If $a, b, c$ are integers, $\gcd(a,b) = 1$ then $\gcd (a,bc)=\gcd(a,c)$
How to show $a,b$ coprime to $n\Rightarrow ab$ coprime to $n$?
in QGis 1.8 were lots of predefined coordinate systems (national projections). Where have they all gone in qgis 2.0? I specifically need the Gauß-Krüger-Projections Zone 1 to 5.  Is there any plugin to load those coordinate systems? How do I fix overscan on my HDMI HDTV? I've connected a HDTV with HDMI on my Radeon 4670, using the open source ati driver. However there is "overscan" which cuts off a bit of each edge of the screen, about the size of the gnome-panel. How can I fix this so it displays the full resolution? Is the size of a Form in Visual Studio designer limited to screen resolution? How to get the EXIF data from a file using C# I would like to write a small program in C# which goes through my jpeg photos and, for example, sorts them into dated folders (using MY dating conventions, dammit...).   Does anyone know a relatively easy way to get at the EXIF data such as Date And Time or Exposure programatically? Thanks! Proving an identity for Bernoulli polynomials Consider the Bernoulli polynomials $B_n(x)$ given by the expansion  $$\frac{te^{xt}}{e^t-1} = \sum\limits_{n=0}^{\infty}B_n(x)\frac{t^n}{n!}.$$  I want to prove the identity $$B_n(1-x)=(-1)^nB_n(x).$$  This formula is given in a lot of books, but unfortunately without any proof. Therefore I'm wondering how to prove it. Do you have any idea?  I tried for instance:  $\sum\limits_{n=0}^{\infty}B_n(1-x)\frac{t^n}{n!}=\frac{te^{-xt}}{e^t-1}e^t=\left(\sum\limits_{n=0}^{\infty}B_n(-x)\frac{t^n}{n!}\right)\sum\limits_{n=0}^{\infty}\frac{t^n}{n!}=\sum\limits_{n=0}^{\infty}\left(\sum\limits_{l=0}^n\binom{n}{l}B_{l}(-x)  \right)\frac{t^n}{n!}$.  Hence  $B_n(1-x)=\sum\limits_{l=0}^n\binom{n}{l}B_{l}(-x)$.  But I do not know how to proceed.  On the other hand I know that $B_n(x+1)-B_n(x)=n x^{n-1}$. Hence  $B_{n}(1-x)=B_{n}(-x)-n(-x)^{n-1}$. But again, I do not know how I can go further.   Hopefully someone can help me.  Best regards Drift velocity of electron JavaScript raises SyntaxError with data rendered in Jinja template Why should I install hardware drivers if my computer is working fine? I've just installed Windows 7 on my PC.  I went online to look for drivers. For , Foxconn offers LAN drivers, on-board audio drivers, and chipset drivers. And for , Nvidia offers a graphics card driver, obviously.  But then I stopped and wondered... what benefit would I gain from installing them?   The LAN cable is plugged in and I'm online; no problems there.  The on-board audio is working fine.  A quick Google tells me that a chipset is responsible for moving data around my motherboard – which seems to be happening OK!  And my graphics look perfect.    So why should I install the drivers? What would they give me, other than badly designed "management" interfaces, and pointless "notifications" whenever I plug in an audio jack?  If there are different considerations for each of the components – chipset, audio, LAN, and graphics (and any others you think I should be aware of) – please break it down in your answer.     Update: I've had a couple of answers already, saying that I'll get better stability, bugfixes, etc.   Can I slightly expand my question... Can I get all these same benefits if I install the drivers via Windows Update? And if I do this, would I avoid having to install the vendor-specific UIs, which I find so annoying? Why is it that     All right angles are equal to each other    -a postulate in Euclid's Elements (). Shouldn't it be a congruence rather than an equivalence? Isn't this just a special case of the definition of congruent angles? Why is my Wi-Fi process using 3GB of memory? Find $\lim_{n \to \infty} \left[\frac{(n+1)^{n + 1}}{n^n} - \frac{n^{n}}{(n-1)^{n-1}} \right]$ (a question asked at trivia) How to remove GNOME Shell from Ubuntu 20.04 LTS to install other desktop environment from scratch? I have just installed the final version of Ubuntu 20.04 LTS desktop system from ubuntu-20.04-desktop-amd64.iso. After two days of testing I completely do not like its graphical interface - the GNOME Shell. Even installation of GNOME Flashback does not help either.  So I want to completely remove the GNOME Shell with all its trails from my system without re-installation. Just want to start from scratch like it was approximately if I have installed Ubuntu from . How to add a script to Startup Applications from the command line? Recently I've changed jobs, and consequently the field of research has also changed. But I'm still getting proposals for publications in the previous field, and it is interesting for me to accept some of them. My concern is about which affiliation to use: on one hand I gain all the knowledge in that field while working on the previous employer, also I hope they can cover my expenses related to the publication. On the other hand, I do not work for them anymore, and my current company has very little interest in my old field of research, and probably will not support it financially, but I think it is wise to mention them as well. As a compromise I'm thinking to put the previous employer in the affiliation, and the name of my new employer in the footnote, something like "currently at XYZ". Surely I'll discuss this issue with both, although the opinion of the community is also very valuable. What bizarrities lurk within the decimal expansion of an irrational number? i.e. can we write  $\pi = 3.14159\dots X\dots$  where $X$ consists of (say) $10^{100}$ consecutive zeroes?  [Originally asked on reddit without response :-( ] I often find myself wanting to install something, but I'm unsure of what package it's in. This is a very common occurrence for me:  $ make html sphinx-build -b djangohtml -d _build/doctrees   . _build/html make: sphinx-build: Command not found make: *** [html] Error 127  $ sudo apt-get install sphinx E: Unable to locate package sphinx  $ sudo apt-get install sphinx-build E: Unable to locate package sphinx-build   googles  $ sudo apt-get install python-sphinx   Is there a better way? Uniformly convergent implies equicontinuous I'm trying to prove that if I have a sequence of continuously differentiable functions $f_n$ that converge uniformly on $[a,b]$, then $\{f_n\}$ is equicontinuous for all $x_0 \in [a, b]$. My idea is to use uniform convergence to deal with the &quot;tail&quot; and then use continuity to deal with the finitely many $f_n$'s left. But I'm having trouble writing it down. Numbering figures and tables by their section I am having trouble formatting my list of figures and list of tables.  Currently, my tex document is printing the list in the following manner:     1 Foreign Key Constraint Problem Due   to Improper Table Design. . . . 3      2 Application Flow of Territory   Maintenance Tool. . . . . . . . . . .   . . 4   But I want to number figures by their sections:     3.1.2 Foreign Key Constraint Problem Due to Improper Table Design Print floating point values without leading zero Simple Past vs. Present Perfect: "was" vs. "has been" Possible Duplicate:          As a English non-native speaker it is difficult for me to understand when I must use present perfect or past simple because in my official language there isn't the present perfect tense. I know that present perfect is related to something that happened in the past but its result is important now. But for instance:        The message has been sent.   The message was sent.      Please help me point me out any examples/contexts where I should use 1. instead of 2.