Q. If $m$ and $n$ are two positive integers and $f:A\to A,$ where $A=\{0,1,2,3,...,m-1\},$ is defined by $f(x)= nx\pmod m.$ Find the condition on $n$ and $m$ which ensures that $f$ is one-to-one and onto.   My attempt: Here $f^{-1}$ must exist if $f$ is a bijection. Now as $f$ maps $x$ to $nx$ under modulo $m$ so $f^{-1}$ must map $x$ to $n^{-1}x$ under modulo $m.$ But in $\mathbb Z_m$(or here $A$), $n^{-1}$ will exist iff $\gcd(m,n)=1.$ So I think the required condition should be that $m$ and $n$ are relatively prime. (I don't know what the actual answer is!) I have just started with number theory and discrete mathematics this semester. So I don't have much knowledge to deal with such type of questions involving moduler arithmetic. So kindly guide me saying that whether my approach is correct or not. If not, please suggest me a way to reach the answer to this question or if it is correct, then can we reach the same conclusion using a different approach? I would love to know about your insights. Thanks in advance...
Say we have two integers $n$ and $k$, such that $0 &lt; k &lt; n$ and $gcd(n, k) = 1$.  Why is it that if we take every integer $x\in\{0,\dots, n-1\}$ and compute $k\cdot x \ (mod \ n)$, we get as remainders all numbers from $0$ to $n-1$, with no repeats?  In other words, why is $\ f:\{0,\dots,n-1\}\rightarrow\{0,\dots,n-1\}$, $\ f(x)=k\cdot x\ (mod\ n)$ a bijection for $k,n\in\mathbb{N},\ gcd(k,n)=1$?
I have an ArrayList that I want to output completely as a String. Essentially I want to output it in order using the toString of each element separated by tabs. Is there any fast way to do this? You could loop through it (or remove each element) and concatenate it to a String but I think this will be very slow. I have a default Blender cylinder positioned upright atop a ground plane:    How can I rig the cylinder so that can be animated to wiggle/jitter (perhaps procedurally with noise f-curve modifiers on two of its rotation axis) without penetrating the floor?    A simple parent-child relationship as in the image above is sufficient to rotate (and translate) the cylinder on 1 axis in either a positive OR negative direction.   I'm looking for a setup that would allow me easily to rotate the cylinder on two or even three axis in both a positive and negative direction.  I imagine this would be possible with some drivers applied to the XYZ coordinates using trigonometric equations but the math escapes me. It might also be possible using a particular bone setup with the correct constraints but my attempts to find something that works so far have not succeeded.  Thanks for your assistance. My problem is simple. I think that the strange deformations on the top are there because there are so many vertices in one face. How do I fix this while maintaining nice topology?  Although I triangulated the top polygon, the problem still persists. I once already asked a very similar question; . But this is some what different.  I have the following code;  \documentclass{standalone} \usepackage{graphics} \usepackage{tikz,pgfplots} \pgfplotsset{compat=1.6} \begin{document} \begin{tikzpicture}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot1]  \end{axis}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.2, ymax=1.2, ylabel={$h(t)$}, ymajorgrids, yminorgrids, name=plot2, at=(plot1.right of north east), anchor=left of north west]  \end{axis}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{o}}(t)$}, ymajorgrids, yminorgrids, name=plot4, at=(plot1.below south west), anchor=above north west]  \end{axis}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=2000, ymax=12000, ylabel={$p_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot5, at=(plot4.right of north east), anchor=left of north west]  \end{axis}  \end{tikzpicture} \end{document}   This creates the following plot:    I think there is to few vertical/horizontal spacing between the plots. I want to increase this. Currently I use x- yshift, this gives the following result:  \documentclass{standalone} \usepackage{graphics} \usepackage{tikz,pgfplots} \pgfplotsset{compat=1.6} \begin{document} \begin{tikzpicture}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot1]  \end{axis}  \begin{axis}[% xshift=1cm, width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.2, ymax=1.2, ylabel={$h(t)$}, ymajorgrids, yminorgrids, name=plot2, at=(plot1.right of north east), anchor=left of north west]  \end{axis}  \begin{axis}[% yshift=-1cm, width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{o}}(t)$}, ymajorgrids, yminorgrids, name=plot4, at=(plot1.below south west), anchor=above north west]  \end{axis}  \begin{axis}[% xshift=1cm, width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=2000, ymax=12000, ylabel={$p_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot5, at=(plot4.right of north east), anchor=left of north west]  \end{axis}  \end{tikzpicture} \end{document}     I want to know if there is a more consistent manner to do this. 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