Quick proof with Euclid's Algorithm
Show $GCD(a_1, a_2, a_3, \ldots , a_n)$ is the least positive integer that can be expressed in the form $a_1x_1+a_2x_2+ \ldots +a_nx_n$
How can I fix an out-of-focus blurred photo in Photoshop? How can I fix this picture?  Somebody else took this picture and well he wasn't able to do it very well, and it's blurred.    I have photoshop Why multiple PTR records in DNS is not recommended? I   that using multiple PTR records in a DNS configuration is not recommended.   However, the reasons are often vague, or not so obvious, naming:    "it can cause problems",  "can trigger bugs in programs expecting a single answer" : it's the software's problem then, isn't it?! "can make DNS answer packet too large" : isn't this fixed with ?   Are these good reasons? Do you know of any other (good) reasons? All this kinda looks like a "legacy fear"... Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series Bump right-aligned text to next line iff no room I've always managed my iPhone/iPad/iPod Manually, I recently updated to iTunes 11 and now I can see no way to get my music onto my device.  I used to be able to just drag songs from my library to my device, now there is seemingly no way to do this.  I can view the music on my device by clicking on it on the top right corner of iTunes, but I can't seem to get to a state where I can drag music from my computer onto my device.  I'm using iTunes 11.0.1.12 for Windows if that makes a difference. Interpreting spline results Prove that $U_{pq}$ is not cyclic if $p$ and $q$ are distinct odd primes.   I am a self taught person. I just learned this and tried this on my own and came up with this.  $x \equiv 1 \pmod{p}$ and $x \equiv -1 \pmod{q}$ has a solution $[a]_{pq}$, since $p$ and $q$ are relatively prime. Because $q$ is an odd prime, $[-1]_{pq}$ is not a solution, so $[a]_{pq}\neq [-1]_{pq}$. But $a^2 \equiv 1 \pmod{p}$ and $a^2 \equiv 1 \pmod{q}$, so $a^2 \equiv 1\pmod{pq}$ since $p$ and $q$ are relatively prime, and thus $[a]_{pq}$ has order 2.  Can someone please tell me if this proof this correct? Please help with proof as I learned it just now as a self taught person. Windows registry entries for default explorer view I'm making an automated installer for Windows XP, and I want the default view to be the "details" view on all folders - that is, the effect gotten by setting one folder to the details view, then going to Tools -> Folder Options -> View -> "Apply to All Folders". I also want the status bar to be showing, same as going View -> Show Status Bar.  What registry entries should I modify, and to what, to have this be the case? No such file or directory but I can see it! I'm trying to run a python script, on a headless Raspberry PI using winSCP and get the following error message:  Command '"./areadetect_movie_21.py"' failed with return code 127 and error message /usr/bin/env: python : No such file or directory.   When I try and run from terminal, I get:  : No such file or directory.   I try a similar python script, in the same directory, with the same python shebang, the same permissions and using the same user pi, and it works.  I also do a ls and I can see the file, so I don't know why it will not run. I use Kali Linux and discovered the following "problem":  root@kali:/proc/self# ls -l cmdline -r--r--r-- l root root 0 May 27 13:43 cmdline root@kali:/proc/self# chmod 744 cmdline chmod: changing permissions of `cmdline': Operation not permitted   I am not allowed to change the permissions of that file... but I am root? Why is that? Are there ways to work around this? I allready tried some things like moving the file (what is obviously not possible), change the runlevel etc. How to delete all lines that do NOT contain a certain word in Vim? In vim I can delete all lines that contain the word "price" with this  :g /price/d   How can I delete all lines that do NOT contain the word "price"? Referencing without captions appearing but keeping numbering My 3-month-old son has started sleeping through the night, with a vengeance. He sleeps for hours. And when he finally wakes up and I expect him to be starving, he's often still ready to happily play for a bit - say 1/2 hour to 45 min. Now, I know this isn't such normal behavior for a 3-month-old, and I'm definitely appreciating having my nights back, but I'm also a bit concerned - how long is the longest I should let him go without eating?  Edit (more details):  I used to try to stretch him and make his last feeding right before I went to sleep, so that I could take full advantage of his sleep-time. So we used to put him to bed about 12:30-1:00, and then head to sleep ourselves. He started off going about 6-7 hours... But a couple of weeks ago he started sleeping longer, and longer, and longer... Now he goes nearly 12 hours. He'll go to sleep at 12:30 AM and wake up at 12 PM or later. And he'll then happily play for another half hour, making about 12 hours of no-eating. I generally trust that I can rely on him to tell me when he's hungry, but I was wondering if there's a limit to how long I can let him go without eating... Why have header files and .cpp files? Manually installing a program on Debian I have installed Debian on my pc, but the problem is that at the moment I have no Internet connection to this pc. So, to install a program I have to manually downloaded from a pc with internet and then install it to the linux-debian computer.  The question is this: Is there a standard way to make the installation manually? Note that when I connect a usb flash with the program I want to install(that is:   cdparanoia_3.10.2+debian-10.1_i386.deb) and I try opening it I get the message:     Failed to execute child process “ar” no file or directory. Javascript function only works if there's an alert Question:      Let $p$ be any prime and $n \geq 3$. Show that there exists a non-abelian group of order $p^n$.    Attempt:   Take $n = 3$. Writing $\mathbb Z_p \times \mathbb Z_p = \{e, \alpha_1, \ldots, \alpha_{p^2 - 1}\}$ and considering $$\begin{align}\tau_{jk} : \mathbb Z_p \times \mathbb Z_p&amp;\to \mathrm {Aut} (\mathbb Z_p)\,\\e &amp;\mapsto id\\\alpha_j &amp;\mapsto id \\\alpha_i &amp;\mapsto \rho_k  \end{align}$$  $\forall i \neq j$ where $\rho_k : \mathbb Z_p \to \mathbb Z_p$ is such that $\rho_k (1) = k$, with $k \in \{2, \ldots, p-1\}$. Then we take the group $(\mathbb Z_p \times \mathbb Z_p) \ltimes_{\tau_{jk}} \mathbb Z_p$.   Then $$(\alpha_i , 2) \ltimes_{\tau_{jk}} (\alpha_i, 1) = (2\alpha_i, 1  + \tau_{jk} (\alpha_i)(2)) = (2\alpha_i , 1  +2 k)$$  and $$(\alpha_i , 1) \ltimes_{\tau_{jk}} (\alpha_i, 2) = (2\alpha_i, 1  + \tau_{jk} (\alpha_i)(2)) = (2\alpha_i , 1  +k)$$  taking $k = 1$ for example we have that the group $(\mathbb Z_p \times \mathbb Z_p) \ltimes_{\tau_{jk}} \mathbb Z_p$ is not abelian.    I was wondering if this is a good aproach. If not, is there an easier way to reach the result?  If this is the "best" way then would the higher cases ($n &gt; 3$) be more of a notation play game? Bijection between an open and a closed interval Recently, I answered to this problem:     Given $a&lt;b\in \mathbb{R}$, find explicitly a bijection $f(x)$ from   $]a,b[$ to $[a,b]$.   using an "iterative construction" (see below the rule).  My question is: is it possible to solve the problem finding a less exotic function?  I mean: I know such a bijection cannot be monotone, nor globally continuous; but my $f(x)$ has a lot of jumps... Hence, can one do without so many discontinuities?    W.l.o.g. assume $a=-1$ and $b=1$ (the general case can be handled by translation and rescaling). Let:  (1) $X_0:=]-1,-\frac{1}{2}] \cup [\frac{1}{2} ,1[$, and  (2) $f_0(x):=\begin{cases}     -x-\frac{3}{2} &amp;\text{, if } -1&lt;x\leq -\frac{1}{2} \\ -x+\frac{3}{2} &amp;\text{,     if } \frac{1}{2}\leq     x&lt;1\\ 0 &amp;\text{, otherwise} \end{cases}$,  so that the graph of $f_0(x)$ is made of two segments (parallel to the line $y=x$) and one segment laying on the $x$ axis; then define by induction:  (3) $X_{n+1}:=\frac{1}{2} X_n$, and  (4) $f_{n+1}(x):= \frac{1}{2} f_n(2 x)$  for $n\in \mathbb{N}$ (hence $X_n=\frac{1}{2^n} X_0$ and $f_n=\frac{1}{2^n} f_0(2^n x)$).  Then the function $f:]-1,1[\to \mathbb{R}$:  (5) $f(x):=\sum_{n=0}^{+\infty} f_n(x)$  is a bijection from $]-1,1[$ to $[-1,1]$.  Proof: i. First of all, note that $\{ X_n\}_{n\in \mathbb{N}}$ is a pairwise disjoint covering of $]-1,1[\setminus \{ 0\}$. Moreover the range of each $f_n(x)$ is $f_n(]-1,1[)=[-\frac{1}{2^n}, -\frac{1}{2^{n+1}}[\cup \{ 0\} \cup ]\frac{1}{2^{n+1}}, \frac{1}{2^n}]$.  ii. Let $x\in ]-1,1[$. If $x=0$, then $f(x)=0$ by (5). If $x\neq 0$, then there exists only one $\nu\in \mathbb{N}$ s.t. $x\in X_\nu$, hence $f(x)=f_\nu (x)$. Therefore $f(x)$ is well defined.  iii. By i and ii, $f(x)\lesseqgtr 0$ for $x\lesseqgtr 0$ and the range of $f(x)$ is:  $f(]-1,1[)=\bigcup_{n\in \mathbb{N}} f(]-1,1[) =[-1,1]$,  therefore $f(x)$ is surjective.  iv. On the other hand, if $x\neq y \in ]-1,1[$, then: if there exists $\nu \in \mathbb{N}$ s.t. $x,y\in X_\nu$, then $f(x)=f_\nu (x)\neq f_\nu (y)=f(y)$ (for $f_\nu (x)$ restrited to $X_\nu$ is injective); if $x\in X_\nu$ and $y\in X_\mu$, then $f(x)=f_\nu (x)\neq f_\mu(y)=f(y)$ (for the restriction of $f_\nu (x)$ to $X_\nu$ and of $f_\mu(x)$ to $X_\mu$ have disjoint ranges); finally if $x=0\neq y$, then $f(x)=0\neq f(y)$ (because of ii). Therefore $f(x)$ is injective, hence a bijection between $]-1,1[$ and $[-1,1]$. $\square$ Coincidence, purposeful definition, or something else in formulas for energy In the small amount of physics that I have learned thus far, there seems to be a (possibly superficial pattern) that I have been wondering about.  The formula for the kinetic energy of a moving particle is $\frac{1}{2}mv^2$.  The formula for kinetic rotational energy is $\frac{1}{2}I\omega^2$.  The formula for energy stored in a capacitor is $\frac{1}{2}C \Delta V^2$.  The formula for energy delivered to an inductor is $\frac{1}{2}LI^2$.  Finally, everyone is aware of Einstein's famous formula $e=mc^2$.  I realize there are other energy formulas (gravitational potential energy, for example) that do not take this form, but is there some underlying reason why the formulas above take a similar form? Is it a coincidence? Or is there a motivation for physicists and textbook authors to present these formulas the way they do? How to make an embedded question from "What time is it?"