So I have the following proposal:    $a$ $\in$ $\mathbb{Z}_{+}$ : $a$ = $2k +1$ $&gt;$ $2$ : 3 $\not$ |  $a$ and $&gt;2$ then $12$ $|$ $(a^2 -$1 ).  My initial idea was to show:  $a^2 -1\equiv 0 \mod 12$  Logically it follows:  $(2k+1)^2 -1\equiv 0 \mod 12$  $4k^2 + 4k\equiv 0 \mod 12$  However I do not know where to go from here. This proof also doesn't incorporate the condition 3 $\not  |$ a . Any tips? Sorry for formatting, this is my first time using MathJax.
Show that if $n$ is not divisible by $2$ or by $3$, then $n^2-1$ is divisible by $24$.   I thought I would do the following ... As $n$ is not divisible by $2$ and $3$ then $$n=2k+1\;\text{or}\\n=3k+1\;\text{or}\\n=3k+2\;\;\;\;$$for some $n\in\mathbb{N}$. And then make induction over $k$ in each case.$$24\mid (2k+1)^2-1\;\text{or}\\24\mid (3k+1)^2-1\;\text{or}\\24\mid (3k+2)^2-1\;\;\;\;$$This question is in a text of Euclidean Division I'm reviewing, and I wonder if there is a simpler, faster, or direct this way.
What is the term for taking it out on someone even though you will be harmed by this too? When you are so mad at someone that you make a bad decision, even though you will be harmed by it as well, in order to harm that person; is there a good term for this?   For example burning $100 because he owns $20 of it. How to include audio file in PDF How do I include audio file, or should I say, link to audio file in PDF?   Audio should be playable "from PDF directly", without opening any additional window. Is it possible to attach audios to PDF     then so I can manipulate only with     that file, without being bothered     with copying folders where those     audio files would reside?   I saw that 1. is definitely doable (but don't know how) and I'm interested if second one is also possible (while compiling .tex file) since I know that some files (like graphs and tables) can be attached in PDF, but after it's generated.   Button for playing audio from document that uses this trick. The equals() method in Java works unexpectedly on Long data type FontAwesome font not found? I have an unlocked North American (Canada) iPhone 5s, where can I buy a data only SIM card in Japan ? I'm not interested in renting nor in a voice plan, I'll be staying for 4 weeks.  I'll be landing in Tokyo, so preferably around there. Can I get them from the airport somewhere ? I'm assuming I won't have problems with the network/frequencies, am I correct ? "batch" tag turns into "batch-file" Why a tesselation of the plane by a convex polygon of 7 or more sides is not possible? I read in several places, including Wikipedia, that a tessellation of the plane by a single, convex, $n-$sided polygon is not possible for $n\geq7$. I was not able to locate a proof, or a paper that discusses this. Maybe it is too trivial, but I am not able to figure it out myself.   So I would like to ask you for help. Where can I find a proof for this?  Also, I am interested in the mathematical background behind plane tiling (especially periodic), If you can suggest papers on the topic it would really help. how to fix ioctl permissions denied error? I searched in lots of websites/forums but did not find a solution for this problem, and this site is last chance, this is problem: This problem have several other users too. my phone is Galaxy S7 Edge , exynos, Oreo 8.0 (Stock ROM), rooted with magisk latest version. root works well in other applications. In pentesting applications like csploit, Network Spoofer, WifiKill and others  root works well(grants permission)  but when start using options of app(for example, arp spoofing, network sniffer, MITM, etc) drops same error: [ERROR] Unable to retrieve local hardware address libnet ( ioctl: Permission denied ) what to do? Can this issue fixed? or exactly which permission is it?  Also one interesting thing, when I installed old version of WifiKill app it worked - killed wifi connection. Remove blank attributes from an Object in Javascript How do I remove all attributes which are undefined or null in a JavaScript object?  (Question is similar to  for Arrays) I find in many places mentions of pre-amplifiers being used to boost audio signals prior to feeding to a power amplifier.  Wikipedia's article on  gives some hints at their usage, but it doesn't really explain why it would be necessary to have two amplifiers in series rather than a single one with greater gain.  If a pre-amp is just a power amp with a smaller gain, what's the point, and why the distinction between the two kinds? Is the distinction in function, or simply purpose? Centre of $GL_n(\mathbb R)$ How to prove that center of $GL_n(\mathbb R)$ is scalar matrices?  Again what can you say about the center of set of all upper triangular matrices and center of set of all diagonal matrices. Solvability and reducibility of a polynomial in a "chain" of finite fields Trying to run , embedded Tomcat instance fails.  Error seems to appear here:    Apr 17, 2015 2:13:47 AM org.apache.catalina.startup.HostConfig deployDescriptor  INFO: Deploying configuration descriptor /opt/OpenForisCalc/calc-server/tomcat/conf/Catalina/localhost/calc.xml  Apr 17, 2015 2:13:58 AM org.apache.catalina.core.StandardContext startInternal   SEVERE: Error listenerStart   Then if messes up with the rest (this repeats a lot):  Apr 17, 2015 2:13:58 AM org.apache.catalina.core.StandardContext startInternal  SEVERE: Context [/calc] startup failed due to previous errors How to reformulate differential equation problem as OrnsteinUhlenbeckProcess How do you end with a single quote followed by a double quote? I have a paragraph as follows:  ``This is a quote. `Here is a quote within a quote.'''   But the formatting works out to be a double quote (") followed by a single quote (') instead of the other way around. How can I solve this problem? Calling 32bit Code from 64bit Process I have an application that we're trying to migrate to 64bit from 32bit.  It's .NET, compiled using the x64 flags.  However, we have a large number of DLLs written in FORTRAN 90 compiled for 32bit.  The functions in the FORTRAN DLLs are fairly simple:  you put data in, you pull data out; no state of any sort.  We also don't spend a lot of time there, a total of maybe 3%, but the calculation logic it performs is invaluable.  Can I somehow call the 32bit DLLs from 64bit code?  MSDN suggests that I can't, period.  I've done some simple hacking and verified this.  Everything throws an invalid entry point exception.  The only possible solution i've found so far is to create COM+ wrappers for all of the 32bit DLL functions and invoke COM from the 64bit process.  This seems like quite a headache.  We can also run the process in WoW emulation, but then the memory ceiling wouldn't be increased, capping at around 1.6gb.  Is there any other way to call the 32bit DLLs from a 64bit CLR process? 90s (or earlier) Japanese anime movie: two planets at war, young man is in love with princess from the other planet, and floods his own village I would like to find out the title of a Japanese anime feature film (not a series) that I saw on Romanian TV in the early 90s.   It was about a feud between the peoples of two planets (one very technologically advanced, the other not). The two planets go to war against each other, and a young man from one of the planets falls in love with a princess from the other planet.  I recall a scene where the army of the second planet take off in hot air balloons when they go to war.  The only other thing I recall is that there is a dam next to the village where the young man lives, and for some reason he opens the dam gates and floods his own village.   I do not recall there being robots in the movie, however I remember very vaguely that the technology level was probably at the level of late 19th century Europe on one planet and pre-industrial on the other one.  What it's not:   Nobunaga the Fool Aldnoah Zero Queen Millennia Uchuu Senkan Yamato (movie) Since when did kidnapping come to include adults too? As per , the word 'kidnap' originated to denote nabbing away of a child. When and how did kidnap come to denote nabbing of adults?  Update: Just found a link to a  that mentions kidnapping of souls. Now not sure how correct the above link is. Is there a way to export large multi-layered Illustrator file to Photoshop?  When I export with preserve layers and text edit-ability, the complex layers are flattened. I've reorganized my layers to have simple groupings but that's time consuming.  Is there a more effective way of doing this? Or script to automate it? I have a Python module installed on my system and I'd like to be able to see what functions/classes/methods are available in it. I want to call the help function on each one. In Ruby I can do something like ClassName.methods to get a list of all the methods available on that class. Is there something similar in Python? eg. something like: from somemodule import foo print(foo.methods)  # or whatever is the correct method to call