Let $a$ and $b$ be integers, not both zero. Prove that for a positive integer $d$, $d=gcd(a, b)$ If and only if  1) $d$ divides $a$ ($d\vert a$) and $d$ divides $b$ ($d\vert b$) 2) whenever $c$ divides $a$ and $c$ divides $b$, then $c$ divides $d$ ..   How to deduce this theorem? I'm new to this stuff. Can somebody give a proper direction within this mathematics field?
Context: I was looking at . I didn't know the result referred to in Brian M. Scott's answer is hard enough to require a separate proof   ― .  ― .  (1)  ― What's the intuition? Is Bezout Identity supposed to be intuitive?  (2) — Why doesn't this try work? Let $c$ be any common divisor of $a, b$. Therefore $c\mid a, c\mid b \iff cj_1 = a, cj_2 = b$ for some integers $j_1, j_2$.     Call $g = \gcd$. By definition $\gcd(a,b)\mid a$ and $\gcd(a,b)\mid b$ and $c \le \gcd(a,b)$. Therefore $g\mid a, g\mid b \iff gk_1 = a, gk_2 = b$ for some integers $k_1, k_2$.   The two paragraphs together precipitate $cj_1 = a = gk_1,  cj_2 = b = gk_2$ but this only proves $c\mid gk_1, c\mid gk_2$?    Origin - Elementary Number Theory, Jones,  p9, Exercise 1.8
Hide Command Window of .BAT file that Executes Another .EXE File How to get the center line between two linestrings Where is the track of a ship when it is shipping always in the center of the   between the coasts of Great Britain and France?   I started with 2 linestrings and I created a lot of points on the linestring with a . After that I created a distance matrix with Raster/Analysis/Proximity in QGIS. Finally I created an aspect raster from the output raster.     Is the red line really the "center line" between these two linestrings?  Is there a more efficient way to do this? TV series with pull out phone screens and pink/purple space ships I’m looking for a TV show where an alien race comes to Earth and sets up big purple/pink cylindrical or cone shaped spaceships in various cities around the world. The Aliens are not hostile but actually created peace and eliminated hunger on Earth.    I believe the aliens had some human features and were wise and calm the main actor had a phone with a pull out screen I believe the main actor worked for or with the aliens I believe that in the last episode some humans and aliens go into some kind of pods and perhaps leave Earth? one of the space ships was located in a major US city. Maybe Washington DC or another East Coast city   Could someone please tell me what TV series this is? I have been searching everywhere and cannot find it. Thanks! How to switch from Natbib to Biblatex in pre-made document class/template? I am using  to write my thesis. The template uses Natlib for references, whereas I really need Biblatex.   When I simply try to change \usepackage[natbib] for biblatex I get the error message  This is BibTeX, Version 0.99d (TeX Live 2016/Debian) Capacity: max_strings=100000, hash_size=100000, hash_prime=85009 The top-level auxiliary file: aagot-1-report.aux The style file: acm.bst Illegal, another \bibstyle command---line 80 of file aagot-1-report.aux  : \bibstyle  :          {acm}  I'm skipping whatever remains of this command Database file #1: Mendeleytry.bib.bib Warning--I didn't find a database entry for "" You've used 14 entries,code here   The solution  \usepackage[backend=bibtex, natbib=true, style=numeric]{biblatex}   didn't work.  Another possible solution was said to be to recompile from scratch to delete the .aux file, but that resulted in  /usr/share/texlive/texmf-dist/tex/latex/biblatex/biblatex.sty:462: LaTeX Error:  Command \bibhang already defined.            Or name \end... illegal, see p.192 of the manual.  See the LaTeX manual or LaTeX Companion for explanation. Type  H &lt;return&gt;  for immediate help.  ...                                                l.462 \newlength{\bibhang}   I have tried to use this "marco" to override the \requirePackage settings, but that also didn't work and resulted into  LaTeX Error: Two \documentclass or \documentstyle comm ands.  See the LaTeX manual or LaTeX Companion for explanation. Type  H &lt;return&gt;  for immediate help.  ...                                                l.45 \documentclass[whitelogo]{                            tudelft-report}    Where I can't seem to delete any of the document classes without getting the functionality of the tudelft-package and the biblatex citing (or in fact getting it to work at all!)  (no line number): [0] Config.pm:343&gt; INFO - This is Biber 2.5 [0] Config.pm:346&gt; INFO - Logfile is 'aagot-1-report.blg' [62] biber:290&gt; INFO - === Tue Apr 24, 2018, 22:24:06 [246] Utils.pm:165&gt; ERROR - aagot-1-report.bcf is malformed, last biblatex  run probably failed. Deleted aagot-1-report.bbl [246] Biber.pm:113&gt; INFO - ERRORS: 1   Is there any way I can still make this template work with Biblatex? Natlib seems to deeply rooted in the file and I don't know how to change it/access the settings/the ".sty"-file. Can I use the example documentclass combined with biblatex (instead of the standard natbib) without losing the great parts of the rest of the document class?  Thanks very much :) I had always kind of wondered why we didn't see interference in things like windows -- we were taught that the condition is that the thickness of the film/slab/medium just has to be an integral number of wavelengths, $d=m\lambda$. My logic was, if visible light has wavelengths ~400-780nm, any window pane has got to be an integral number of wavelengths (or close to integral) for some of the wavelengths in the visible.  For example, if a window is, $6mm = 6\times10^6nm$ thick, then you should see interference for wavelengths (and some range around them) 200nm, 250nm, 300nm, 400nm, 500nm, 600nm... Because 6mm is an integral number of each of those.  Obviously that would be really annoying and isn't the case. I found  (warning: PDF) source that seems to explain it and I was wondering if what they're saying is sound. On the 10th page (p171 for them), it says that the reason we don't see this interference is that no window is that smooth, it has a variance in the thickness of at least several wavelengths, so the interference effect is effectively average out among all those thicknesses, which usually gets rid of it. They do the math and show that.  Is that the reason?  If so, that leads me to two questions: Could we, if we were really careful and used something like MBE, make a glass window with that sort of accuracy, and then see interference effects in the optical range? I know that quartz crystal monitors can easily give you nanometer accuracy for evaporators and sputterers, but they also have pretty limited growth rates.  My second question is, then, do we see interference effects in glass windows with electromagnetic radiation of a comparable wavelength, like microwaves?  Thanks! How to install HPLIP on my Ubuntu to support my HP printer and/or scanner? Let's assume that one has some HP printer and/or scanner and checked its support status on the corresponding page of HPLIP site - .  So one have the following information: the device Model name, Min. HPLIP Version and Driver Plug-in need.   What should one install to make Ubuntu fully support the HP printer and/or scanner? LibVLC.NET in GTK# Dual Canadian and British citizen - healthcare coverage in the UK? Over time I realized a very annoying property of the Stack Overflow reputation system: Easy answers to popular question are rewarded with a lot more reputation points than actually great answers. In case anybody actually needs examples for this, here are two examples from :   gained me 420 rep, I invested 2 minutes writing it, and it only was a wild guess.   took me about an hour for the research and the math and gained me 40 rep.  Now I know that you are bound to get more rep for answers that are useful to more people, and I'd say that's fine. But the current situation is quite extreme: 30% of my reputation is from 5 of my 330 answers, and they are definitely far from being my best ones.  A final example that is not from my profile, but more extreme: Do you really think  is worth 40000 reputation?  My proposition to improve that situation would be to make the reputation gain from a single answer non-linear, maybe even have it capped at a certain point. I'm not sure on how that non-linear function should look exactly, and would leave good proposals for that to users with more experience than me. If a question is in all-uppercase or all-lowercase, our quality filters should automatically reject it Proposal If a question on an English-language Stack Exchange site does not contain at least one uppercase and one lowercase letter, our quality filter should reject it. (We can make exceptions for sites which don't use a mixed-case alphabet, such as .) Justification The quality filter currently rejects neither all-uppercase nor all-lowercase posts. I've checked. The best Stack Exchange users use mixed case. If an asker is too lazy to use their Shift key, their question tends not to be of very good quality. Don't take my word for it. You can see proof. Sam Saffron helped me write  which lets you see with your very own eyes that what I am saying is true. Single-case askers tend to ask poor-quality questions such as:      In addition, posts which use only uppercase or only lowercase — even if they aren't riddled with typos or grammatical errors — tend to be hard to read. Capitalization aids readability. Why is this proposal not a duplicate? This proposal is not a duplicate of the older proposal . This is because it also deals with all-lowercase questions. All-lowercase questions are 50 times more common than all-caps questions.  Stack Overflow gets only about 100 all-uppercase questions per year; see .  Yet it gets about 5,000 all-lowercase questions per year; see . (These statistics are for Stack Overflow only; the rest of our network wasn't considered.) I suspect that the older proposal was declined for two reasons:  All-uppercase questions are rare.  Therefore, the effort required to block all-uppercase questions is too much.   This proposal is better. For just a bit more effort, the Stack Exchange developers can get a much richer reward. Conclusion What do you think of my proposal? Agree? Disagree? Thoughts? PS: This idea is modified and adapted from  by Stack Exchange user Gilles. My dear sir Gilles, I thank you. Note There's even a  which automatically detects questions with all-caps titles and alerts interested users and moderators. Archimedes' principle for two liquid layers Problem: I have a cylindrical vessel of height $H$ and radius $R$. There are two liquid layers in the vessel. The first has density $D_1$ and height $h_1$, the second has density $D_2$ and height $h_2$. The second liquid is floating on the first liquid (thus $D_2 &lt; D_1$) and they are both entirely within the vessel ($h_1+h_2 \le H$). Now I have a cube of density $D$ and edge length $n$. The cube fits into the vessel ($n \le \sqrt2R$) but it might not be completely inside it ($n&gt;H$ is possible). I place it into the vessel such that one side is horizontal (the cube is not lopsided). Some liquid might overflow as a result. How do I use Archimedes' principle to calculate the topmost liquid level after placing the cube? I can ignore damping and other physical effects. EDIT: Note that the only values I know are $H, R, D_1, h_1, D_2, h_2, n$ and their abovementioned constraints.  My workings: I tried using the formula directly, where $\text{weight of cube}=\text{weight of displaced liquid}$. I first get $Dn^3=D_2n^2s_2$ where $s_2$ is the height submerged in the second liquid. If $s_2 \le h_2$ then the cube is only submerged in the second liquid. So the answer is $\min(H, h_1+h_2+{n^2s_2\over\pi R^2})$, accounting for overflows. This case is simple. But if $s_2&gt;h_2$ then the cube can be submerged in the first liquid too. I thought of subtracting the weight of the second liquid displaced to obtain the height submerged in first liquid, but realize that I don't know that. The weight displaced depends on the final liquid height, which can be affected by overflows, whether the cube fits totally within the two liquids, etc. In fact, it can be the case that after the liquid level rises, the cube now displaces more of the second liquid, causing it to not be submerged in the first (is that even possible?) It seems very messy and I have no idea how to start. :( Does anyone have a nice solution to this problem? Can Gmail automatically base "send from" address on the "to" address? What is the status of pdfsync? The pdfsync homepage states that:     At the end of this summer (2007), pdfsync is slowly migrating to pdftex itself. We are working at giving pdftex all the pdfsync ability and more...   The pdfsync README file discourages people from using pdfsync:     pdfsync uses extremely sensible code. You should not use pdfsync on final documents because it can change the layout rather significantly.   From your knowledge and experience, what is the actual status of pdfsync?   Is it safe to use pdfsync? Is there an alternative for pdfsync that is built into pdftex? Is there a different tool for pdflatex that offers DVI-like functionality (forward and backward search)? \section*[toc]{normal section title} not working for me for my thesis I want to have a section appear differently in the ToC than in the text.  After googling a bit, I found \section[short]{long} as a proposed answer, but this does not seem to work for me.  Here is a MnotWE:  \documentclass[12pt,a4paper,reqno]{amsbook} \usepackage{fontspec} \setmainfont[SlantedFont={Latin Modern Roman Slanted},          SmallCapsFont = {Latin Modern Roman Caps},          ]{Latin Modern Roman}    \usepackage[activate=true,nocompatibility},final,tracking=true,factor=1100,stretch=10,shrink=10]{microtype}  \usepackage[bindingoffset=1cm]{geometry} \usepackage{polyglossia} \setdefaultlanguage{english} \usepackage{csquotes}  \title{\textsc{Some Title}} \author{me}  \begin{document} \maketitle \frontmatter \tableofcontents \mainmatter  \part[toc part]{normal part} \section*[toc section]{normal section} some text  \end{document}   It just takes "toc part" as normal and as ToC heading for the \part command.  For the section I am interested in the starred version. Here the [toc section] part of the command seems to be completely ignored.  I use LuaLaTeX to build my document. Does anyone have a suggestion how to get a \section* appear with a different title in the ToC? Viewshed Analysis With GDAL I am looking to perform a simple (one observer location) viewshed analysis on a standard DEM using GDAL (either python or .NET bindings).  I have searched, but haven't found any readily available implementation.  Can anyone point me in the right direction? Should Community Moderators be "elected for life", or have terms? While I applaud the excellent work of , the question has come up:     How long will be the elected moderator’s “mandate”? Is it for life, or are we going to have “moderator elections” regularly?   Of course, any community moderator who wishes to recuse him or herself can certainly do that at any time.   Should we have term limits that require community moderators be cycled out periodically, so others get a chance to moderate? Should we hold regular (say, yearly) moderator elections? Is it better to have a larger pool of community moderators, and let the normal ebb and flow of turnover get us "natural" variability in moderation?   What are your thoughts on this? Don't heavier objects actually fall faster because they exert their own gravity? The common understanding is that, setting air resistance aside, all objects dropped to Earth fall at the same rate. This is often demonstrated through the thought experiment of cutting a large object in half. The halves clearly can't then fall more slowly just by being sliced in two.  However, I believe the answer is that when two objects fall together, attached or not, they do "fall" faster than an object of less mass alone does. This is because not only does the Earth accelerate the objects toward itself but the objects also accelerate the Earth toward themselves. Considering the formula: $$ F_{\text{g}} = \frac{G m_1 m_2}{d^2} $$  Given $F = ma$ thus $a = F/m$, we note that the mass of the small object doesn't seem to matter as when calculating acceleration the force is divided by the $m$ term, its mass. However, this overlooks that the force is actually applied to both objects, not just to the smaller one. The acceleration on the second, larger object is found by dividing $F$, in turn, by the larger object's mass. The two objects' acceleration vectors are exactly opposite, so closing acceleration is the sum of the two:  $$ a_{\text{closing}} = \frac{F}{m_1} + \frac{F}{m_2} $$  Since the Earth is extremely massive compared to everyday objects, the acceleration imparted on the object by the Earth will radically dominate the equation. As the Earth is $\sim 5.972 \times {10}^{24} \, \mathrm{kg} ,$ a falling object of $5.972 \, \mathrm{kg}$ (just over 13 pounds) would accelerate the Earth about $\frac{1}{{10}^{24}}$ as much, which is one part in a trillion trillion.  Of course in everyday situations, we can for all practical purposes treat objects as falling at the same rate because of this negligible difference—which our instruments probably couldn't even detect. But I'm hoping not for a discussion of practicality or what's measurable or observable, but what we think is actually happening.  Am I right or wrong?  What really clinched this for me was considering dropping a small Moon-massed object close to the Earth and a small Earth-massed object close to the Moon. This made me realize that falling isn't one object moving toward some fixed frame of reference, but that the Earth is just another object, and thus "falling" consists of multiple objects mutually attracting in space. Cam I have two GFCIs connected to one double gang box in my laundry room and if so can I do a single line connection so that they both can work independently?  I have a 12/2 going from the panel through conduit (copper wire) to the double gang box. I want to have each GFCI to run independently without using the load. Why, in case one shuts off the other isn't affected. the double gang box is standard. the device is a Leviton 20 amps. I was told that you can run 12/2 from the panel to the first GFCI black and white. Then connect the second GFCI using a jumper and attaching it to he second GFCI. Is this acceptable by code. we use a washer,dryer in the laundry and the garage heavy machinery. so why should this be a problem running two independent? Why are "south" and "southern" pronounced with different vowels? I was wondering why we pronounce the vowels in the words south and southern. They seem to be very closely related to each other. Both refer to the same direction. South is a noun and Southern is an adjective, which I don't think caused them to have a different vowel. Oxford Dictionary shows that south is /saʊθ/ and southern is /ˈsʌð(ə)n/. In short, south has a long vowel and southern has short. Unable to add comments on closed questions or their answers in the Android app