I have trouble proving the above statement.  Here is my approach.  First, I prove that if $n$ and $k$ are not relatively prime, it implies that $[k]_n$ is not invertible.   If $n$ and $k$ are not relatively prime, there exist $g \neq 1$ such that $nx+ky=g$ for some integers $x$ and $y$. (Bézout's identity)  Multiply the equation by, $n/g$  $(n^2x)/g+(nky)/g=n$  We can simplify this by taking the $(mod \hspace{0.1cm} n)$  $(nky)/g=0(mod \hspace{0.1cm} n)$  I don't understand how to proceed from here!. I sense that it has something to do with $k=0$ is the only solution here.  The converse proof is fairly straightforward (I assume.)  Here is my approach to prove the converse, if $k$ and $n$ are relatively prime then $[k]_n$ is invertible.  Again by the Bézout's identity (if $k$ and $n$ are relatively prime), there exist integers $x$ and $y$ which satisfy $nx+ky=1$. We can simplify this to $k\times(y+nx/k)=1$. It is clear that $(y+nx/k)$ is the inverse of $k$.
I think the following is true:  If $q$ is coprime to $a$ then $a\mid (nq-1)$ for some $n\in\mathbb Z$.  I have started to first reducing $n\in\mathbb Z$ to $n\in\{1,\ldots,a-1\}$ and then considering all the cases $q = ka + j$ for $j=1,2,3,\ldots,a-1$. I stopped at $j=3$ and found that this is kinda messy. Does anyone have an elegant proof for this (maybe using some results from number theory)?  [Note: question has been generalized from prime $q\nmid a$ to $\,q$ coprime to $a$]
I really enjoyed that site. I've only posted on it once, but I enjoyed seeing other people's posts and appreciated reading about entrepreneurial dilemmas. For $q, p$ odd primes such that $p \neq q$, there is not primitive root modulo $pq$. How does "failure" work in Dungeon World? How does it move the story forward? I am having trouble understanding the specifics of failure in Dungeon World. According to the rules, a GM should "make a move" against players when they fail, and that's alright, I get that.  What I don't get is how a GM is supposed to advance the plot by doing so. Let's take an example:  Rick the Fighter is climbing a massive chain that serves to hold down the wizards tower, preventing it from drifting off. He makes the move Defy Danger, to avoid slipping of, since it's raining hard, and the chain is slippery with algae and moss. He rolls and get a score of 5, meaning he fails.  In the above situation, he fails climbing the chain, so the GM should make a move. I have a hard time seeing what move would advance the story here, since I feel that the failure would imply that he doesn't get to the tower.  Is this one of those situations, where you just skip making a roll, and just let the players climb the chain, or does anyone have any idea how to handle the move, so that the story is advanced somehow?  I just want to be able to give the players some dangers that aren't traps and monsters, and I thought that a flying wizards tower was cool. I'm having a hard time seeing how I can advance the story on a failure in this situation.    Let me elaborate on the problem. I will not say that it's wrong to make the players succeed with some "cost", even on a failed roll, it just makes true "failure" impossible. That's my beef about it. The players will eventually succeed at anything that doesn't outright kill them. Given that not all employees have a diamond:     All Stack Overflow employees aren't given a moderator diamond; the Community team has tightened up internal policies regarding who gets one and who doesn't. [...] —      can we have some visual indication that they are an employee?  This was  back in 2011, but declined. However, the situation appears to have changed now with fewer employees being given a diamond.  That way when they post on a meta site we know that it's somewhat "official".  Failing that please ensure that everyone includes the fact they're an employee in their profile given that  to add employee to their profile pages was also declined. Short story about a group of sci-fi writers sitting around discussing their profession Date comparison : which date format? I know that hardly any information is passed to the PDF when a .tex file is compiled.  But is there a tool that can convert a PDF document back to (La)TeX? Evaluating $\int_{0}^{1}\int_{0}^{1}\sqrt{x^{2}+y^{2}}dxdy$ using polar coordinates. I want to evaluate the integral $\int_{0}^{1}\int_{0}^{1}\sqrt{x^{2}+y^{2}}dxdy$ using the substitution $x=r\cos\theta$ and $y=r\sin\theta$ where the Jacobian is given as $|J(r, \theta)|=r$ and thus $dxdy=rdrd\theta$.  I need to set the range for $r$ and $\theta$. For $\theta$, it seems obvious that it ranges from $0$ to $\frac{\pi}{2}$ to cover all parts of the rectangle. But I am confused with the range for $r$. Would it be OK to set minimum $r$ as $0$ and maximum $r$ as $\sqrt{2}$, which is the diagonal of the square domain? I think I am confused with the fundamentals of integration. Hi i want to put this simbol "%" in the tick labels only in the x axis how can i do? A bit of background, I am doing research in atmospheric physics and detection systems  Recently, I received a paper submission back with revisions suggested by a couple of reviewers.  For the most part, the revisions were definitely constructive, justified and most importantly, helpful.    That is, except on one particular crucial aspect of the paper where both reviewers made suggestions, that they labelled as critical for the paper's success.  However, the suggestions were directly opposite.  i.e. Both suggestions are mutually exclusive, but equally feasible.    What is an effective means to address this conflict?  I have spoken to my academic supervisor and he is not sure how to proceed either and we have double checked and both suggested paths are feasible. Investment options for f1 visa students in USA The differences between TeX engines Why does an error appear every time I try to open the Ubuntu Software Center? There is a glitch on the Ubuntu Software Center and whenever I open it an error appears and it keeps loading and never opens.  Why does this happen?  I have installed Ubuntu 13.04 on a disk and partitioned it.  Please help me and ask for excess information if you need it.  If you know of any duplicates please show me them!!  I get the following error in a terminal  SystemError: E:Type '&lt;!DOCTYPE' is not known on line 1 in source list /etc/apt/sources.list.d/medibuntu.list   The /etc/apt/sources.list.d/medibuntu.list file contains a HTML document. Is it possible to somehow mark a System.Array as immutable.  When put behind a public-get/private-set they can't be added to, since it requires re-allocation and re-assignment, but a consumer can still set any subscript they wish:  public class Immy {     public string[] { get; private set; } }   I thought the readonly keyword might do the trick, but no such luck. How to change camera movement speed on Path during animation Im making a animation in blender that requires the camera to change speeds while moving along a path. I'm using the Curve/Path for the path. I need the camera to move at a certain speed going between 2 places, and then slow down while moving around something its focusing on. Adobe Reader and xcolor within a table breaks the line render banner set interval Direct product of Sylow subgroups Proposition II.7.5 of Hungerford's Algebra goes     Proposition 7.5. A finite group is nilpotent if and only if it is the direct product of its Sylow subgroups.   Let $G = (\mathbb{Z}_6, +)$. $G$ is abelian, so it is nilpotent and thus by the proposition is the direct product of its Sylow subgroups.  The Sylow subgroups of $G$ are $H = (\mathbb{Z}_3, +)$ and $K = (\mathbb{Z}_2, +)$, so the proposition says $G = H \times K$. In particular, $0 \in H$ and $0 \in K$, so $(0,0) \in H \times K = G$... but that's not true, since $G = \{0,1,2,3,4,5\}$.  Where is/are my misunderstanding(s)? "Newtonian limit" property of special relativity Books say that special relativity is indistinguishable from Newtonian mechanics when the speed of the primed frame ($v$) is small compared to the speed of light ($c$). This is what I mean by the "Newtonian limit" property of special relativity. (I don't know the correct name for it). But the transformation of the time coordinate  $$t' = \gamma \left(t + \frac{vx}{c^2}\right)$$  involves the spatial quantity $x$ which, if large enough, could balance out the smallness of $v/c^2$. So why doesn't this observation mess up the "Newtonian limit" property? Image of Jacobson Radical is the Jacobson Radical I'm stuck on the following problem:      Let $R$ be a commutative ring with $1$, and suppose $R$ is semilocal with maximal ideals $\mathfrak m_1, ... , \mathfrak m_s$.  Let $\mathfrak a$ be an ideal of $R$. The problem is to show that under the usual surjection $\pi:R \rightarrow R/\mathfrak a$, the Jacobson radical of $R$ is sent to the Jacobson radical of $R/\mathfrak a$.   My attempt: suppose that $\mathfrak m_1, ... , \mathfrak m_t$ contain $\mathfrak a$, but $\mathfrak m_{t+1}, ... , \mathfrak m_s$ do not.  Then $Jac(R/\mathfrak a) = (\mathfrak m_1/ \mathfrak a) \cap \cdots \cap (\mathfrak m_t/ \mathfrak a)$.  Obviously $\pi Jac(R) \subseteq Jac(R/ \mathfrak a)$, and the converse inclusion is equivalent to the following claim: if $x \in \mathfrak m_1 \cap \cdots \cap \mathfrak m_t$, then there exists a $y \in \mathfrak m_1 \cap \cdots \cap \mathfrak m_s$ such that $x - y \in \mathfrak a$.    This should be some application of the Chinese remainder theorem, but I'm not seeing it.  All I have to go on is that there exist $m_i \in \mathfrak m_i$ ($t+1 \leq i \leq s$) with $m_i \not\in I$.  I also haven't found out a way to use the fact that $\mathfrak m_1 ,... , \mathfrak m_s$ are all the maximal ideals of $R$.  I was also thinking about looking at some explicit maps.  If we let $\Gamma$ be the composition $$\frac{(R/\mathfrak a)}{Jac(R/ \mathfrak a)} \xrightarrow{\cong} \frac{(R/\mathfrak a)}{(\mathfrak m_1/\mathfrak a)} \oplus \cdots \oplus \frac{(R/\mathfrak a)}{(\mathfrak m_t/\mathfrak a)} \xrightarrow{\cong} R/\mathfrak m_1 \oplus \cdots \oplus R/\mathfrak m_t$$ then we have a commutative diagram $$\begin{matrix} R/Jac(R) &amp;  \rightarrow &amp; \frac{(R/\mathfrak a)}{Jac(R/\mathfrak a)} \\ \downarrow &amp; &amp; \downarrow \Gamma \\ R/\mathfrak m_1 \oplus \cdots R/\mathfrak m_s &amp; \rightarrow &amp; R/\mathfrak m_1 \oplus \cdots \oplus R/\mathfrak m_t \end{matrix}$$ where the vertical arrows are isomorphisms, and the horizontal arrows are surjections.