Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so?    Addendum by LePressentiment on 11/29/2013: (in the interest of  and averting a duplicate)  What's the intuition behind this result? I only recognise the proof and examples solely due to algebraic properties and formal definitions; I'd like to apprehend the result naturally.
How can I prove that $\gcd(a, b) = \gcd (a, b + ma)$?  I have tried this:  let $g = \gcd(a, b)$, then $g \mid a$ and $g \mid b$. This means that $g \mid ax+by$.  I don't know what to do next.  Thanks.
How to loop over the lines of a file? Say I have this file:  hello world hello world   This program  #!/bin/bash  for i in $(cat $1); do     echo "tester: $i" done   outputs  tester: hello tester: world tester: hello tester: world   I'd like to have the for iterate over each line individually ignoring whitespaces though, i.e. the last two lines should be replaced by  tester: hello world   Using quotes for i in "$(cat $1)"; results in i being assigned the whole file at once. What should I change? Right-Click menu awareness on Web Application The web application I am building has a custom right-click menu. I'm concerned that the user might not find this menu as normally right-clicking will only display the default browser's menu. Is there a standard way to make the user aware of custom right-click menus? Is there another way to show this information? Find out BIOS Version from Windows Canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism Number agreement between subject and object For example, on this page:   I see the following:   What does this 499 mean? I can't seem to correlate it with upvotes, posts, edits or anything else. Why does the right leg's bone in my rig move when I move the left? For some reason, when I moved the left leg's bone, the right leg has also moved and the bone of the right leg doesn't control anything at all.  I think this is because I used Mirror to make the model, but I'm not sure I'm a complete beginner. Is there a PPA with the latest version of LibreOffice? There was recently another release of , but it is  a pain to have to manually go to their website and download the packages every time there is a new release. Is there a PPA that always has the latest version of LibreOffice? Quick way to subtract white outline from knotwork pattern in Illustrator? I am working with a pretty complex traced AI that I received, and I want to basically make it so that all the visually black stuff is shapes, and all the white outlines (invisible here) are removed. Most of the spaces when a line crosses over another is created by a white outline.   I know about expanding appearance and the pathfinder tool, but because there are so many lines that are crisscrossing, I haven't had an easy way of minusing the fronts away. I could expand appearance and sort every object carefully, but I feel like there may be a quicker way of doing this.  If I put it in Photoshop as a smart object in 'screen' mode, it looks right.. I could put that back in Illy and trace it, but I don't want to complicate the paths. Test if a website is alive from a C# application I am looking for the best way to test if a website is alive from a C# application. Background My application consists of a Winforms UI, a backend WCF service and a website to publish content to the UI and other consumers. To prevent the situation where the UI starts up and fails to work properly because of a missing WCF service or website being down I have added an app startup check to ensure that all everything is alive. The application is being written in C#, .NET 3.5, Visual Studio 2008 Current Solution Currently I am making a web request to a test page on the website that will inturn test the web site and then display a result. WebRequest request = WebRequest.Create(&quot;http://localhost/myContentSite/test.aspx&quot;); WebResponse response = request.GetResponse();  I am assuming that if there are no exceptions thown during this call then all is well and the UI can start. Question Is this the simplest, right way or is there some other sneaky call that I don't know about in C# or a better way to do it. elementary row operation change the eigenvalue of a matrix We have the matrix: $$\left[\begin{array}{rrr} 0.8&amp;0&amp;0\\ 0&amp;1.2&amp;0\\ 0&amp;0&amp;1\\\end{array}\right]$$ Finding the eigenvalues and eigenvectors are easy. But finding ${\bf M}^n$ when $n \rightarrow \infty$ is the hard part. Any suggestions? Showing that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set Problem 4.3.32 in Linear Algebra, Lay:  Let $V$ and $W$ be vector spaces, let $T:V\to W$ be a linear transformation, and let $\{\mathbf{v}_1, \dots , \mathbf{v}_p \}$ be a subset of $V$. Suppose that $T$ is a one-to-one transformation [...]. Show that if the set of images $\{T(\mathbf{v}_1), \dots ,T(\mathbf{v}_p)\}$ is linearly dependent, then $\{\mathbf{v}_1, \dots , \mathbf{v}_p \}$ is linearly dependent. This fact shows that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set.  All I can think of doing: If $\{T(\mathbf{v}_1), \dots ,T(\mathbf{v}_p)\}$ is linearly dependent, then there exist weights $c_1, \dots, c_p$, not all zero, so that $$c_1 T(\mathbf{v}_1) + \cdots + c_p T(\mathbf{v}_p) = \mathbf{0}$$ Since $T$ is linear: $$c_1 T(\mathbf{v}_1) + \cdots + c_p T(\mathbf{v}_p) = T(c_1\mathbf{v}_1) + \cdots + T(c_p \mathbf{v}_p)=T(c_1 \mathbf{v}_1 +\cdots + c_p \mathbf{v}_p)=\mathbf{0}$$ So the set $\{\mathbf{v}_1, \dots , \mathbf{v}_p \}$ must be linearly dependent. Am I missing something? And why must $T$ be on-to-one for this to be true? An example showing that it is not true if $T$ is not one-to-one would be great! Showing that $e^z=z$ has infinitely many solutions One of our review problems for my complex analysis final is the following:     Show that $e^z=z$ has infinitely many solutions in $\mathbb{C}$.   I've seen a few solutions online, but none of them have used material that was covered in our introductory complex analysis course. Would anyone be able to provide a simple solution?  I'm trying to think of ways to invoke bounding in some way but I can't seem to come up with anything. Imputing missing observation in multivariate time series MicroSD card is set to Read-only state. How can I write data on it? I have a MicroSD card reader in my notebook. When I insert a MicroSD memory card into it, I can't write any data on it - it's read-only. The same thing happens when I try to do it under root. And I can't change the permission policy for it. This problem happens for all microSD cards (I've tested 2 cards). I've encountered it on Ubuntu 12.04 LTS, now I'm on Ubuntu 12.10, the problem is still on it's place. When I try to insert such a "read-only" card in my mobile phone, everything's OK and I can write data on it. So, can anybody tell me, how can I write a data onto the card?  UPD: here's fdisk -l output:  Disk /dev/sdc1: 2013 MB, 2013135360 bytes 62 heads, 62 sectors/track, 1022 cylinders, total 3931905 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x00000000   Device Boot      Start         End      Blocks   Id  System   The file system of the card is msdos. As to me it's weird.   UPD2: I've found a small "lock" switch on my card adapter, and it turns out, that when I insert it in my notebook, it switches on. Perhaps, I have to find some glue or something to get that switch stuck...  UPD3: There was no glue near me, so I've decided to buy another card-reader. Flip x and y axes in Excel graph This should not be very difficult, but I cannot figure out how to do it.  I have a table similar to this     %low %high  0   0   12  1  13   26  ... 19  90   94 20  95  100   When I graph it, excel defaults to having the first column on the x axis and plotting the second and third column as y values. I want the first column to be on the y axis instead. I assume there is an easy way to do this, but I cannot figure it out. Most of the things I have found from searching have suggested the "Switch Row/Column" button, but that does something else.  Thanks for the help. Proof of $\gcd(a,b)=ax+by\ $ [Bezout's identity] This question was phrased as a bug report for LaTeX and I thought it would be helpful to be presented here (including an answer).  The person had a lot of "common words" in typewriter font sprinkled around his text and he was surprised that this text would not automatically hyphenate and he had to manually insert \- to mark hypenation points. fetching list of friends using facebook 2.0