Prove that gcd(a,b)=dgcd(a/d,b/d).
For $D$ a GCD domain , let $a,b,x \in D \setminus \{0\}$ , then is it true that $\gcd (ax,bx)=x \cdot \gcd (a,b)$?
In several different contexts we invoke the  to justify whatever statistical method we want to adopt (e.g., approximate the binomial distribution by a normal distribution). I understand the technical details as to why the theorem is true but it just now occurred to me that I do not really understand the intuition behind the central limit theorem.  So, what is the intuition behind the central limit theorem?   Layman explanations would be ideal. If some technical detail is needed please assume that I understand the concepts of a pdf, cdf, random variable etc but have no knowledge of convergence concepts, characteristic functions or anything to do with measure theory. On SO, when I'm scrolling through "questions" some have a yellow background.  Is this intuitive to everyone but me?  What is the yellow background trying to tell me about this particular question? Let $A = \{ 1,2,3,4 \}$ Let $F$ be a set of all functions from $A \to A$.  Let $S$ be a relation defined by : $\forall f,g \in F$ $fSg \iff f(i) = g(i)$ for some $i \in A$  Let $h: A \to A$ be the function $h(x) = 1 $ for all $x \in A$.   How many functions $g \in F$ are there so that $gSh$ ?  My solution : $gSh$ means that $g(i) = h(i)$ for some $i \in A$. So $g(i) = 1$.  So number 1 always needs to connect to some x - 4 choices.  Then we have $3$ left-over numbers that can connect to $4$ numbers each.  So solution is $4 \cdot 4 \cdot 4 \cdot 4$. Is this correct at all? Thanks in advance !! :) What does the word "SARGable" really mean? Is there any easy way to open the document library files in a new window or popup depending on the file types?  e.g. PDF - In Adobe reader (currently opens in IE)  Image file (currently opens in IE) My Alt + Tab menu looks and works oddly How do I change / spoof my MAC address and easily switch between multiple ones? My router can do port-forwarding based on MAC addresses. That is, a specific MAC will get a specific IP, for which I can configure a set of ports to be forwarded. In order to easily change that set of ports, I'd like to have different connections in the Network manager.  How do I change the MAC address for a network connection? Add Word Limit to Posts LIGO has announced the detection of gravitational waves on 11 Feb, 2016. I was wondering why the detection of gravitational waves was so significant?   I know it is another confirmation of general relativity (GR), but I thought we had already confirmed GR beyond much doubt. What extra stuff would finding gravitational waves teach us? Is the detection of gravitational waves significant in and of itself, or is there data which can be extracted from the waves which will be more useful? Can .exe and .msi files (Windows software) be installed in Ubuntu? What exactly is a photon? Consider the question, .  The answers say, "an elementary particle" and not much else. They don't actually answer the question. Moreover, the question is flagged as a duplicate of,  &ndash; the answers there don't tell me what a photon is either. Nor do any of the answers to  mentioned in the comments. When I search on , I can't find anything useful. Questions such as,  look promising, but bear no fruit.  say things like, "the photon is an excitation of the photon field." That tells me  nothing. Nor does the , which says:     The photon is the quantum of the electromagnetic four-potential, and therefore the massless bosonic particle associated with the electromagnetic force, commonly also called the 'particle of light'...   I'd say that's less than helpful because it gives the impression that photons are forever popping into existence and flying back and forth exerting force. This same concept is in the  too - but it isn't true. As , "Virtual particles only exist in the mathematics of the model." So, who can tell me what a real photon is, or refer me to some kind of authoritative informative definition that is accepted and trusted by particle physicists? I say all this because I think it's of paramount importance. If we have no clear idea of what a photon actually is, we lack foundation. It's like what :     Photons seem to be one of the foundation ideas of quantum mechanics, so I am concerned that without a clear definition or set of concrete examples, the basis for understanding quantum experiments is a little fuzzy.   I second that, only more so. How can we understand pair production if we don't understand what the photon is? Or the electron? Or the electromagnetic field? Or everything else? It all starts with the photon.   I will give a 400-point bounty to the least-worst answer to the question. One answer will get the bounty, even if I don't like it. And the question is this:    What exactly is a photon? Removing spaces between "words" in math mode What is the main difference between StringBuffer and StringBuilder? Is there any performance issues when deciding on any one of these? Maximum length of tags Possible Duplicate:        Is there a maximum limit on the length of tags? I noticed that the tag for SQL Server Management Studio is truncated to sql-server-management-stu.  It seems like this should be an easy thing to correct, if not to allow the creation of new longer tags, but to correct existing tags that have been truncated. ssh into an account which has no home directory How to delete a full stop on reference ending According to APA rules, after each reference there must the digital object identification (doi). I use JabRef as a reference manager and for some reason, if I put an article doi on its "doi" field, it won't show up after the build on my editor, but if I put the doi on its "note" field it shows up. However, it's adding a period after the note field. I need a way to have the reference without an ending period.  Here's the LaTex document. I use TexPad as my editor.  \documentclass[12pt,a4paper,oneside]{report} \usepackage[utf8]{inputenc} \usepackage[portuguese, english]{babel} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage[sort, longnamesfirst]{natbib} \usepackage[strings]{underscore} \bibliographystyle{newapa}  \usepackage{sectsty} \chapterfont{\fontsize{14}{15}\selectfont} \sectionfont{\fontsize{12}{15}\selectfont} \subsectionfont{\fontsize{12}{15}\selectfont} \renewcommand\bibsection{\section*{Referências}} \bibpunct[:\ ]{(}{)}{;}{a}{,}{,}  \title{} \date{} \makeindex  \begin{document}  \cite{aldous-newviewsgrandparents-1995}  \bibliography{/Users/Ricardo/Documents/Ricardo/library2.bib}  \end{document}   It produces this:     Aldous, J. (1995). New views of grandparents in intergenera- tional   context. Journal of Family Issues, 16(1), 104–122. doi:   10.1177/019251395016001006.   I need that last period out.  Thanks. I am seeking advice on which memory to use with my motherboard and CPU. Below are my components and some relevant information extracted from their official websites.   NOTE: Although this is a very specific case, a general answer is equally good for me — and also alot more useful for everyone else!  CPU: AMD Phenom II X4 810   ASUS M4A785TD-V EVO listed as one of many  motherboards.   Motherboard: ASUS M4A785TD-V EVO   4 x DIMM, Max. 16 GB, DDR3 1800(O.C.)*/1600(O.C.)/1333/1066/800 ECC,Non-ECC,Un-buffered Memory Dual Channel memory architecture (Complete list of specs: ).   NOTE II: I wouldn't want to do any overclocking as I have no experience with that. So I think the 1333 MHz is the most appropriate. Since the higher frequencies are marked with "O.C."  Questions:   How can I be certain that a specific RAM will work well (or not work) together with my CPU and motherboard? (The Qualified Vendor List is a good starting point, but it's a bit limited and sometimes a bit outdated). Is it true that choice of memory is not independent of the processor? How?    I'm worried I'll be missing something essential when buying my next RAM chips :) Resuming and Finishing the flow Created a flow in the sandbox to capture a survey.  I have enabled the ability to Pause the flow and added the component to Home page.   All the screens for the flow is complete and when I test run below 2 issues are found.  1. In the last screen I do not find the Finish Button. There is a record update after last screen. When I add a screen after record update It shows the finish screen But if click the previous button present there it moves to the first and create a new record. Is there any possibility to finish without going on the previous screens   When I resume a Paused flow, after clicking on the last screen, instead of redirected to the Account page, the flow is restarting. Appreciate your help.   On pause going to first screen is normal functionality , Is there any modification that can be done. sort but keep header line at the top I am getting output from a program that first produces one line that is a bunch of column headers, and then a bunch of lines of data. I want to cut various columns of this output and view it sorted according to various columns. Without the headers, the cutting and sorting is easily accomplished via the -k option to sort along with cut or awk to view a subset of the columns. However, this method of sorting mixes the column headers in with the rest of the lines of output. Is there an easy way to keep the headers at the top? "Fake" 3d effect in Photoshop/Illustrator with raster images