So when I want the gcf of a,b I could use the euclidean algorithm for example.  But why is gcf (a mod b, a) = gcf (a,b)    (because 0&lt;=a&lt;=b and in a mod b, a is always smaller than the rest from a mod b?)   So essentially my question is: how does the gcf(a mod b, a) work.  A dumb question:   why is the result of a mod b even smaller than a and how is that linked to the greatest common factor of a,b?    Thank you.
Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so?    Addendum by LePressentiment on 11/29/2013: (in the interest of  and averting a duplicate)  What's the intuition behind this result? I only recognise the proof and examples solely due to algebraic properties and formal definitions; I'd like to apprehend the result naturally.
How to host a single website on multiple servers? I'm planning on releasing an iOS app that downloads video from my server. If hundreds of people on the launch day try to do this, then my website and the services that my server provides will clearly go down. How do you host a website on multiple servers? When someone tries to access my site, how is their request passed to an available server? How do Apple and Google do it? Thanks for your help. I am trying to use a Bluetooth headset on an Ubuntu PC. I am able to connect phone to pc via BT. I can stream music to PC. But cannot hear call audio. I have checked in Android phone logs, and BT sco link is established. But when making a call or virtual call (play audio through sco path in Android phone) the audio is not audible in PC headset. SCO packets are NOT appearing in btmon logs either. Anyone successfully done this?  version- ubuntu 16.04 Installed- bluez, ofono, pulseaudio pactl sink: Name- bluez_sink.22_22_8E_DE_B9_2B,             Name: alsa_output.pci-0000_00_1b.0.iec958-stereo pactl source: Name-bluez_sink.22_22_8E_DE_B9_2B.monitor               Name: alsa_input.pci-0000_00_1b.0.analog-stereo The foreach identifier and closures Many of the UK visa refusals we see here share a common pattern and the prevailing reasons refer to V 4.2 (a) and (c).    I do understand that, while the applicants may describe very different individual circumstances, there's a consistent pattern and, broadly, fall into specific categories:    Credibility (lifestyle, lack of ties, visit history, lies &amp; omissions) Funding (insufficient funds, provenance of funds, funds parking) Sponsorship (family, friends, employer)   Question: What most commonly triggers a UK visa refusal where V 4.2 (a) and (c) are given as the grounds?  Secondarily: Given that there is a clear pattern, to what extent are these refusals predictable? Does sponsorship make a difference? Is there a set of personal circumstances, however abstract, where a refusal is all but guaranteed? For example, why do those we see here on TSE who sought entry for the PLAB or British Army appear to be refused with a common theme? Is there a uniform shortcoming or is it just discrimination?   PLAB: What is about applications for a visa, for the purpose of sitting the Professional and Linguistic Assessments Board (PLAB) exam, that seem to invite refusals. After all, the test is given so that international medical graduates can show that they qualify to practise medicine in the UK. The first part, PLAB 1, can be taken in centres outside of the UK. However, PLAB 2 can be taken only in the UK. Why can't you can't get a visa just to sit the exam, promising that you'll leave immediately after?    British Army: Since Commonwealth citizens are eligible to apply online to join, even those who don't reside in the UK, why is it so difficult to get a visa just to attend an interview to see whether you are suitable? Even with an invitation from the Army, such visa applications seem to be unsuccessful. Isn't a career in the British Army a valid reason?     Lastly:  After such a refusal, what approaches would increase the chance of a successful application? Replacement character being used question URL Some questions, for example    Instead of using a correct URL, such as  http://meta.stackexchange.com/questions/79543/٩●̮̮̃-̃۶-forbid-unicode   are using  https://superuser.com/questions/516363/what-is-the-��-character   What is causing this behavior? I have a JavaScript object like the following:  var p = {     "p1": "value1",     "p2": "value2",     "p3": "value3" };   Now I want to loop through all p elements (p1, p2, p3...) And get their keys and values. How can I do that?  I can modify the JavaScript object if necessary. My ultimate goal is to loop through some key value pairs and if possible I want to avoid using eval. Create tar archive of a directory, except for hidden files? I'm wanting to create a tar archive of a specific directory (with its subdirectories of course). But when I do it, using the tar command, I get a list of files that were included, for example:     a calendar_final/._style.css      a calendar_final/style.css   As you can see, there are two versions of the same file. This goes for every file, and there are many. How do I exclude the temporary files, with the ._ prefix, from the tar archive? Can someone tell me how to get a closed form for  $$\sum_{k=1}^n k^p$$  For $p = 1$, it's just the classic $\frac{n(n+1)}2$.   What is it for $p &gt; 1$? How can I automate the workflow for producing multiple versions of a document? Say you have two versions of the same LaTeX document.    For instance, you have one version of your lecture slides for actually delivering a lecture: as you're really not a big fan of the "wall of text", this version would be parsimonious in terms of text content, allowing the audience to listen to what you're talking  about instead of reading the slides, and giving graphical content centre stage.  Another version of those slides, made available to the audience after the lecture, for self-study, would be more self-contained and include a summary of what you said during that lecture along with the graphical content mentioned above.  Consider that you can toggle between the two versions by either changing a class option or by changing the value of some dedicated counter in your input file; something relatively simple to do "by hand".  To produce two different versions of the same document, my current workflow is   "manually" compile the first version, change the filename of the output produced (in order to prevent overwriting in step 4) make appropriate changes in my input file to switch to the second version, "manually" compile the second version.   However, this workflow is tedious and error-prone, especially when the number of different versions increases. I would like to automate it as much as possible. Is there a way to batch compile all the versions of my document, each getting a different output filename?  I guess that would involve modifying the input file at the command line (I'm on Mac OS X), but I don't know how to do that.  Do you have an automated workflow for compiling multiple versions of the same document? How do you do it?  Edit: I figured it out using  and  on , of which my question is a duplicate. Where to start learning Linear Algebra? I'm starting a very long quest to learn about math, so that I can program games. I'm mostly a corporate developer, and it's somewhat boring and non exciting. When I began my career, I chose it because I wanted to create games.  I'm told that Linear Algebra is the best place to start. Where should I go? How do you scrape AJAX pages? Please advise how to scrape AJAX pages. How can I prevent my mail from being classified as spam? Show a Form without stealing focus? I'm using a Form to show notifications (it appears at the bottom right of the screen), but when I show this form it steals the focus from the main Form. Is there a way to show this "notification" form without stealing focus? $\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}).$ Task is to find $$\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}).$$  I tried to make it with double-angle formula: $\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}) = \frac{\prod_{k=0}^\infty \sin(x2^{1-k})}{2^\infty \cdot \prod_{k=0}^\infty \sin(x \cdot 2^{-k})} $ I'm sad to admit, that I'm stuck with that one. So I would appreciate any help, such as pointing the right direction. Show some indication that account is banned from asking new questions Why is document.write considered a "bad practice"? How do I merge two dictionaries in a single expression (taking union of dictionaries)? Are the following sentences grammatical or not? If not, does substituting a to-infinite make them grammatical? (Optional: designate if the sentence expresses purpose, reason, aim, or function)  1 I have decided to purchase a jet for getting to NYC and back more quickly. 2 I have decided to buy a jet for having a quicker way to get to NYC and back.  This is my only reason and aim; this will be the jet's only purpose and function.  3 I hate hairdryers, so I am buying a space heater for drying my hair more naturally, and that is my only aim &amp; purpose &amp; reason and that will be its only function 4 For having a lasting shine in your hair, buy ShineOn Shampoo. 5 My neighbor and I have decided to cooperate more for having a better life together.   Research done: ELL  (Note the  here is a restatement of Cambridge Dictionary's ) ELL  ELL  BBC English  (Currently the best site I've found.) ELU  All of which have answers that I find  (i) incomplete (ii) contradictory (iii) terribly confusing, if not (iv) wrong. To sort through the haze, I am focusing on one issue. Background Some sources say that you need to use a to-infinitive for expressing purpose, aim, function, reason, etc. I find these to be rather hazy distinctions, given that the ODO defines  as The reason for which something is done or created or for which something exists and the same work lists as : aim, reason, function, etc. It is my assertion that at least some of the following sentences demonstrate that one can use the -ing form (be it a gerund, verbal noun, participle) to express individual purpose (aim, reason, function, intention, whatever you want to call it). Do you agree, or not? Why (not)?  In shorter terms, are the sentences grammatical? or do we have to substitute a to-infinitive (including in order to + infinitive) to make them grammatical? (Optional: designate if the sentence expresses purpose, reason, aim, function, etc.) Replace non-numeric with empty string If a mass moves close to the speed of light, does it turn into a black hole? I'm a big fan of the podcast Astronomy Cast and a while back I was listening to a Q&amp;A episode they did. A listener sent in a question that I found fascinating and have been wondering about ever since.   From the show :      Arunus Gidgowdusk from Lithuania asks: "If you took a one kilogram mass and accelerated it close to the speed of light would it form into a black hole? Would it stay a black hole if you then decreased the speed?"   Dr. Gay, an astrophysicist and one of the hosts, explained that she'd asked a number of her colleagues and that none of them could provide a satisfactory answer. I asked her more recently on Facebook if anyone had come forward with one and she said they had not.  So I thought maybe this would be a good place to ask.