Show that $\gcd(a + b, a^2 + b^2) = 1$ or $2$ if $\gcd(a, b)=1$.  I have absolutely no clue where to start and what to do, please provide complete proof and answer.
If $\gcd(a,b)=1$, how can I find the values that $\gcd(a+b,a^2+b^2)$ can possibly take? I can't find a way to use any of the elemental divisibility and gcd theorems to find them.
I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are used) the better. Thanks More section headings? I am a novice in LaTeX and I have one question.  Can someone tell me what if you want more heading's in LaTeX. Like 5 or 10 or even 15. Is there a package for that? Commutator of a group $g^k$ is a primitive element modulo $m$ iff $\gcd (k,\varphi(m))=1$ Consider the following table:  id  title   genre1  genre2  genre3  genre4  genre5 1   Movie1  Thriller    Crime   NULL    NULL    NULL 2   Movie2  Fantasy NULL    NULL    NULL    NULL 3   Movie3  Political   Philosophical   Historical  NULL    NULL 4   Movie4  Science Fiction NULL    NULL    NULL    NULL   When I unpivot genre1, genre2, ... the NULL values are removed, so that, for example, the rows for Movie1 would be:  id title  genre     original_column 1  Movie1 Thriller  genre1 1  Movie1 Crime     genre2   Is there a way to include NULL values, for example:  id title  genre     original_column 1  Movie1 Thriller  genre1 1  Movie1 Crime     genre2 1  Movie1 NULL      genre3 1  Movie1 NULL      genre4 1  Movie1 NULL      genre5   I can SELECT ISNULL(genre1, '*'), ISNULL(genre2, '*') ..., but suppose I want the values to appear as NULL? Is there a better way than:  WITH  -- SELECT that unpivots table AS unpivoted SELECT unpivoted.id,    unpivoted.title,    ,CASE unpivoted.genre WHEN '*' THEN NULL ELSE unpivoted.genre END    ,unpivoted.original_transaction FROM unpivoted   Also, I notice that the query that returns the source data to be unpivoted MUST have an alias. For example:  SELECT ... FROM   -- *must* give it an alias or it's a syntax error   (SELECT ... FROM) AS source    UNPIVOT(genre for original_column IN (genre1,genre2, genre3,genre4,genre5))   Does anyone know why? I have not found a way to refer to the alias anywhere in the query. Installing 3rd party python libraries for QGIS on Windows Given the following HTML:   &lt;div id="container"&gt;   &lt;!-- Other elements here --&gt;   &lt;div id="copyright"&gt;     Copyright Foo web designs   &lt;/div&gt; &lt;/div&gt;    I would like #copyright to stick to the bottom of #container. Can I achieve this without using absolute positioning? "Started to work" vs "Started working" If I multiclass into 2 or more spellcasting classes, how do I determine my known/prepared spells? Basically, I've designed a booklet on Photoshop CC. It's an informational booklet so it's quite vibrant with lots of colours and pictures in it.   I've finished it and exported all the files into PDF format. Upon doing this, the PDF's are much "duller" for use of a better word. Now I'm not a designer and the whole thing was a learning process but now I'm reading about CMYK and RGB colour models and conversions etc. I designed the whole thing RGB (because CMYK looked duller obviously. The PDF's aren't disastrous, red is still red, black is still black but it's nowhere near as visually appealing anymore.   Is this just because of the monitor and the models each program uses or when I send my brochure to be printed, am I doomed for a much duller booklet? There's obviously a way around it otherwise all print would be dull. This is a common question asked by those who want to root their phones. How exactly do I root my Android device? Are there risks involved? Divergence of $\sum_{n=1}^{\infty} \frac{1}{2+i^n}$? I notice that,  $i^n = i, \quad if \text{ } n \equiv \text{ } 1 \text{ } mod \text{ } 4$  $i^n = -1, \quad if \text{ }n \equiv \text{ } 2 \text{ } mod \text{ } 4$  $i^n = -i, \quad if \text{ }n \equiv \text{ } 3 \text{ } mod \text{ } 4$  $i^n = 1, \quad if \text{ } n \equiv \text{ } 0 \text{ } mod \text{ } 4$  so the series $\sum_{n=1}^{\infty} \frac{1}{2+i^n}$ seems to be an alternating series that does not converge to any point in the complex plane. Is there a rigorous way to prove this divergence? How would you write a proof to this divergence or convergence (if I am wrong)?  Thanks! It seems "See votes, expandable user card" goes on a second line on the activity page of the user profile resulting in a pretty bad look:    You can see it live on  (though of course that may change any day now depending on their rep-growth).  I'm using Chrome Version 67.0.3396.99 (Official Build) (64-bit) I sadly haven't been able to answer this questions using Google. In beamer, I'd like to make the default font for the entire title page, as well as frame titles, to be the default font from the old  (example ). Is there any way to accept only numeric values in a JTextField? Co-Buyer Automobile As a co-buyer on an automobile, can the other party take full possession? Both parties names are listed as the Purchaser's. Only one purchaser signed the retail order for the motor vehicle, and if so, how can one become a no co-buyer? Find $x$ such that $(x^2 + 4x + 3)^x + (2x + 4)^x = (x^2 + 4x + 5)^x$ I have 16GB RAM. Do I need 32GB swap? 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: $\forall x,y>0, x^x+y^y \geq x^y + y^x$ Prove that $\forall x,y&gt;0, x^x+y^y \geq x^y + y^x$   A friend of mine told me none of the teachers in my school have succeeded in proving this seemingly simple inequality (it was asked at an oral exam last year). I tried it myself, but I made no significant progress.