int multiplication with float = WHAT?
Is floating point math broken?
If $\gcd (x,4) = 2$ and $\gcd(y,4) = 2$ then $\gcd(x+y,4) = 4$ I was working my way through some number theoretic proofs and being a newbie am stuck on this problem :     If $(x, 4) = 2$ and $(y, 4) =2$, then $(x + y, 4) = 4$,     where $(a,b)$ denotes the greatest common divisor of $a$ and $b$.   My Solution  (Incorrect)   $x-4 = 2t$ $y-4 = 2p$ $x + y - 8 = 2(t+p) \Rightarrow x + y - 4 = 2f + 4 = 2(f+2)\Rightarrow 2$ divides $(x+y , 4)$   My Question:  My solution is definitely inadequate. Can someone help me out? How can I add a new user as sudoer using the command line? A Particular Two-Variable System in a Group Suppose $a$ and $b$ are elements of a group $G$. If $a^{-1}b^{2}a=b^{3}$ and $b^{-1}a^{2}b=a^{3}$, prove $a=e=b$.   I've been trying to prove but still inconclusive. Please prove to me.  Thanks very much for proof. What are the system requirements for each flavour of Ubuntu Desktop? I'm thinking about installing Ubuntu Desktop, but I don't know what flavor is the best for my system. What are the minimum and recommended hardware requirements? What kind of CPU? How much memory? Should I have Hardware Acceleration? What flavor should I use?  This is an attempt of a canonical answer. My answers have the &quot;official  requirements&quot;, the recommended are a mix of official sources and opinion based (along with the answer it's told the source). You can comment or edit if you feel that the information is obsolete or incomplete. It is a good rule of thumb that any system capable of running Windows Vista, 7, 8, x86 (Intel) OS X will almost always be a lot faster with any Ubuntu flavor even if they are lower-spec than described below. Is there a way to "auto login" in PuTTY with a password? Is there a way to configure a password for a stored session in ?  I know there is the capability to specify an "auto-login username" (under Connection/Data), but is there a way to do the same with the password? Is $\sqrt{x^2}=|x|$ or $=x$? Isn't $(x^2)^\frac12=x?$ $|x|=\sqrt{x^2}$ as  shows. But, as $(x^2)^\frac12=x$, I can't understand where am I wrong interpreting Square-root. This is my last homework problem and I've been looking at it for a while. I cannot nail down what is wrong with this proof even though its obvious it is wrong based on its conclusion. Here it is:     Find the flaw in the following bogus proof by strong induction that   for all $n \in \Bbb N$, $7n = 0$.      Let $P(n)$ denote the statement that $7n = 0$.      Base case: Show $P(0)$ holds.      Since $7 \cdot 0 = 0$, $P(0)$ holds.      Inductive step: Assume $7·j = 0$ for all natural numbers $j$ where $0 \le j \le k$ (induction hypothesis). Show $P(k + 1)$: $7(k + 1) = 0$.      Write $k + 1 = i + j$, where $i$ and $j$ are natural numbers less than $k + 1$.  Then, using the induction hypothesis, we get $7(k + 1) = 7(i + j) = 7i + 7j = 0 + 0 = 0$. So $P(k + 1)$ holds.      Therefore by strong induction, $P(n)$ holds for all $n \in \Bbb N$.   So the base case is true and I would be surprised if that's where the issue is.  The inductive step is likely where the flaw is. I don't see anything wrong with the strong induction declaration and hypothesis though and the math adds up! I feel like its so obvious that I'm just jumping over it in my head. My lecturer has used some notation that I've never seen before: it is a (matrix) product symbol with a left-to-right arrow over the top. Does anybody know what this means?  Thanks in advance.  Edit: It looks like this: Change this behavior to allow for spelling corrections and the like: "Edits must be at least 6 characters" Now, it is easy to workaround this: add 10 characters, plus your one-character change, then remove these 10 characters in the next edit for the same post.  But this can hardly be what the designers intended. Instead, my suggestion is to change it to something like on Wikipedia, where you can mark an edit as "minor". Something like this:     Edits must be at least 6 characters, or select the checkbox "Minor, spelling or punctuational correction."   Edits like that will, of course, not count toward the edit-limit (before it becomes Community Wiki).  The suggestion often seen on Meta Stack Overflow is "just edit more in that post". But sometimes the post is just right, and there aren't any other spelling mistakes to correct. Why force a user to change more than necessary, or use an ugly hack? One page website - Effect on SEO I'm considering re-designing my personal website to promote my web services in my local area.  My issue is whether to do a one page website with a scroll-to navigation system. What effect does this have on SEO. As i understand it, you set stuff like h1's and meta description on specific page content. How does this work when all of your different content is on one page? Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd I read somewhere that     $(a^n - b^n)$         It is always divisible by $a-b$.   When $n$ is even it is also divisible by $a+b$.   When $n$ is odd it is not divisible by $a+b$.      and     $(a^n + b^n)$         It is never divisible by $a-b$.   When $n$ is odd it is divisible by $a+b$.   When $n$ is even it is not divisible by $a+b$.      I wonder what's the proof for this.  First postulate is clear. $(a-b)$ would or would not be a factor. Any light on others? How to use IDM on Mozilla Firefox Quantum I want to use Internet Download Manager for downloading on Firefox. According to this  it can be done with the extension called Flashgot. But Flashgot does not work on new Firefox Quantum. However, I found  that can be used to interrupt browser's downloading jobs by redirecting these jobs to an external download manager like Internet Download Manager (IDM). But I don't know how to integrate IDM with this new extension. I wrote IDM's location (/.wine/drive_c/Program Files (x86)/Internet Download Manager/IDMan.exe) in the box that says "Path to the executable" but when I try to download something, an error is appearing.     The preference page of the extension        The error that is appearing while trying to download files     How can I make it work? Thanks beforehand! How to add readable text to a light and dark image? I'm trying to add text to a photo of a city but the colors in the background make it difficult to read light and dark text. I've tried several light and dark colors but they all looked overpowered by the colors in the image. What colors could I use or what can I do to the text to make it show but also make it so you can still see the background?  Here is the image I'm trying to use: how to properly save an architecture in a fpga ic forever What is the maximum party size and the corresponding monster buffs? What is the maximum party size (if there is one) in Diablo 3?  Additionally, how much of a buff do monsters get when adding party members?     "A new ally has joined, but the minions of Hell grow stronger." Try to solve this puzzle:     The first expedition to Mars found only the ruins of a civilization.   From the artifacts and pictures, the explorers deduced that the   creatures who produced this civilization were four-legged beings with   a tentatcle that branched out at the end with a number of grasping   "fingers". After much study, the explorers were able to translate   Martian mathematics. They found the following equation:   $$5x^2 - 50x + 125 = 0$$   with the indicated solutions $x=5$ and $x=8$. The value $x=5$ seemed   legitimate enough, but $x=8$ required some explanation. Then the explorers    reflected on the way in which Earth's number system developed, and found   evidence that the Martian system had a similar history. How many fingers would    you say the Martians had?      $(a)\;10$      $(b)\;13$      $(c)\;40$      $(d)\;25$   P.S. This is not a home work. It's a question asked in an interview. Check for Process if Same is Running dpkg error: "trying to overwrite file, which is also in..." For example:  $ sudo apt-get install curl Reading package lists... Done Building dependency tree Reading state information... Done You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: bsh : Depends: libjline-java but it is not going to be installed groovy : Depends: libjline-java but it is not going to be installed rhino : Depends: libjline-java but it is not going to be installed E: Unmet dependencies.   Try 'apt-get -f install' with no packages (or specify a solution).   I get the same or similar errors when I attempt to install clojure1.3, leiningen, and several other packages.   When I try the suggestion made in the error message, this is what happens:  $ sudo apt-get -f install  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following packages were automatically installed and are no longer required:   diffstat linux-headers-3.2.0-26-generic linux-headers-3.2.0-26 dh-apparmor dkms html2text libmail-sendmail-perl libsys-hostname-long-perl Use 'apt-get autoremove' to remove them. The following extra packages will be installed:   libjline-java Suggested packages:   libjline-java-doc The following NEW packages will be installed:   libjline-java 0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. 23 not fully installed or removed. Need to get 0 B/72.0 kB of archives. After this operation, 129 kB of additional disk space will be used. Do you want to continue [Y/n]? Y (Reading database ... 226243 files and directories currently installed.) Unpacking libjline-java (from .../libjline-java_1.0-1_all.deb) ... dpkg: error processing /var/cache/apt/archives/libjline-java_1.0-1_all.deb (--unpack):  trying to overwrite '/usr/share/java/jline.jar', which is also in package scala 2.9.2-400 Errors were encountered while processing:  /var/cache/apt/archives/libjline-java_1.0-1_all.deb E: Sub-process /usr/bin/dpkg returned an error code (1)  $ sudo apt-get upgrade Reading package lists... Done Building dependency tree        Reading state information... Done You might want to run 'apt-get -f install' to correct these. The following packages have unmet dependencies:  bsh : Depends: libjline-java but it is not installed  groovy : Depends: libjline-java but it is not installed  rhino : Depends: libjline-java but it is not installed E: Unmet dependencies. Try using -f. During U-substitution, is the $du$ that we use as the infinitesimal in $\int f(u) \ du$ changing size? I am familiar and comfortable with the following definition of an integral (defined as the the Riemann Sum where n approaches infinity).    I have circled in red two terms to emphasize a concept that I was taught: the $\frac{b-a}{n}$ term effectively becomes the $dx$ term. This intuitively makes sense to me.   Additionally, I am familiar with the following statement (as expressed here: )    However, I have boxed in red something that I find rather confusing when trying to interpret it through the understanding that $\frac{b-a}{n}$ sorta kinda equals $dx$...  Specifically, the $du$ value appears to be changing throughout the interval $[ g(a), g(b)]$. To reframe this in the "Riemann Sum" version,we would have something like: $\text{ "value that varies depending on where you are"} * \frac{g(b)-g(a)}{n}$  This is very confusing to me...because, if this is the correct interpretation, I'm not quite certain I understand why this is allowed. Other than portability, are there any other advantages (or disadvantages) to storing apps on external storage?