Quick proof with Euclid's Algorithm I'm a little stuck on an exercise in a textbook of mine, and need some help. Let $J = \{e \in \mathbb{N}^* | e = ar + bs\}$ for some integers $r,s$. and let $d$ be the least element of J. Show that $d | a$ and $d|b$. I see that I'm supposed to divide d into a (stated in the back of the book) but I don't know where to go from there. If someone could lead me in the right direction, that'd be great (and explain step-by-step, cause I have trouble following sometimes.) Thanks!
Show $GCD(a_1, a_2, a_3, \ldots , a_n)$ is the least positive integer that can be expressed in the form $a_1x_1+a_2x_2+ \ldots +a_nx_n$ Given $a_1, a_2, a_3, \ldots , a_n$ not all zero, show $\gcd(a_1, a_2, a_3, \ldots , a_n)$ is the least positive integer that can be expressed in the form $a_1x_1+a_2x_2+ \ldots +a_nx_n$. Also deduce $a_1x_1+a_2x_2+ \ldots +a_nx_n = b$ has solutions $\iff \gcd(a_1, a_2, a_3, \ldots , a_n)|b$.   I know that $ax+by=n$ has integer solutions $\iff \gcd(a,b)|n$. I'll need to use induction to show $a_1x_1+a_2x_2+ \ldots +a_nx_n = \gcd(a_1, a_2, a_3, \ldots , a_n)$ has solutions, and then I will be able to say that as $\gcd(a_1, a_2, a_3, \ldots , a_n) | a_1x_1+a_2x_2+ \ldots +a_nx_n $ for any $x_1,x_2, x_3 \ldots x_n$ then $c \leq \gcd(a_1, a_2, a_3, \ldots , a_n)$ and $c \geq \gcd(a_1, a_2, a_3, \ldots , a_n)$, and thus $c=\gcd(a_1, a_2, a_3, \ldots , a_n)$.
Can someone suggest a way to compare the values of two dates greater than, less than, and not in the past using JavaScript? The values will be coming from text boxes. Occasionally, the Unicode character sequence U+200C U+200B (ZWNJ ZWSP) is inserted into comments In     ... the number "5000" does not paste as such but instead as  5 0 U+200C U+200B 0 0   The HTML source looks like this:  perhaps the easiest would be &lt;code&gt;\edef\tmp{\everypar{\widowpenalty\the\widowpenalty\relax}}\tmp\widowpenalty=-50&amp;zwnj;&amp;#8203;00&lt;/code&gt; that may reset   Or (smaller snippet):  \widowpenalty=-50&amp;zwnj;&amp;#8203;00&lt;/code&gt; that   So in the HTML, we have extraneous &amp;zwnj; and &amp;#8203; in between the two middle digits of 5000.  Upon pasting and trying out this code, this first led to a mysterious compiler error. The same issue happened within a (now deleted) comment thread below ; there, these or similar two characters were inserted after the letter "r" within the string \ref ("\ref").  Note that not all browsers wrap the same way. The above screenshot was for Firefox 19.0.2, in which it breaks between \relax}} and \tmp, with 5000 insidiously left intact on the second line but with the two special characters remaining in the middle. So that is why this took me completely by surprise. Other browsers might indeed only break on the invisible Unicode characters, such as Chrome 25:    Or on a smaller screen with the mobile site, multiple line breaks might show, possibly still leaving the Unicode characters unused: Isomorphism of $S^1$. To env or not to env Basically, I acidentally spawned too many ArmorStands due to a wrong code writing , resulting in the game crashing, and wanted to know how to delete those armorstands from the World save folder. They hava names and I belive they also have a scoreboard set, if that helps.   I have already tried to open the world but it crashes, I tried to open a server (Since it worked the last time I tried it) and killing it all from the console but, different from the last time, it crashed before killing them and now I just tried MCedit but it crashes as well. The easy way: Install Nvidia drivers, CUDA, CUDNN and Tensorflow GPU on Ubuntu 18.04 What does the human body use oxygen for besides the final electron acceptor in the electron transport chain? My biology teachers never explained why animals need to breathe oxygen, just that we organisms die if we don't get oxygen for too long. Maybe one of them happened to mention that its used to make ATP. Now in my AP Biology class we finally learned the specifics of how oxygen is used in the  due to its high electronegativity. But I assume this probably isn't the only reason we need oxygen.   What other purposes does the oxygen we take in through respiration serve? Does oxygen deprivation result in death just due to the halting of ATP production, or is there some other reason as well? What percentage of the oxygen we take in through respiration is expelled later through the breath as carbon dioxide? Isn't the const modifier here unnecessary? I have B1/B2 Visa. Can I travel for Business trip to US with this Visa? As a lab assignment I have to replicate experiment with spring oscillations, and eventually calculate spring constant. The measurements I took are: spring displacement from equilibrium $x$, mass of weights $m$, and period $T$. And presumed uncertainties: $\Delta_\alpha(x)=1[mm]$, $\Delta_e(x)=2[mm]$,  $\Delta_\alpha(t)=0.1[s]$,  $\Delta_e(t)=1[s]$. Where $\alpha$ stands for uncertainty of used equipment, and $e$ is readings uncertainty made by me.  I calculated the standard uncertainty using this formula: $$\sqrt\frac{\Delta_\alpha(x)^2+\Delta_e(x)^2}{3}$$ Which produced $u(x)=1.29[mm]$. Then using the following substitutions:  $$F=kx$$ $$mg=kx$$ $$x=\frac{g}{k}m$$ I assert that:  $$tan(a)=\frac{g}{k}$$  Because in this part of experiment I plotted $x=f(m)$, using excel's linest, I get the slope $s$  $$tan(a)=\frac{g}{k}=s$$  Now, my question is, since I had uncertainty for $x$ introduced, and it is used to compute $tan(a)$, then the latter is affected and should have uncertainty of itself? How in this case should I go about propagating uncertainty to $tan(a)$? Imagine a situation in which the whole place is on fire, a bomb is about to explode, everyone is running for their lives and someone is checking his looks on the mirror... pretty inappropriate for the situation, don't you think? I'm looking for an idiom or colloquial/slang/informal expression to describe such an indifferent stance. Setup for a wine bottle with multiple labels I have modeled a wine bottle and created various wine label designs for a client. The labels were created in Illustrator and so I have PNGs. I want to present a render of the same bottle with each label the client. The labels are not simple rectangles.   Should I:    "model" each label and wrap it on the bottle, or  try to map each label image to a portion of the bottle's surface?    I've seen lots of tutorials showing option #2, but something is telling me that the glass of the bottle should be separate from the "paper" label.  Also, with #1, couldn't I use the same bottle blend file and use a different layer for the each label, then turn on a different layer for each new render?  Maybe there's an even better strategy?  thanks, Some sites are . What additional sites should the chat support?  One suggestion per answer. It seems that $\{e\}, \{e,s\}, \{e,rs\}, \{e,r^2s\},...,\{e,r^{n-1}s\}, \{e,r,r^2,...,r^{n-1}\}, D_n$ are always subgroups of $D_n$. Especially when $n$ is odd, these seem to be the only subgroups. But when n is even, say $n=4$, then there are also $\{e,s,r^2,r^2s\}$ and $\{e,rs,r^2,r^3s\}$. It makes me wonder, is there a general formula/algorithm for finding all subgroups of $D_n$ when $n$ is even? I have a USB drive that I can see/access/modify/etc in Linux/Ubuntu but it doesn't show up on my Mac OSX.  I did sudo fdisk /dev/sdb (where my USB is) and then g to create a new GPT partition table, n to make a new partition, used all default settings, then w to save and exit.  Then I did sudo mkfs.fat -F32 /dev/sdb1 to format the partition with a FAT32 filesystem.  I ejected the USB and put it in my Mac and it doesn't show up / get detected.  What did I miss? It seems like maybe cancelled downvote might be less...weird? How do I make animations a steady speed? Reference undefined I receive the a Reference "LastPage" undefined error when trying to compile a document with this syntax:  \fancyfoot[R]{\thepage/\pageref{LastPage}}   What is necesarry to prevent this error? I'm using TeXworks Version 0.4.3 (MiKTeX 2.9) and pdfLaTeX. Inspired by this Hagon Von Eitzen's answer( ) I started investigating how I could express natural numbers as differences of squares. Using the method explained in the solution, I investigated natural numbers up to $60$ manually. I observed the following:  There is no way to express numbers of the form $4k+2$ as a difference of two squares.  2.There is one and only one way to express prime numbers (except $2$) as a difference of two squares.  The smallest number that can be expressed as a difference of two squares in two different ways is $15$ and the smallest number that can be expressed as a difference of two squares in three different ways is $45$.  As, $15= 8^2-7^2=4^2-1^2$. $45=23^2-22^2=9^2-6^2=7^2-2^2$. I could prove point $2$ easily but could not prove point $1$. Also I am interested in knowing how I could find the smallest natural number that could be expressed as a difference of two squares in four different ways without manual calculation. The Wikipedia article on  mentions two methods of sampling "small" polynomials namely uniform sampling and discrete Gaussian sampling. Uniform sampling is clearly the simplest, involving simply uniformly selecting the coefficients from the set of small coefficients, thereby guaranteeing they will be small. However, the articles mention that security proofs rely on discrete Gaussian sampling.  However, I have a bit of difficulty wrapping my head around    Why discrete Gaussian sampling would be better than uniform, security-wise?   I mean, usually uniform is preferable - it maximizes the entropy/information in each sample, making the outcome more unpredictable. I could understand it if the Gaussian sampling technique would somehow serve to avoid (at least with high probability) "weak" parts of the space of outcomes but I have found no information/justification that this is the case.  Another reason I can see why discrete Gaussian might be better is that there is, after all, a chance (albeit a small one) one would select a non-small coefficient, so the space of outcomes is actually larger. However, given that this probability of hitting this extra space is (by design) quite small but also that one would likely immediately reject such a sample (because the algorithm calls for a small polynomial), this seems to be irrelevant.  It is not because I am against discrete Gaussian sampling I just want to understand why it is more secure. Also, I want to understand if it really IS more secure or it is simply easier to prove security in that model. I would actually consider that very interesting if it is the case because it raises the question of why is it easier to prove the security under conditions that are intuitively worse (i.e. discrete Gaussian vs the intuitively better uniform case)? If it is so intuitively clear that uniform sampling is better, surely it must be possible to transform that intuition into a security proof of RWE also under uniform sampling. Otherwise, I would fear the intuition must with high likelihood be wrong or at least dubious.