Prove that if $a$  is odd, then $a^2\equiv 1\pmod 8$I got this question in my discrete mathematics class, can anyone help me?Thanks.
To prove that if $n$ is odd, then $8 \mid n^2−1$, observe that as $n$ is odd either $n = 4k+1$ for some integer $k$ or $n=4k+3$ for some integer $k$. In the first case $n^2 -1=8(2k^2+k)$ and in the second case $n^2-1 = 8(2k^2 + 3k + 1)$.  Given an odd integer $a$, establish that $a^2 +(a+2)^2 +(a+4)^2 + 1$ is divisible by $12$. Solve by consideration of six cases.
How to print all printable ASCII chars in CLI? Is there PvP in Diablo 3? How does it work? Since Naruto failed the Genin exam so many times, doesn't that mean he's older than Sasuke and Sakura? Gtk skip_taskbar_hint (seen in Glade-ui-Designer) equivalent in Qt With Glade Designer that lets you set up Gtk-Ui's you can check the option that windows are not shown in the launcher. If I take a look at the ui-file generated by Glade, it says &lt;property name="skip_taskbar_hint"&gt;True&lt;/property&gt;. In gtk code this hint is  window.set_property('skip-taskbar-hint', True). However I believe that this option is not exclusively Gtk-specific, and want to apply this one on my PyQt4-app. The Ubuntu window manager must be able to receive that hint, otherwise I would see the window of a gtk-application i use (stickynotes-indicator). So there MUST be a way of doing this in Qt or any other language, right?  Does anybody know such an option? Python or Qt syntax, doesn't matter! Multiple Linked Objects with unique Materials / Group Materials override How does update work with programs installed from .deb file I'm having some difficulties exactly understanding how the the whole update thing work in Ubuntu and good practice for installing software.  I understand I have a  /etc/apt/sources.list file where all my repositories are listed, and that these repositories are queried when I use apt-get update - to later be used with e.g. apt-get upgrade.  This makes perfect sense and I recently installed spotify by adding deb http://repository.spotify.com stable non-free to this "sources" list.    But then I got confused...  When I went to download Google Chrome I merely had to download and grab a .deb file, and Chrome installed with no problem... but I don't see any new entry in /etc/apt/sources.list...  So how does apt-get update know where to query concerning Chrome updates?  Has it somehow been added to one of the already listed repositories in the sources file?  I would like all my installed software to be encompassed by the update function. The following is a homework problem:  Prove that between two roots of the function $f(x) = \exp(x) \sin(x) -1$ there must be at least one root of the function $\exp(x)\cos(x) +1$.  I think it is possible to prove it by simply computing the signs of the functions at $\frac{\pi}{2}k, \pi k, \frac{3\pi}{2}k, 2\pi k$. However, I'm looking for a more elegant way. It is intuitively clear that the roots of $f$ are ˋˋconverging'' to the roots of $sin$ and the ones of $g$ to the ones of $cos$. Do you have any hint how to construct a nice argument based on this observation? Redirect to SSL only if browser supports SNI Where to get QGIS toolbar icons? Does anybody know where to download the QGIS toolbar icons? How to execute code after html form reset with jquery? Is there a way of finding a formula for $\sum\limits_{k=1}^n k^k$? Maybe I'm missing something really obvious, but I've looked around a bit on the Internet and I haven't been able to find anything.  So, what I'm looking for is a formula in closed form to generate the sequence $1,5,32,288,3413,\dots$ Find all integral solutions of $ x^4 + y^4 + z^4 -w^4 = 1995 $ Locking an iPhone if button is broken? My friend's iPhone's sleep/wake button no longer works. Is there any way to turn off the screen besides setting up auto-lock and waiting a few minutes? I have a simple skull model, to which I applied the subdivision modifier. For the most part everything works as expected, except I keep getting these weird sharp vertices and I can't seem to figure out how to flatten them.    I'm using Blender 2.8. Is this some kind of feature I can disable/enable for particular vertices? The distance between the sharp vertices and the gray outline differ.    Any suggestions please? Thank you!  EDIT: I also tried selecting as both - vertexes and edges and then applied Edge->Clear Sharp, but nothing changed. From Windows Vista to 7, can I get rid of folder Windows.old? I've just changed my laptop OS from Windows Vista to Windows 7.  I didn't choose to have both OS together, I just want to have Windows 7.   I was specting Windows 7 instalation program to format/clean my HD and leave me most of its capacity for a fresh start. But I've found, once the OS has been isntalled, that all files and programs from the Vista era are still in my HD, in a folder called Windows.old   I don't need these programs and files, I want to have as much HD space as possible and Windows 7 as clean as possible, what should I do?  Can I just delete Windows.old folder? Would that affect the how Windows 7 works?  Maybe is because I ended up angry with Vista, but I don't like the fact that there wasn't any cleaning/formatting ..   Thanks a lot Add an application to the command search path in csh Recently, I have installed an application (gnuplot) under user directory (/home/usr/bin). I want to run the application whenever I type 'gnuplot' in any directory. What could I do?  I tried to set the path in ~/.cshrc by  setenv gnuplot /home/usr/bin/gnuplot   but it doesn't work. It reports "command not found" as I type gnuplot in the terminal. How to show $a,b$ coprime to $n\Rightarrow ab$ coprime to $n$? Given the key, the plain text and the cipher text can I calculate the IV used in CBC mode? If I have the plain text, the ciphertext and the key for an AES-128 CBC operation, can I determine the IV, even if I don't know the padding (assuming the padding follows one of the more common formats)?  I believe it should be possible since my understanding is that the IV is only used as an initial XOR of the plain text in encryption, and should, therefore, be able to be available like this:  AESDecrypt(key, ciphertext) =&gt; (PlainText XOR IV); IV = ((Plaintext XOR IV) XOR (Plaintext))   Is this correct? I installed Android studio some time ago using the following suggested method:     To set up Android Studio on Linux:      Unpack the downloaded Tar file,   android-studio-ide--linux.zip, into an appropriate location   for your applications. To launch Android Studio, navigate to the   android-studio/bin/ directory in a terminal and execute studio.sh. You   may want to add android-studio/bin/ to your PATH environmental   variable so that you can start Android Studio from any directory.   Follow the links to install the SDK outside of the Android Studio   directories.   For whatever reason it has become corrupt and insists I must un-install and re-install. I can't for the life of me find how to un-install it however.  I didn't use apt-get so I can't do apt-get remove.  Does anyone have any suggestions? What is the definition of "creature" and is it used consistently?