the former is grammatically correct, whereas in the latter the "ages" doesn't seem to serve as any grammatical component of this sentence. So when I see an APA paper use "ages" in that way, I am just quite surprised: I just can't believe such an authoritative and rigourous academic institution will let such an explicit (and indeed naive, even to a non-native speaker's eyes) grammatical mistake survive in their publications. I'm aware that in daily language usage one ought not to become so fussy over insignificant matters like this, but I do wonder whether it should be paid attention to in academic writings.  OR, using "ages" in that way is no longer considered a grammatical mistake? Honestly I don't think this'd be true, but if it really is, I'd be grateful to be informed. Thanks! Is it possible to match encrypted documents using user-defined search terms? Suppose I am storing a number of encrypted documents in a database. I would like to make it possible to identify the subset of documents whose contents match user-specified search terms without a) maintaining a plaintext index or b) decrypting documents on the fly. Is there any way to accomplish this securely?  By the same token, I'm wondering if a collection of encrypted documents can be sorted according to an encrypted attribute (e.g., document title) without first decrypting all attribute values. How do the review limits work? I got the brand new Custodian badge today for reviewing posts. Inspired by that, I reviewed some more suggested edits. Now I ran into a limit:     You have no more suggested edit votes today; come back in x hours.   How is that limit calculated? I made 61 reviews today which seems to be some calculated limit.  I searched for an answer on meta but found nothing helpful. Are Love Potions legal? $f(x)f(1/x)=f(x)+f(1/x)$ Find a function $f(x)$ such that:  $$f(x)f(1/x)=f(x)+f(1/x)$$  with $f(4)=65$.  I have tried to let $f(x)$ be a general polynomial: $$a_0+a_1x+a_2x^2+\ldots a_nx^n$$  which leaves $f(1/x)$ as:  $$a_0+a_1{1\over x}+a_2{1\over x^2}+\ldots + a_n{1\over x^n}$$  On comparing the coefficients of both sides, we see that:  $$2a_0=(a_0)^2+(a_1)^2+(a_2)^2+\ldots+(a_n)^2$$  And  $$a_1=(a_0a_1)+(a_1a_2)+ \ldots +(a_{n-1}a_n)$$  I don't know how to proceed further. I know I need to compare coefficients and come to a conclusion based on their values, but I don't see what to do next. What's the best way to communicate between view controllers? Being new to objective-c, cocoa, and iPhone dev in general, I have a strong desire to get the most out of the language and the frameworks.  One of the resources I'm using is Stanford's CS193P class notes that they have left on the web.  It includes lecture notes, assignments and sample code, and since the course was given by Apple dev's, I definitely consider it to be "from the horse's mouth".  Class Website:   Lecture 08 is related to an assignment to build a UINavigationController based app that has multiple UIViewControllers pushed onto the UINavigationController stack.  That's how the UINavigationController works.  That's logical.  However, there are some stern warnings in the slide about communicating between your UIViewControllers.  I'm going to quote from this serious of slides:   Page 16/51:     How Not To Share Data         Global Variables or singletons         This includes your application delegate      Direct dependencies make your code less reusable         And more difficult to debug &amp; test         Ok.  I'm down with that.  Don't blindly toss all your methods that will be used for communicating between the viewcontroller into your app delegate and reference the viewcontroller instances in the app delegate methods.  Fair 'nuff.  A bit further on, we get this slide telling us what we should do.  Page 18/51:     Best Practices for Data Flow         Figure out exactly what needs to be communicated   Define input parameters for your view controller   For communicating back up the hierarchy, use loose coupling         Define a generic interface for observers (like delegation)         This slide is then followed by what appears to be a place holder slide where the lecturer then apparently demonstrates the best practices using an example with the UIImagePickerController.  I wish the videos were available! :(  Ok, so... I'm afraid my objc-fu is not so strong.  I'm also a bit confused by the final line in the above quote.  I've been doing my fair share of googling about this and I found what appears to be a decent article talking about the various methods of Observing/Notification techniques:   Method #5 even indicates delegates as an method!  Except.... objects can only set one delegate at a time.  So when I have multiple viewcontroller communication, what am I to do?  Ok, that's the set up gang.  I know I can easily do my communication methods in the app delegate by reference's the multiple viewcontroller instances in my appdelegate but I want to do this sort of thing the right way.  Please help me "do the right thing" by answering the following questions:   When I am trying to push a new viewcontroller on the UINavigationController stack, who should be doing this push.  Which class/file in my code is the correct place? When I want to affect some piece of data (value of an iVar) in one of my UIViewControllers when I am in a different UIViewController, what is the "right" way to do this? Give that we can only have one delegate set at a time in an object, what would the implementation look like for when the lecturer says "Define a generic interface for observers (like delegation)".  A pseudocode example would be awfully helpful here if possible. Make locking posts more granular Unwanted Lines when rendering using GPU and OpenCL (RX 580) Resetting crontab file to default So I know umask can restrict privileged users, using this format umask ugo.  I understand that the read = 4, write = 2, and exec = 1. However, when I type umask, it returns 4 digits which is 0022 or 0073. I have no understanding of how does this work now because there is an extra digit. What is that extra digit and what does 0022 mean? Is there a way for me to download the full list of tags (and synonyms) used on Stack Overflow? My intent is to import these tags into an SQLite database as part of a process to convert all of my delicious bookmarks and tags to the ones used here on Stack Overflow. Brasero says it won't burn the CD because the file is too big, what's going on? Recently, my iPhone 5 was stolen, and while I have a new iPhone 6, I received a message from a number of someone claiming to be in possession of the device, with a new number.   All images, contacts, Outlook mail, you name it, are still on the device.  While I realize it's my fault for not having it connected to iCloud to wipe manually, is there anything I can do to have this sensitive information wiped from the device? Apply function conditionally