Prove that $2730$ divides $n^{13} - n$ for all integers $n$.   What I attempted is breaking $2730$ into $2, 3, 5$, and $7, 13$. Thus if I prove each prime factor divides by $n^{13} - n$ for all integers $n$, it will prove the original problem.   I already proved $2, 3$, and $5$ work. Struggling with how to do $91$?  Other solutions are welcome, but they should be associated with mod and solutions of this kind, because my answer should pertain to what was learned in class.  (Edit: Fixed the 91 problem, being slow today...In any case, I've got this problem now, so no answers are necessary. Posting my own soon.)
Show that $2730\mid n^{13}-n,\;\;\forall n\in\mathbb{N}$   I tried, $2730=13\cdot5\cdot7\cdot3\cdot2$  We have $13\mid n^{13}-n$, by Fermat's Little Theorem.  We have $2\mid n^{13}-n$, by if $n$ even then $n^{13}-n$ too is even; if $n$ is odd $n^{13}-n$ is odd.  And the numbers $5$ and $7$, how to proceed?
UIToolbar tint on iOS 4 math behind complex numbers C||RLC I have a mathematical problem with the following formula. The goal is to transform this equation to have a seperated real and imaginary part. I have already done this with several other equasions, but for this particualar one I don't know where to start.  I dont expect you to do the complete work of transforming for me, it would already be very nice if you could give me a hind on how you would start.  Thank you in advance!  C||RLC      This is my current progress:    My question is:  How can I get rid of the imaginary part jωC2*R?  When the denominator is completely real I can just split up the formula and have an imaginary and a real part. How do i install drush for a drupal site which is on shared hosting? How do I create a decimal field in Access with Alter Table? Video showing up as white space On  question, there is a YouTube video in , on the Android app the video shows up as a white space. There is no indication that it is a video.  Would it be possible to show the video in the app or link to the video on YouTube instead? When to use a hyphen to coin a new word and when to omit a hyphen? Someone has asked for answer to these topics. However, I still want someone to provide me with simple and universal answers. I recently read a sentence from the English-speaking person. It is in an article. It is, ' The available options for mitigating pad cratering are not universally effective (see link below), but they may be helpful on a case by case basis until more crack resistant laminates are developed and become available.  '  If I am to write this, I will use 'a case-by-case basis until more crack-resistant laminates'   Can anyone tell me when to use a hyphen to coin a new word? Canonical Isomorphism Between $\mathbf{V}$ and $(\mathbf{V}^*)^*$ For the finite-dimensional case, we have a canonical isomorphism between $\mathbf{V}$, a vector space with the usual addition and scalar multiplication, and $(\mathbf{V}^*)^*$, the "dual of the dual of $\mathbf{V}$." This canonical isomorphism means that the isomorphism is always the same, independent of additional choices.   We can define a map $I : \mathbf{V} \to (\mathbf{V}^*)^*$ by $$x \mapsto I(x) \in (\mathbf{V}^*)^* \ \text{ where } \ I(x)(f) = f(x) \ \text{for any } \  f \in \mathbf{V}^*$$  My Question: what can go wrong in the infinite-dimensional case? The notes I am studying remark that if $\mathbf{V}$ is finite-dimensional, then $I$ is an isomorphism, but in the infinite-dimensional case we can go wrong? How? What's the fastest connection speed / shortest elapsed time using Migration Assistant? How do I create a shortcut for a script, and run it as a .desktop file? I have installed Teamspeak onto my Ubuntu 15.10 system. Teamspeak requires a script to be ran in-order to start. How can I create a symbolic link that is a .desktop so I can open the .desktop file, and have it run the script? I have looked at a few methods, but they all seem like hotfixes. Is there a proper way to do this? If I remote wipe my iPhone, can I still locate it remotely? If I lose my iPhone, and I use remote wipe to prevent my confidential data from being stolen, then can I still use Find My iPhone to locate it on a map?  I'm assuming not (since it resets to factory settings), but I wanted to double-check. Find all functions $f$ such that $(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$ Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ such that $$(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$$  I've tried subbing in heaps of values but I keep getting things like $f(0)=f(0)$ and other such useless results.  Any help would be hugely appreciated. I have a Dirichlet distribution from which I'm maximizing $\alpha$ by calculating the log likelihood with the following equation  $p\left(s|\alpha\right) = \sum\limits_{i=1}^O\log\left(\frac{n_i!}{\prod\limits_{j=1}^kn_{ij}!}\frac{\Gamma\left(\alpha_0\right)}{\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)} \prod\limits_{j=1}^k \frac{\Gamma\left(n_{ij}+\alpha_j\right)}{\Gamma\left(\alpha_j\right)}\right)$  I have a python script to calculate the log likelihood. It works for small values, but when $n_{ij}$ gets high $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ python runs into overflow error (the max for math.gamma is between 171 and 172).   Is there a way to change the likelihood function so that $\Gamma\left(\sum\limits_{j=1}^k\left(n_{ij}+a_j\right)\right)$ and $\Gamma\left(n_{ij}+\alpha_j\right)$ don't run into overflow errors. Can I just divide the values in $\Gamma$ by for example 100, if I then compare the likelihood between $p(s|\alpha)$ with different $\alpha$ values do the best $\alpha$ values still have the highest likelihood? Which punctuation is correct in this case (colon, semicolon, or period)? How is Cycles different from Blender Internal? How does Cycles work and how is it different from Blender Internal?  Which is better? Image looks embossed when converted to SVG Prove that the additive groups of $\Bbb Z$ and $\Bbb Q$ are not isomorphic.    It is hard to create a map that will show homomorphism. Rendering order in an Entity System Say I use a , and also inside Systems I hold lists of all entities that Systems are required to process.  How do I maintain this list of entities in desired rendering order, i.e. for a dumb 2D RenderingSystem?   I saw , and what they suggest is to do something like Z ordering - what I would probably do is just to store a "layer" int in DrawableComponent and then, inside RenderingSystem, just sort entities by mentioned "layer" whenever the entity list for RenderingSystem changes.  They also say we could just delete and recreate the entity whenever we want it on the top, but it seems too inflexible to me.  How is this problem usually solved? What are the similarities/differences between the behaviors of Quantum particles and bouncing droplets? What are the minimum requirements to run Windows 10? I am looking to upgrade my system to Windows 10 RTM, but I'm afraid of losing performance.  I researched on the site but found details (maybe I have not sought duty) on the hardware minimum.  What are the minimum and recommended requirements for running Windows 10 RTM? How can I bind arguments to a Python method to store a nullary functor for later invocation?  Similar to C++'s boost::bind.  For example:  def add(x, y):     return x + y  add_5 = magic_function(add, 5) assert add_5(3) == 8