Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$ I'm having some trouble proving the following proposition:  Let $m \in \mathbb N$ and $a,b,k \in \mathbb Z$, then: $$ka \equiv kb \pmod{m} \iff a \equiv b \pmod{\frac{m}{\gcd(k,m)}}$$  I began by trying to prove that $a \equiv b \pmod{\frac{m}{\gcd(k,m)}} \implies ka \equiv kb \pmod{m}$ but I got stuck with: $$a\gcd(k,m) \equiv b \gcd(k,m) \pmod{n}$$ Then I also tried to prove that $ka \equiv kb \pmod{m} \implies a \equiv b \pmod{\frac{m}{\gcd(k,m)}}$ But I also was not able to conclude anything. This is my first time learning modular arithmetic and I'm still confused with the concepts and have no intuition with this at all, this is perhaps why I'm having some trouble solving these types of questions. How can I prove this?
$ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form] I came across this theorem:   For all integers a,b,c and m&gt;0, if d = GCD(c,m) then a*c ≡ b*c (mod m) iff a ≡ b (mod m/d).  The =&gt;proof states this:  a*c ≡ b*c (mod m) =&gt; m|(a-b)*c                    =&gt; m/d | (a-b)*c/d                   =&gt; m/d | (a-b)                   =&gt; a ≡ b (mod m/d)   I don't understand how to go from step 2 to step 3. The proof states that this is possible because m/d and c/d are coprime. To me this means GCD(m/d, c/d) = 1 but how does this cancel out the whole term?
Was there any reason why Dr Soong chose for Data's eyes to be yellow?  I can understand that he wasn't going for a completely human-looking android based on his skin color, but why yellow eyes? Given that $x^y=y^x$, what could $x$ and $y$ be? It's not too difficult to figure out that $x$ and $y$ can both be 1, and also $x$ can be 2 and $y$ can be 4 (and vice versa). But I can't rule out if there are other solutions. Does it have anything to do with inverse functions? Is there a way to see the solutions by graphing, or algebraically? As far as I understand, whatever can be done using DeclareMathOperator can also be achieved using newcommand, but not the other way around. If this is the case, what are the circumstances where DeclareMathOperator is the better choice? When should I take solely newcommand?  I found a question regarding , and I wonder what other aspects should be taken into consideration when it comes to choosing which method to use when defining math operators. I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework. The problem is stated as follows. Assume we have an infinite tetration as follows $$x^{x^{x^{.^{.^.}}}} \, = \, a$$ With a given $a$ find $x$. The next part of the problem was to discuss the convergence radius of a. If a is too big or too small the tetration does not converge. Below is my humble stab at the problem.  My friend said you would have to treat the tetration as a infinite series, and therefore could not perform algebraic manipulations on it before it is know whether it converges or diverges. However my attempt is to first do some algebraic steps, then discuss the convergence radius. I) Initial discussion At the start it is obvious that the tetration converges when $a=1$ (just set $x=1$) Now after some computer hardwork it seems that the tetration fails to converge when a is roughly larger than 3. II) Algebraic manipulation $$ x^{x^{x^{.^{.^.}}}} \, = \, a$$ This is the same as $$ x^a \, = \, a$$ $$ \log_a(x^a) \, = \, \log_a(a)$$ $$ \log_a(x) \, = \, \frac{1}{a}$$ $$ x \, = \, a^{\frac{1}{a}}$$ Now, if we let $a=2$ then $x = \sqrt{2}$. After some more computational work, this seems to be correct, which makes me believe this formula is correct. III) Discsussion about convergence By looking at the derivative of $ \displaystyle \large a^{\frac{1}{a}} $ we see that the maxima occurs when $a=e$, which also seems to correspond to the inital computational work. Now I think, that the minima of $\displaystyle \large a^{1/a}$ is zero by looking at its graph, studying its derivative and the endpoints. So that my &quot;guess&quot; or work shows that it converges when $$ a \in [0 \, , \,  1/e] $$ VI) My questions Can my algebraic manipulations be justified? They seem rather sketchy taking the a`th logarithm and so on . (Although they seem to &quot;magically&quot; give out the right answer) By looking at Wikipedia it seems that the tetration converge when $$ a \in \left[ 1/e \, , \, e \right] $$ This is almost what I have, why is my lower bound wrong? How can  I find the correct lower bound? Editing older revision (not most recent) no longer includes older title and tags, and automatic rollback note Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$ I am looking for the combinatorial argument for the identity:  \begin{equation} k\binom{n}{k} = n\binom{n-1}{k-1} \end{equation}  This is easy to show algebraically as:  \begin{equation} \binom{n}{k} = \dfrac{n(n-1)(n-2)(n-k+1)}{k(k-1)!} \end{equation}   What is the combinatorial argument? What are some general ideas to get started?   Here is a clarification of 2. From what I have seen so far, proving (combinatorially) an identity with an addition sign usually implies that we need to partition a set (this makes sense because of the addition rule and provides a nice visual). On the contrary, the previous observation leads me to believe that multiplication in identities can be resolved with the multiplication principle, but what is the "visual/interpretation" for this? Could someone provide such an interpretation for the example identity given above? Can't preview smoke simulation on Blender 2.82 Whenever I add a Quick Smoke effect, I can't preview it on the viewport unless I bake it.   And Yes, I added the Smoke Domain on the object.   I've seen tutorials like this  and this  that clearly shows the possibility of previewing the smoke before baking it.   I'm using Blender on a Mac with AMD GPU. I would guess that this is something related to OpenGL, as thoroughly explained here -  but that's for Windows.   Does anyone have any idea how to fix this? Community, please show some holiday spirit! Why does ? And more importantly: why does she not wear them?    Please, show some goddamn holiday spirit! Can we get Community to wear her hats, either randomly or through popular vote? I'm trying to make a longtable that is wider than the textwidth at a certain moment. I've tried different things always ending up with a different problem:  First possibility: If I use the adjustwidth environment the longtabu does not align correctly. The left margin stays fixed and the right margin gets out of the page. This in contrast, if I do the exact same thing to make a tabu (instead of longtabu), it works perfectly. I think using the adjustwidth is the best way to go, but I can't manage to get it working.  Second possibility: I've found a solution using the fullwidth package. That package does the trick on small longtables, but freezes when I'm trying to do it on my big longtable. The command \setlength\LTleft{-2cm} didn't work as well.  The final option I tried was to use geometry. But that changes the margin of the whole page thus making a page break which I'm not interested in. The code I used for this was  \newgeometry{left=1.2cm} \begin{longtable} long table stuff \end{longtable} \restoregeometry   Here's an example of what I want contra what I get  \documentclass[a4paper]{article} \usepackage{longtable} \usepackage{chngpage} \usepackage{lipsum} \usepackage{tabu}  \begin{document} \section{This is the original textwidth.} \lipsum[1]  \section{This is the width I need for my longtabu} \begin{adjustwidth}{-1in}{-1in} \begin{tabu} to \linewidth {*{13}{|X[l]} |} 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ \end{tabu} \end{adjustwidth}  \section{This is what happens} \begin{adjustwidth}{-1in}{-1in} \begin{longtabu} to \linewidth {*{13}{|X[l]} |} 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ \end{longtabu} \end{adjustwidth}  \end{document} Finding the ID of a process and killing it When ever I need to kill a background process I do ps -e | grep &lt;process_name&gt;  Which prints something like this 1766 ? 00:00:13 conky , Then I use the process ID to kill it like so kill 1766 .   Is there any way I can simplify this ? Make it quicker ? reduce the amount of typing ? Solutions for INSERT OR UPDATE on SQL Server Assume a table structure of MyTable(KEY, datafield1, datafield2...).  Often I want to either update an existing record, or insert a new record if it doesn't exist.  Essentially:  IF (key exists)   run update command ELSE   run insert command   What's the best performing way to write this? With the announcement of the Ubuntu Phone OS I'd like to know what phone (and tablet) models are supported at this time.  Note from foss &amp; Oli: We are making this the master question for all future "Will this work on &lt;insert random tablet/phone/device here&gt;?!" questions Do electrons move around a circuit? We could imagine a simple electronic circuit composed by a power source and a resistor.   It is usual find descriptions as "The moving charged particles in an electric current are called charge carriers. In metals, one or more electrons from each atom are loosely bound to the atom, and can move freely about within the metal. These conduction electrons are the charge carriers in metal conductors." (wikipedia ). That made me imagine electrons going from power -, running by the wires, crossing the resistor and returning to power +, where they suffer a new kick to power -. All that at near the speed of light.  In this model, it is easy to define "electrical current intensity" as proportional to the number of electrons per second that cross one specific point of the circuit.  However, looking for the physics of resistivity, I can read the phrase "The actual drift velocity of electrons is very small, in the order of magnitude of a meter per hour." (wikipedia ). This is a complete different point of view, where electrons are more or less fixed around a fixed position (or moving very slowly), an electricity is a wave of "collisions" or "jumps" of electrons, like in a Newton's cradle (or "do the wave" as public in a football match). Obviously, we do not need severals hours from the switch on to the light on.  So, the question is, do electrons move around a circuit ? References to band theory are welcome.  A related question is: if electrons doesn't moves along the circuit, what is "electrical current intensity"?  Note: I known there are similar questions already posted, but answers of the ones I've read are not very detailed (could be I've not found the best ones). Please, be a few patient before "close vote". Analysing the "kind(s) of problem(s) are [ones]/[to be]" construction A. These kinds of problem are to be avoided. B. These kind of problems are to be avoided.  Are both A and B grammatical? If not, why not? If so, after having noticed the parallelism between A and C, and between B and D, could we say that C and D are &quot;standard&quot; English, too?  C. These kinds of problem are ones to avoid. D. These kind of problems are ones to avoid. How much to tip in the United States? I created a test MVC sharepoint app succesfully in a solution. Now I need to integrate this into a main solution project.   But when I added a SharePoint MVC App into the solution and hit F5, I am getting following error:  Error occurred in deployment step 'Install app for SharePoint':  Failed to install app for SharePoint. Please see the output window for details.   Detailed stack trace:  CorrelationId: 50583239-71cb-4b07-8bcb-4bbc51a22631      ErrorDetail: There was a problem with activating the app web definition.      ErrorType: App      ErrorTypeName: App Related      ExceptionMessage: Microsoft.SharePoint.SPException: Exception from HRESULT: 0x81070964 ---&gt; System.Runtime.InteropServices.COMException: Exception from HRESULT: 0x81070964 at Microsoft.SharePoint.Library.SPRequestInternalClass.EnableModuleFromXml(String bstrSetupDirectory, String bstrFeatureDirectory, String bstrUrl, String bstrXML, Boolean fForceUnghost, ISPEnableModuleCallback pModuleContext) at Microsoft.SharePoint.Library.SPRequest.EnableModuleFromXml(String bstrSetupDirectory, String bstrFeatureDirectory, String bstrUrl, String bstrXML, Boolean fForceUnghost, ISPEnableModuleCallback pModuleContext) --- End of inner exception stack trace --- at Microsoft.SharePoint.Administration.SPElementDefinitionCollection.ProvisionModules(SPFeaturePropertyCollection props, SPSite site, SPWeb web, SPFeatureActivateFlags activateFlags, Boolean fForce) at Microsoft.SharePoint.Administration.SPElementDefinitionCollection.ProvisionElements(SPFeaturePropertyCollection props, SPWebApplication webapp, SPSite site, SPWeb web, SPFeatureActivateFlags activateFlags, Boolean fForce) at Microsoft.SharePoint.SPFeature.Activate(SPSite siteParent, SPWeb webParent, SPFeaturePropertyCollection props, SPFeatureActivateFlags activateFlags, Boolean fForce) at Microsoft.SharePoint.SPFeatureCollection.AddInternal(SPFeatureDefinition featdef, Version version, SPFeaturePropertyCollection properties, SPFeatureActivateFlags activateFlags, Boolean force, Boolean fMarkOnly) at Microsoft.SharePoint.SPFeatureCollection.AddInternalWithName(Guid featureId, Int32 compatibilityLevel, String featureName, Version version, SPFeaturePropertyCollection properties, SPFeatureActivateFlags activateFlags, Boolean force, Boolean fMarkOnly, SPFeatureDefinitionScope featdefScope) at Microsoft.SharePoint.SPFeatureCollection.AddInternal(Guid featureId, Version version, SPFeaturePropertyCollection properties, Boolean force, Boolean fMarkOnly, SPFeatureDefinitionScope featdefScope) at Microsoft.SharePoint.SPFeatureCollection.Add(Guid featureId, Boolean force, SPFeatureDefinitionScope featdefScope) at Microsoft.SharePoint.SPUserSolutionCollection.EnsureWebFeaturesActivated(SPUserSolution solution) at Microsoft.SharePoint.Packaging.SPUserCodeSolutionDeploymentGroup.ActivateSolution(SPWeb web, SPUserSolution solution) at Microsoft.SharePoint.Packaging.SPUserCodeSolutionDeploymentGroup.Deploy() at Microsoft.SharePoint.Administration.SPAppTask.DeployOperation() at Microsoft.SharePoint.Lifecycle.MonitoredTaskExecution.DoTask()      Source: AppWeb      SourceName: App Web Deployment   Can somebody shed a light on this issue? :) Integrated circuits seem to have standard voltages of 5 V, 3.3 V, 2.5 V, 1.8 V, etc.  Who decides these voltages? Why do smaller devices require lower voltages? Securing SSH server against bruteforcing I have a little SVN server, old dell optiplex running debian. I don't have that high demands on my server, because its just a little SVN server... but do want it to be secure.  I just renewed my server to a newer and better optiplex, and started looking a bit into the old server. I took it down after experiencing problems. When I check the logs, its full of brute-force attempts and somehow someone has succeeded to enter my machine. This person created some extra volume called "knarkgosse" with two dirs "root" and "swap1" or something.  Don't really know why and what they do, but sure do want to prevent this from happening again. I find this a bit strange though because I change my password ever few months or so, and the passwords are always random letters and numbers put together... not easy to brute-force.  I know I can prevent root from logging in, and use sudoers... and change the SSH port, but what more can I do?  So I have a few questions:   How can I prevent logging in for 5 minutes after X amount of incorrect tries. Or slow tries down after each incorrect try? Is there some kind of central blacklist which a server can connect to? A blacklist that keeps track of IP addresses that are "unsafe" and should never be granted access? What more can I do to apply safety to my server?   Like I said earlier, I am running Debian 5 with Apache (www-data user problem?), svn, mysql, php, phpmyadmin, hudson. It is on a home network with port forwarding on 80, 443, 8080, 8180, 23 and 22. Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$} Here we use infinity norm as metrics for $C[0,1]$. The professor claims that this set is compact. I can show this set is relative compact by Arzela Ascoli, i.e., for each subsequence there is a further sub-subsequence that uniformly converges to a continuous function on $[0,1]$. But it is not intuitive to me why this set is compact, i.e. the limiting function also satisfies $|f'|_\infty \le M$. Is it true that $A$ is close and therefore compact? How to use custom icons with mapKit framework?