I have the following theorem:  If $a = p^{n_1}_1 p^{n_2}_2$ ... $p^{n_k}_k, b = p^{m_1}_1 p^{m_2}_2$ ... $p^{m_k}_k$, then  $gcd(a, b) = p^{min(n_1, m_1)}_1 ... p^{min(n_k, m_k)}_k$  How can I prove it? Any solution or hint?
How should I approach obvious proofs like these    I have been trying but couldn't find an elegant way to work these. Any help is highly appreciated ! Especially looking for a nice proof/hint for first problem
Proof of $Y=F_X(X)$ being uniformly distributed on $[0,1]$ for arbitrary continuous $F_X$ Including the interaction but not the main effects in a model Is it ever valid to include a two-way interaction in a model without including the main effects?  What if your hypothesis is only about the interaction, do you still need to include the main effects? Installing Kali on a USB drive - NOT Live USB or Persistence partition Greeting's, to clarify; my question is not creating a Live USB drive or a persistence partition but creating a USB with GRUB bootloader (or other) and Kali which can be used on any machine. This is similar to running other linux OS directly from USB. I am currently running Ubuntu from a pen drive. I installed the OS (Ubuntu) and the bootloader on the pen drive and I can boot off it anywhere. Here is a video explaining the same (video is not by me: )  Since, Kali's installer doesn't give option to select which drive to write the booloader to and directly writes on the HDD (sda1). This means even if I partition and install Kali on a pen drive since the bootloader will be installed on the machines fixed primary drive, the USB drive will fail to boot on a different device.  I have two questions:   Anyone has any idea how to write bootloader to a USB drive and what parameters should be given so that it boots with Kali installed on it. I am using graphic install mode for installing Kali on the USB drive, however there is no option for the install to be encrypted (Full disk encryption or at least /home folder to be encrypted). Does anyone know how can I install Kali with full disk / home folder encryption? Or enable is post install?   Thank You! The real fastest gun in the west Are the questions in the about page taken from real posts? If not, who designs them?  Particularly, why does this specific answer have a date from before the question was asked? Switching back to last *used* tab on Chrome understanding "mount" as a concept in the OS I want to understand what mounting is. It is used in different contexts and situations and I can't find resources which:   Describe the mount concept Explain the actions taken by the computer/OS/utility when a mount is performed How and in which situations mount is used Which features in the Linux mount command are of frequent use and some examples   ( I hear mount applied to diverse entities directories, flash drives, network card, etc ) A physical explanation for negative kelvin temperatures Android: fragments overlapping issue What Happens to electrons after Alpha Decay and Nuclear Fission? Where do the electrons go? In alpha decay do 2 electrons follow the alpha particle and make stable Helium or does the larger daughter nucleus become an anion?  Also what do the electrons do in the mixture of fission and alpha decay? With Beryllium-8 it decays into two alpha particles and 4 lonely electrons or do you get two Helium atoms?If someone could add an entry to Holocron it would be greatly appreciated, thanks. Can 'AMD-only' RAM really be restricted to AMD chipsets, and if so, why? I was browsing for a RAM upgrade earlier for my system, and came across 'AMD-only' RAM (aka, RAM that only works on AMD chipsets).  This RAM can , and seems to be slightly cheaper than 'normal' RAM.  The RAM prompts a question: is this RAM really restricted to AMD chipsets?  When considering RAM for a system, there are certain considerations to make such as:   Speed (1066MHz, 1600MHz, etc); CAS (latency, lower is better); Generation (DDR2, DDR3, DDR4, etc)   However there is usually no such consideration as chipset compatibility. Sure, it's possible to guarantee that a given stick of memory will work if it's within the bounds of the chipsets listed in the information given by the seller/manufacturer.  For instance if a series of chipsets supported RAM between 1600MHz and 2400MHz, with 9-12 CAS, and on DDR3, it would be possible to list a product as being fully compatible with that series of chipsets if the said RAM was 2000MHz, 10 CAS, and DDR3.  I suspect that's partially the story here. However, is it possible at all for a given stick of memory to be 'AMD-only' or 'Intel-only', and if so, how? Well, you can see it for yourself. The badge multiplier is a bit misaligned. Can you fix this please? (I know it's kinda a minor issue but it's bugging me and it's somehow off)    If it helps, I'm using Chrome version 44.0.2403.125 m. (Also repro'ed on FireFox 39.0 and )    For no good reason, that's a screenie from 's profile page. Differences in meaning when the verb tense changes (headlines) What is the difference between the examples below?  Generally, in a newspaper, news is based on past tense (things which have already happened).   Then why are headlines written in ways that says: He Kills (shouldn't it be: He Killed)?  For example:  Parliament passes bill vs. Parliament passed bill  Also  He kills his friend vs. He killed his friend  Please help. Concurrently interleaving execution of Lua and TeX in LuaTeX The usual way to interleave Lua and TeX in generating documents is by invoking Lua from Tex through the \directlua facility and generate Tex from Lua using tex.print().  For example, consider the Plain LuaTeX:  \directlua{     es=""; someluatex = function () tex.everypar="PAR!" end     tex.print("one", es, "two")     someluatex()     tex.print("three", es, "four") } five\bye   Looking at the document LuaTeX generates for that input, we see that someluatex() is run before the Tex backend does anything with the output: the PAR!s will be inserted before "one" and "two", as well as after "three".  Is there best practice for ensuring that TeX code issued by some Lua code is issued before some other LuaTeX code?  There are several ways to do this: with some code I am working with I am trying the following scheme:   Wrap the code executed by the \directlua in a coroutine, call it myco; At the point we want to ensure the Tex so far tex.print-ed is executed, use:  tex.print("\\directlua{coroutine.resume(myco)}") coroutine.yield()  If everything is in order, the Lua will resume after the Tex is processed.   This is not ideal, however: we must ensure that the TeX backend is in the right state to understand the \directlua command, which may not always be the case.  Is there a better way to achieve this? Unit ball in $C[0,1]$ not sequentially compact This question is taken from Saxe K -Beginning Functional Analysis.   Show that the closed unit ball in $C[0,1]$ is not compact by proving that it is not sequentially compact.   (It's assumed that we are using the uniform norm).  I've been working on this for ages but I could not come up with any sequence $\{f_n\}$ in the unit ball such that there exists $N\in \mathbb{N}$ such that for all $m,n\geq N$ we have that $d(f_n,f_m)&gt;c$. Should be a nice example of this, please help me! How do you programmatically set an attribute? Suppose I have a python object x and a string s, how do I set the attribute s on x? So:  &gt;&gt;&gt; x = SomeObject() &gt;&gt;&gt; attr = 'myAttr' &gt;&gt;&gt; # magic goes here &gt;&gt;&gt; x.myAttr 'magic'   What's the magic? The goal of this, incidentally, is to cache calls to x.__getattr__(). Does it make any difference to SEO ratings what-so-ever if your url has a trailing slash:  http://www.example.com/some/slug/paths/   as opposed to one that does not:  http://www.example.com/some/slug/paths How do I remove a directory and all its contents? In bash all I know is that   rmdir directoryname   will remove the directory but only if it's empty. Is there a way to force remove subdirectories? Gauge fields and strings: Loop equations Drush from GitHub stopped working I have a  that has been working since it moved there.  I did a git pull this morning, and am now getting errors:     $ drush      Warning: require(/home/mpd/drush/vendor/autoload.php): failed to open   stream: No such file or directory in   /home/mpd/drush/includes/bootstrap.inc on line 1007      Fatal error: require(): Failed opening required   '/home/mpd/drush/vendor/autoload.php' (include_path='.:') in   /home/mpd/drush/includes/bootstrap.inc on line 1007   How can I fix this? Suppose $X_1,\ldots,X_n$ are a random sample of $U(-\theta,\theta)$ with $\theta&gt;0$. How can find $$ E\left(|X_1|\ \big|\max |X_i| \right) $$ that $1\leq i\leq n$?