$\frac{1}{5}n^5+\frac{1}{3}n^3+\frac{7}{15}n$ is an integer for every integer n.
Using congruences, show $\frac{1}{5}n^5 + \frac{1}{3}n^3 + \frac{7}{15}n$ is integer for every $n$
Let $A$ be a square matrix of order $n$ such that $A^2 = I$.  Prove that if $1$ is the only eigenvalue of $A$, then $A = I$. Prove that $A$ is diagonalizable.   For  (1), I know that there are two eigenvalues which are $1$ and $-1$, how do I go about proving what the question asks me? What is Java's answer to WPF? Can a single player build a clan dojo? Is it possible for an one-or-two-person clan to build a dojo? If so, what are the resources involved?   I'd like to experience the whole Dojo building experience so I don't plan to join an existing Clan with a dojo already. Alright, when I run certain commands the wrong way, (misspelled, etc.) The terminal outputs this: &gt; instead of computername:workingfolder username$, and when I type enter it goes like this:  &gt;  &gt;  &gt;   That would be if I pressed enter 3 times. CentOS 7 DVD ISO vs Everything ISO different On my mac I have 2 accounts and the both are normal account. So in this moment I don't have a Administrator account, only one that was in the past a administrator, but now is not.  How I can have again a Administrator account? I bought a new MacBook Air and was able to connect to the internet at first but now it won't connect.  I have other devices as well as other computers on wifi but my Mac won't connect.  When I try and click on the wifi, it tells me that it can't be joined.  How can I troubleshoot? Expression for a choice which isn't really one Why doesn't the Enterprise use stealth/invisibility technology like the Romulans and Klingons? The stealth mode of starships plays a huge role in many Star Trek films and episodes and is a strong advantage in combat. I remember an episode revealing that the Federation developed a secret stealth technology (afair on Riker's former ship with another captain appearing in this episode) that even made it possible to fly with the ship through matter.   Nonetheless, while stealth technology seems standard for Romulans, Klingons (Ferengi, Vulcans?) and does not break any intergalactic laws (do these exist at all?), the Federation doesn't use it? Let moderators navigate the review queues at will This post is an expansion of  For many graduated sites (eg Physics), moderators do not want to apply the unilateral close vote in all cases--it is better to let some community votes pile on first.  We only use the unilateral in obvious cases then. This renders the CV queue useless to us. If we &quot;skip&quot; something, we can't get back to it. If we say &quot;do not close&quot;, it's removed from the queue and our community can't see it. If we use &quot;close&quot;, we are using a unilateral mod vote--we don't want to do that. What I propose is a set of &quot;next/previous&quot; buttons, that let us navigate the queues, looking for closeable questions, and coming back to the others later. Thoughts? Update and bounty reason Due to the recent Great Flag Overhaul, mods no longer see close flags in the flag queue. Which means that the only way to get to them is via the review queue 1 Shog's solution is ideal in this case; let mods view the close queue as a list. (In fact, a list similar to the old flag queue -- which shows a list of mods who have seen the post and what mod actions have already been taken -- would be great) On Physics, the majority of the flags that we get are VTC flags. And the community isn't always able to close things on their own, so we have to step in. However, powering through the close queue isn't an option; because &quot;skip&quot;ing a post2 leads to it being dumped into the void (I can't get back to it unless I have a link). Extremely inconvenient. Could the workflow here be improved? 1. The /tools listing for posts with outstanding close votes doesn't work because it doesn't list posts which have been flagged but not VTCd 2. which I may want to take a closer look at later; gotta be careful with the binding votes. How does a wheelie work? So I've been trying to create a mathematical model for an electric motorcycle and began to wonder about the maximum possible torque that could be supplied to the rear driven wheel without having the bike begin to lift up on one wheel.  I found ways to calculate this value online; however, the basic concept as to how the bike actually lifts escapes me.    My problem started when I began to think about what axis the bike will rotate about when it is doing the wheelie.  My first intuition was that the frame and front wheel, together, rotate about the rear axle.  But when I drew a free body diagram of the frame/front wheel system just at liftoff (see below) I made note that the only forces acting about the rear axle, point O, is the force of weight.  This means that an increase in applied torque and subsequently, the applied force Fa, should not effect the rotation about the rear axle.  I know that the applied force on the back wheel is indeed correlated with the propensity for a bike to wheelie, so I considered that the axis of rotation I was choosing was wrong.  If we take the free body diagram above and sum the moments about the center of mass, we would find that an increased applied force would in fact cause the solid body to rotate.  The problem with this understanding is that during rotation, the center of mass of the drawn system should actually rises relative to the surface the bike is moving across.  If the bike were to be truly rotating about its center of mass, then the back wheel would begin to dip below the surface of the road like you might see in a glitchy video game.    So I suspect that the bike is in fact rotating about the back axle, but I don't understand why, please help!  edit: I added the external torque from the back wheel to the frame, which would allow the bike to rotate about the back axle  edit 2:  I suppose that the torque acting on the frame via the engine should not effect the rotation as the movement of a motorcycle can be perfectly replicated by applying a force at the back axle.  A third possible axis of rotation might be the lowest point of the back wheel. According to general relativity, inertial mass and gravitational mass are the same, and all accelerated reference frames (such as a uniformly rotating reference frame with its proper time dilation) are physically equivalent to a gravitational field of the same strength.  Refer: Einstein's thought experiment of a Physicist in an Accelerated Box.  I find that the 'How' part of this question is somewhat easy to understand Mathematically but hard to visualize physically and on the 'Why' part I am totally blank. I've watched the Vampire Knight anime. I really like the story so far but find it frustrating that the anime doesn't finish off the story. I'm trying to figure out if it is worth reading the manga.  Does the manga contain more story than the anime? Does it finish the story off? Why do you cast downvotes on answers? This is a question which started off with a , then turned into an , and now it's a question in its own right (as suggested by Jeff). I'm not sure where it could go from here :)  It's obvious that different people cast downvotes for different reasons. (Upvotes are less controversial, generally.) Here are some possible reasons:   Answer is wrong Answer is misleading (may be technically accurate, but will lead to readers making mistakes) Answer doesn't match the question (e.g. a C# answer to a question about Java) Answer doesn't really address the question, e.g. suggesting a completely different solution, even if the question specifies that there are good reasons why the current approach has to be followed Answer is okay, but not as good as another one which currently has fewer votes Answer appears to be plagiarised from existing answers. Answer is by someone I don't like. Answer is abusive (profanity, spam etc). No reason, I just felt like it. I'm a , and prefer down to up. Have you considered down? Ideological grounds: answer suggests a technology I don't like, or expresses a negative opinion of a technology I do like   Personally I go with 1, 2 and 8 - although I'll add comments suggesting changes in various other situations.  Have I missed anything? What do you do, and why? What are the net benefits (to you, the world in general, and the answerer) from your approach? What are preimage resistance and collision resistance, and how can the lack thereof be exploited? What is "preimage resistance", and how can the lack thereof be exploited? How is this different from collision resistance, and are there any known preimage attacks that would be considered feasible? I would like to install Ubuntu  for almost 120 users. But it's very difficult in using USB and drives.  Any suggestions? Wireless antenna does not appear despite its driver being installed ffmpeg - how to apply filters without losing quality NHibernate Get objects without proxy Teleporting Castle I remember seeing a movie when I was younger, but can't find it online because I can't remember a whole lot about the story.  What I do remember is there was a group of soldiers trying to catch this castle that appeared only during the day and at the start of each new day it would teleport to a new random location.  I also remember the main character having this neat little flying blade disk that reminds me of something from the Predator movies that he could control with his mind.  Again I can't recall a lot about the movie, but if it sounds familiar I would be grateful.  Also it seems like at one point there were horses that ran so fast they caught fire!