Show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$
How can I find the possible values that $\gcd(a+b,a^2+b^2)$ can take, if $\gcd(a,b)=1$
Should an exactly equal edit be dropped? I just improved an answer which was one second later edited by another user with the same actions, so that the diff is completely empty. I think the later edit should been dropped automatically.  Here is the question I'm talking about: I am currently learning electromagnetism and I am getting really annoyed using a physical hand to accomplish the right hand rule. I get it, i've seen diagrams, and i've solved problems with it but still somehow I mess up because I bend a finger or twist my wrist a strange way. I am tired of using my hand.  So my question is can I find an entirely mathematical basis for this?  Lets say for any given diagram I define the positive $x$ axis as going to the right, the positive $y$ axis as going up and the positive $z$ axis as going into the page. Then I just did unit vectors for everything to determine the missing component?  As I understand it $\vec{F} = q (\vec{V}\times\vec{B})$ and with that definition I am confident I could find out the direction of the electric field given the velocity of a particle of charge and the magnetic field but how can I re-arrange this so I can solve for other variables? For example what does $\vec{B}$ equal in terms of a cross product involving $\vec{V}$ and $\vec{F}$? Seems like a stupid question, but I'm after a nice solid answer if possible.  I joined SO a while ago to ask the odd question, but recently I'm starting to use it more often.  I've been reviewing the 40 max items (20 first posts/20 late answers) per day, and editing anything I see wrong with them (lots of +2's)... flagging them... etc.  I've also been  per day.  But other than commenting where I can and answering the odd question, I'm only  (1k at the moment).  I might just be overeager, but I was hoping some high-rep users might be able to write a little something on how they earned their stripes?  I've , and to be honest I was hoping I'd get a hit, because it sounds like a stupid question... anyways, here you go!  Also, I'm not referring to how to gain rep on Meta, since it's a bit of a fluke I've earned the rep I have on here. I'm asking about main SO. What is the smallest viable reproducing population, such as in a human population. By viable I mean a population which keeps genetic defects low (enough).  A very strongly related question: what is the expected number of generations a given population can survive? Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$ dual boot missing files on ntfs I have 3 partitions: one for Windows 7 (ntfs), one for Ubuntu (ext4) and one just for data (ntfs so both operating systems can see them).  My problem is that I had stored some files on the data partition using Ubuntu and when I booted Windows 7 all that data was gone! After that I couldn't find the files even when using Ubuntu.  My files were simply GONE :(  Is there something I can do in Ubuntu or is it just windows problem? Please provide a mobile-optimized captcha page Set limit to one author when using "et al." in biblatex In the header of my document I set the maximal author names to 2:  \usepackage[style=authoryear-icomp, maxbibnames=9, maxcitenames=2, backend=biber]{biblatex}   Now, when I cite texts with two authors, LaTeX gives me the following:     (Author_A/ Author_B 2012: 232)   When I have three authors or more, LaTeX makes it:     (Author_A/ Author_B et al. 2012: 232)   But I want LaTeX to just name the first author if there are more than two authors ... like this:     (Author_A et al. 2012: 232)   Any tips on how to achieve this ? Is there any way I can 'undo' the factorial operation? JUst like you can do squares and square roots, can you do factorials and factorial roots (for lack of a better term)?   Here is an example: 5! = 120.  Is there a way I can work out the number that must be factorialed (??) to give the answer 120? Incorrect icons on hot network questions I see the same problem has occurred in the past, the most-linked one being   Is this happening for anyone else?   I've marked the icons I could recognize as incorrect.  This exists on every page, is on both chrome and firefox (from Ubuntu 16.04), and persists after hard-refreshes in each.  Edit: The icons in Meta's Hot Network Questions tab appear to be consistently accurate. Every other page I checked (including Unix, Workplace, English Language &amp; Usage, Money, and Sci-fi) have a handful of incorrect ones, even for some of the same questions that are displayed correctly in Meta.  Further note: if it matters, the icons are also incorrect in the drop down community-search feature on the right end of the header. Again, they appear correct on this Meta page, and incorrect on all other pages I've checked. Some of the ones that are consistently incorrect:   computational science constructed languages movies and TV raspberri pi rpg russian language science fiction and fantasy sitecore skeptics super user latex workplace   That distribution seems skewed, but I may just be disproportionately familiar with the sites at the end of the alphabet. Many others could be wrong, those are just the ones I could recognize.    Last edit: apparently my Google skills were lacking -  was asked an hour prior. How disk space is used on Android device? Why does the definition of limits of a function have strict inequality? Definition (As written in Michael Spivak's Calculus)     The function $f$ approaches a limit $l$ near $a$ means: for every $\epsilon &gt;0$ there is some $\delta &gt; 0$ such that, for all $x$, if $0&lt;|x-a|&lt;\delta$, then $|f(x)-l|&lt;\epsilon$.   my question is: why can't it be: $$0&lt;|x-a|\leq \delta,|f(x)-l|\leq \epsilon$$ After looking at limits of functions for a long time just to grasp it's meaning and using the definition quite a lot solving homework I realized I keep writing the same inequality without really understanding why.  The only explanation given in Spivak's book for this part of the definition goes over it without explaining the inequality. I tried looking for an explanation myself but wasn't really able to find anything wrong with it. Is it also possible to write the definition like that or is there a problem with that?   (first non-homework related question :p) What are the summary rules for when to use image stabilization and when not to? When I got my DSLR, I took a lot of pics and all of them came excellent, even though I had VR at the "Off" position.   Since those came out so well, I wonder when should I turn VR on, and when I should just leave it off.   I found  which answer this question in detail, but that's too much for me to understand easily. Can someone explain the strict rules in short?  Should I use VR with tripod? When should I use it when I'm not using a tripod, and when should I leave it off? What are the advantages and limitations of dynamic type languages compared to static type languages?  See also:  (a far more argumentative thread...) I was writing a toString() for a class in Java the other day by manually writing out each element of the class to a String and it occurred to me that using reflection it might be possible to create a generic toString() method that could work on ALL classes. I.E. it would figure out the field names and values and send them out to a String.  Getting the field names is fairly simple, here is what a co-worker came up with:  public static List initFieldArray(String className) throws ClassNotFoundException {      Class c = Class.forName(className);     Field field[] = c.getFields();     List&lt;String&gt; classFields = new ArrayList(field.length);      for (int i = 0; i &lt; field.length; i++) {         String cf = field[i].toString();         classFields.add(cf.substring(cf.lastIndexOf(".") + 1));     }      return classFields; }   Using a factory I could reduce the performance overhead by storing the fields once, the first time the toString() is called. However finding the values could be a lot more expensive.  Due to the performance of reflection this may be more hypothetical then practical. But I am interested in the idea of reflection and how I can use it to improve my everyday programming. Every word is marked as misspelled in TeXstudio Why is that most of the words, almost 90 percent, are marked as misspelled words when there is no error? And the spellchecking suggestions don't even look like English.  I am using TeXstudio as my editor. How to define servlet filter order of execution using annotations in WAR Why does the review system disable the "First Posts" and "Late Answers" if I reached daily upvote limit? I was reviewing on , and I faced a weird thing. I can review "Close Votes", "Low Quality Posts", and "Suggested Edits" if I reached my daily vote limit, but I can't for "First Posts" and "Late Answers". Why?    If they just deserve an upvote/downvote then they should not include in the review system. But if we can improve them through leaving comments and by editing the posts then why can't we access them? The more I read about this, the more confused I become. Illuminance, as I understand it,  measured in lux, is the perceived brightness of light on a surface. It's what you'd measure with an incident light sensor. Luminance, then, is how much reflected light you'd perceive looking at that surface. This is what the light meter in a camera measures. Am I correct so far?  In both cases, the word "perceived" is important, because the scales are weighted using a  to map wavelengths to . However, in the case of illuminance, there's only actual perception if your eye happens to be the surface in question.  I can basically grasp this, but then I come across charts saying things like "family living room: 50 lux". Wait, hold on! Does that really mean that the typical lights in a house are that bright, or is it just confused and wrong, or am I confused and wrong?  If you're not taking pictures of light sources directly, why would an incident light meter reading ever be useful in photography? Reflected light recorded on film or by a sensor is usually what makes a photograph. So, if I have an incident light meter, how does that reading relate meaningfully to my camera settings?  Since incident light meters are sold and used, this implies that there must be some useful conversion. But this is where my brain explodes. Google tells me that 1 lumen equals 1 candela, so therefore 1 lumen/m² (i.e. 1 lux) must equal 1 candela/m2 (i.e. 1 nit). But clearly something is missing from this. There's something called a "steradian". Cones are involved. I'd never heard of this before; how does it fit in? I can see how it might be determined when calculating the usefulness of LED home lighting, but for a photo, I'm at a loss.  Some part of my (exploded, from the last paragraph) brain is trying to relate this to the difference between flash metering with TTL and with an incident meter. But how can the incident meter work without knowing the reflectance properties of the objects in the scene? Is this what the "C" in the  is all about? Is it just an average-this-probably-will-work value, or is there more to it than that? And if it is just an average, what knowledge is necessary for compensating for off-averages scenes? (As with the , where the photographer simply judges if the scene should be rendered brighter or darker than the average given by the meter.)  So yeah, so much confusion. In short:   What's the difference? Can one meaningfully convert between the two? When and how are illuminance / incident light measurements useful for photography?     Update: I appreciate Stan's answer, which covers the third point of when and how pretty well. And I think I've basically got the first point figured out, as described above. But I'd appreciate some answers covering the issue of conversion as well, both in the mathematical abstract and as the practical for photography. And I wouldn't mind more why and how, either. I keep getting an error in line 38 saying "Extra alignment tab has been changed to \cr." \documentclass [12pt]{article} \oddsidemargin=0pt \topmargin=0pt \textwidth=6.5in \textheight=8.5in \begin{document} \begin{center} {\large\bf\LaTeX\ Example 6} \end{center} We can typeset array structures \begin{equation} \mathbf{A}\times \mathbf{B}=\left| \begin{array}{ccc} \mathbf{i} &amp; \mathbf{j}  &amp; \mathbf{k} \\      a_1  &amp;      a_2        &amp; a_3 \\       b_1  &amp;      b_2        &amp; b_3  \end{array}  \right| \end{equation} \begin{quote} \begin{verbatim} \begin{equation} \mathbf{A}\times \mathbf{B}=\left| \begin{array}{ccc} \mathbf{i} &amp; \mathbf{j}  &amp; \mathbf{k} \\  a_1  &amp;  a_2   &amp; a_3 \\  b_1  &amp;  b_2   &amp; b_3 \end{array} \right| \end{equation} \end{verbatim} \end{quote} Or \begin{equation} \mathbf{A}=\left( \begin{array}{ccc} a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\ a_{21} &amp; a_{22} &amp; \cdots &amp;  a_{2n} \\ \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ a_{n1} &amp; a_{n2} &amp; \cdots &amp; a_{nn} \end{array} \right) \end{equation} \end{document}