If $\gcd(m,n) = 1$, show that $m^{φ(n)} + n^{φ(m)} ≡ 1 \mod {mn}$   The only similar problem I found to this in my notes was "Find all positive integers $n$ such that $φ(n) = 12$." Unfortunately this was no help and I'm really lost, any help is really appreciated.
I do not know how to prove this question: $m^{\phi(n)}+n^{\phi(m)}\equiv 1 \pmod {mn}$ where $m$ and $n$ are relatively prime.  Can anyone help?
How do I get a module path? I'm creating a module in Drupal 8 and I need to define a relative path to mymodule to pass it on a JavaScript file like this.  $scope.getPlatformIcon = function (_platform) {      var modulePath = "/modules/mymodule"      switch (_platform) {          case "youtube":          case "twitter":          case "instagram":          case "facebook":              return modulePath+"/img/"+_platform+".png";      }      return false; };   In Drupal 7 I could use the following.  var modulePath = Drupal.settings.mymodule.basePath;   For the Drupal 8 root directory, I could use the following.  drupalSettings.path.baseUrl   I didn't find how to get the module path.  How can I do it with Drupal 8? I've been stuck on this problem for about an hour.     In an ideal transformer, the same flux passes through all turns of the primary and of the secondary . Show that in this case $M^2=L_1L_2$, where $M$ is the mutual inductance of the coils, and $L_1$, $L_2$ are their individual self-inductances.   My first instinct was the following: If we isolate the response of each coil to the other one, we get the equations $N_1\Phi_1=MI_2$ and $N_2\Phi_2=MI_1$, where $\Phi$ is magnetic flux going through a single turn of the coil and $I$ is current. Likewise, if we isolate the response of each coil to itself, we get the equations $N_1\Phi_1=L_1I_1$ and $N_2\Phi_2=L_2I_2$. I would now naturally want to say "Now just multiply $N_1\Phi_1$ and $N_2\Phi_2$ both ways, and get the desired result!" i.e.,  $$(N_1\Phi_1)(N_2\Phi_2)=(MI_2)(MI_1)=(L_1I_1)(L_2I_2)\implies M^2=L_1L_2$$  But that's incorrect, because the response of the coils to themselves and to the other coil are different quantities (i.e. not necessarily equal).  Realizing this, I write down the more general equations.  $$N_1\Phi_1=MI_2+L_1I_1$$ $$N_2\Phi_2=MI_1+L_2I_2$$  Now even with the hypothesis that $\Phi_1=\Phi_2=\Phi$, I still can't see how the identity $M^2=L_1L_2$ follows. Could you help clear up my confusion?  *Taken from D. Griffiths Introduction to Electrodynamics (4th ed.), Problem 7.58. Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$ Let $G$ be a finite abelian group and let $\phi: G \rightarrow G$ be a group homomorphism. I am trying to show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$.  I know that since $G$ is abelian we have that $\ker(\phi^{k})$ and $\phi^{k}(G)$ are normal subgroups of $G$ for any $k$  I also suspect we have the following towers which stabilize at $m$ and $m'$  $$\phi(G) \unrhd \phi^{2}(G) \unrhd \cdots \unrhd \phi^{m}(G)=\phi^{m+1}(G)=\cdots$$  $$\ker(\phi) \unlhd \ker^{2}(\phi) \unlhd \cdots \unlhd \ker^{m'}(\phi)=\ker^{m'+1}(\phi)=\cdots$$  I know that if given a group of the form $HK$ where $H$ and $K$ are normal subgroups of $HK$ and $H\cap K=1$ then $H \times K \cong HK$. I want to apply this to this situation but I am unable to show how write $G$ as a product $HK$  Resolution One of the comments directed me to the following article:  The last part of which answers the question. Below is what is says:  Choose $n=\max(m,m')$, then we have for $x \in \ker^{n}(\phi) \cap \phi^{n}(x)$, this means that $x=\phi^{n}(y)$ for some $y \in G$. This gives:  $0=\phi^{n}(x)=\phi^{2n}(y)$ which means that $y \in \ker^{2n}\phi=\ker^{n}{\phi}$. and then we have that $0=x=\phi^{n}(y)$.  The answer below kindly points out that every element $x$ is contained in one of the cosets of $G/\ker^{n}(\phi)$, this means that $x=k+g$ for some $k \in \ker^{n}(\phi)$ and $g \in G$. We also have that $g=\phi^{n}(h)$ for some $h \in G$. Writing $x=k+\phi^{n}(h)$ essentially shows that $G=\ker(\phi^{n}) + \phi^{n}(G)$. Using the fact above, the claim follows. Trouble plotting arctan with pgfplots I'm trying to plot arctan(x) with pgfplots but I don't get a good output. Here is my code:   \begin{cfigur}\begin{axis}[      width=160pt,compat=1.5.1,grid style={ultra thin},every axis plot post/.append style={thick},      x tick label style={font=\tiny},y tick label style={font=\tiny},      scale only axis,grid=major,axis lines=middle,      xlabel={$x$},      ylabel={$y$},      xmin=-200,      xmax=200,      domain=-200:210,      ymin=-5.5,      ymax=5.5,      xtick={-150,-100,...,150},      ytick={-5, -4,...,5},      restrict y to domain=-20:20,      legend style={at={(0.5,-0.05)},anchor=north,nodes={right}},  ]  \addplot[mark=none,color=blue, samples=500]{atan(x)};  \addlegendentry{$y = \tan^{-1}x $};  \end{axis}\end{cfigur} How do I concatenate const/literal strings in C? I'm working in C, and I have to concatenate a few things.  Right now I have this:  message = strcat("TEXT ", var);  message2 = strcat(strcat("TEXT ", foo), strcat(" TEXT ", bar));   Now if you have experience in C I'm sure you realize that this gives you a segmentation fault when you try to run it. So how do I work around that? If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$, prove that $\mathrm{Spec}(A_f)=\mathfrak{V}((f))^c$.   Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in \mathrm{Spec}(A): P \supset (f)\}$  My attempt:   On one hand, if $P \in \mathrm{Spec}(A)$ and $P \cap S$ is empty then we identify $\mathrm{Spec}(A_f)=\mathrm{Spec}(S^{-1}A)=\{P \in \mathrm{Spec}(A): P\cap S= \emptyset\}$.  On the other hand, $\mathfrak{V}((f))^c=\{P \in \mathrm{Spec}(A): P \nsupseteq (f)\}$.   So the exercise is equivalent to prove: $$\{P \in \mathrm{Spec}(A): P\cap S =\emptyset\}=\{P \in \mathrm{Spec}(A): P \nsupseteq (f)\},$$ but I can't continue so I'd appreciate if somebody could help me.   Thanks in advance. I am travelling to Budapest and therefore have a Schengen Visa. I hold an Indian passport and a valid US visa. I'm travelling from Chicago-Heathrow-Budapest and returning by Budapest-Dublin-Chicago.  Would I need a transit visa for UK/Ireland?  UPDATE  Hoping this will help someone someday.   I was not allowed to board my return flight (BUD-DUB-ORD) because I do not hold an overnight Irish Transit Visa. They said I could have transited happily if the connecting flight was in the same day, but because I would have to originally land at 10 PM and catch the connecting flight next morning at 10 AM, I needed a transit visa. Well, I thought it was BS, but couldn't help comply with the rules. I booked another last minute ticket and flew via Warsaw via LOT. Does tourist visa (90 days) apply after a long-term visa ends in Schengen countries? What are the obj and bin folders (created by Visual Studio) used for? A positive integer is divisible by $3$ iff 3 divides the sum of its digits I am having trouble proving the two following questions:    If $p|N$, $q|N$ and gcd(p,q)=1, then prove that $pq|N$ If $x$ is non zero positive integer number, then prove that $3|x$ if and only if 3 divides the sum of all digits of $x$.   For both questions I tried to use theorems of discrete mathematics, but I could not find the way to solve them. Suppose that I create Material001  and Material002  in Material Properties and set them to some color. Can I use this 2 items in other buttons like Texture Properties, Object Data Properties etc. ? Where and how do I intertwine them with this button ? There's more gravitational force in our galaxy (and others) than can be explained by counting stars made of ordinary matter. So why not lots of dark planetary systems (i.e., without stars) made of ordinary matter? Why must we assume some undiscovered and unexplained form of matter? Understanding hard drive space Finding files by only knowing extension? Hat duplicates when info is expanded Why hash the message before signing it with RSA? A non-trivial, non-negative, function bounded below by its derivative with $f(0)=0$? I did not know what to search to see if this existed elsewhere. But, I could not find it. Here's the question:  Does there exist a continuously differentiable function, $f: [0,1] \rightarrow [0, \infty)$, such that the following hold? (i) $f(0) = 0$ (ii) $f'(x) \le c f(x)$ for all $x$ ($c$ is a fixed constant) (iii) $f \not\equiv 0$  This was a fun problem someone asked me a long time ago. I have always been convinced there is no such function, but I cannot prove it. I have mainly tried using the mean value theorem (in multiple forms), and re-wording the problem in terms of the integral of a continuous function to no avail. I spent some time trying to use (ii), the definition of derivative, and squeeze theorem, but no luck. After much time trying to prove non-existence, I spent a little time trying to find such a function... also with no luck. This problems ability to avoid being solved has since taken away the original fun and replaced it with a twinge of annoyance. Can I connect to SQL Server using Windows Authentication from Java EE webapp? I am currently investigating how to make a connection to a SQL Server database from my Java EE web application using Windows Authentication instead of SQL Server authentication. I am running this app off of Tomcat 6.0, and am utilizing the Microsoft JDBC driver. My connection properties file looks as follows:  dbDriver              = com.microsoft.sqlserver.jdbc.SQLServerDriver dbUser                = user dbPass                = password dbServer              = localhost:1433;databaseName=testDb dbUrl                 = jdbc:sqlserver://localhost:1433   I have zero problems with connecting to a SQL Server database in this fashion when using SQL Server authentication.  Is there any way I can retrieve the credentials of the user's Windows Authentication and use that authentication for SQL Server?  UPDATE: I know in ASP.net there is a way to set up Windows Authentication for access to the webapp, which is exactly what I am looking for, except I want to pass that token off to SQL Server for access to the database. Reduction Integration of $I_n=\int{\frac{x^n}{\sqrt{x^2+a^2}} dx}$ How do I derive this using reduction formula?  $$I_n=\int{\frac{x^n}{\sqrt{x^2+a^2}} dx}$$  I got $\sqrt{x^2+a^2}(x^n) - \frac{1}{n-1}(x^{n-1})\sqrt{x^2+a^2} + \frac{1}{n-1}(I_n) $ . I'm not too sure if this is correct. I would appreciate any form of help or correction to my answer. Thank you. While using the mobile site on my Android phone, I saw a comment that I wanted to flag but there was no flag button. After searching, I saw  that says flagging comments is not supported on mobile site.  Could the ability to flag comments be added to the mobile site?