Showing that if $d=\gcd(n,m)$ then the integers $\frac{n}{d}$ and $\frac{m}{d}$ are coprime I'm having trouble understanding the indicated line from the solutions  Do a proof by contradiction by assuming there exists an $a &gt; 1$ such that $a \mid \frac{n}{d}$ and $a \mid \frac{m}{d}$.  This implies (I don't understan why!) $ad\mid n$ and $ad\mid m$,   but $ad&gt;d = \gcd(m,n)$, a contradiction.
$\forall a,b,c \in \mathbb{N}\!:\ a\mid b \Rightarrow ac\mid bc. $ Divisibility Proof Could someone verify this proof?  Given the following definition of divisibility $(\forall{a,b \in \mathbb{N}})[a \mid b \equiv (\exists{c \in \mathbb{N}})(ac=b)]$.  Statement: $(\forall{a,b,c \in \mathbb{N}})[(a\mid b) \rightarrow (ac\mid bc)] $ .  Proof:  Let $a,b \in \mathbb{N}$ be arbitrary.  Assuming that $a\mid b$, we can conclude that $\exists{d \in \mathbb{N}}$ such that $ad = b$.  By Algebra we can deduce the following sequence of statements.  If $(ad)=b$, then $(ad)c=bc$.  Therefore $(ac)d=bc$.  Hence by the definition of divisibility we know that $ac\mid bc$.
I am using Windows 10. I am trying to install 1 open source software.  In the readme.txt of that software, it says it requires Visual C++ 2008 Redistributable. I found that my Windows already has Visual Studio 2012 Redistributable installed.  After I install the open source software, it seems that it is working, even though I haven't installed Visual C++ 2008 Redistributable.  Do I need to install Visual C++ 2008 Redistributable if my Windows already has Visual Studio 2012 Redistributable? if $(x_0,y_0)$ is local extrema in $ax^2 + by^2 + cxy + dx + ey + l$ then its global too. An exercise on a book asks me to prove that if a point $(x_0,y_0)$ is a local extrema for the function $$f(x,y) = ax^2 + by^2 + cxy + dx + ey + l$$ then it's also a global extrema.  The exercise asks me to consider that $g(t) = f(x_0+ht,y_0+kt)$ is a parabola, but I couldn't understand how it helps. Maybe since it's a parabola, I know that this derivative has to be $0$ in some point, and also that this function $g(t)$ has a global extrema point, but I couldn't derive anything from this. I also tried to find the derivatives, equal them to $0$ and solve the system. I got:  $$x_0 = -\frac{2 b d - c e}{4 a b - c^2}, y_0 = -\frac{   c d - 2 a e}{-4 a b + c^2}$$  But when I try to find   $$f\left(-\frac{2 b d - c e}{4 a b - c^2},-\frac{   c d - 2 a e}{-4 a b + c^2}\right)$$ so I can try to prove:  $$f(x,y)-f(x_0,y_0)$$  is either greater or less $0$ depending on the case (so I can do $f(x,y) \le f(x_0,y_0)$ and $f(x,y) \ge f(x_0,y_0)$)  things get hairy. I have decided to switch from Powerpoint to Beamer, and for my first presentation I'd like to use the , and the same colors as in this goggle slide template:  .  I tried to do some research about it, the closest answers I found are the question  and the . I tried this code:  \documentclass{beamer} \usepackage[frenchb]{babel} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc}  \usetheme{metropolis}  \definecolor{bluegreen}{rgb}{3, 166, 155} \definecolor{pitchblack}{rgb}{0, 0, 0} \definecolor{lightbeige}{rgb}{255, 251, 241} \definecolor{mediumgray}{rgb}{183, 183, 183}  \setbeamercolor{background canvas}{bg=pitchblack} \setbeamercolor{normal text}{fg=lightbeige} \setbeamercolor{headline}{bg=bluegreen, fg=mediumgray}  \begin{document} \begin{frame}     \frametitle{The title}     Lorem ipsum... \end{frame} \end{document}   I does everything I want, except that the title bar's background color is lightbeige:    What did I do wrong? Or what argument should I pass to \setbeamercolor to get it right? Thanks in advance :-). How do I calculate the date six months from the current date using the datetime Python module? I am using the datetime Python module.  I am looking to calculate the date 6 months from the current date. Could someone give me a little help doing this?  The reason I want to generate a date 6 months from the current date is to produce a review date.  If the user enters data into the system it will have a review date of 6 months from the date they entered the data. I noticed there is a square knob sticking out of my basement wall. Does anyone know what this is? I have come across this .  Has anyone tried anything similar or does it work for you? Can you tell if an email customer / user enters is correct &amp; exists? Drawing a triangle from medians Is it possible to draw a triangle, if the length of its medians $(m_1, m_2, m_3)$ are given only?  Someone asked me this question, but I can not see it. Is it really possible?  UPDATE  Apart from the algebraic solution given by JimmyK4542, can anyone give me a direct construction?  I mean, it should sound like:     Draw a line segment sufficiently long. Cut the length of $m_1$ from it. Then $\ldots$ Categorize my data into 3 groups Is it true that $aH = bH$ iff $ab^{-1} \in H$ Let $H$ be a subgroup of a group $G$. The I know that for $a,b\in G$ we have $aH = bH$ if and only if $a^{-1}b \in H$.  My question is if it is also true that $aH = bH$ if and only if $ab^{-1} \in H$? Does it matter where the $-1$ goes?  I am guessing that this is not true since we keep putting the $-1$ on the left element, but I am also thinking that it might be true because they look like the same.  If $G$ is abelian, then it doesn't matter, so a possible counter example would have to involve a non-abelian group. If $H$ is a normal subgroup, then it should also be true because then $a^{-1}bH = Ha^{-1}b$. Approximating $\arctan x$ for large $|x|$ Is it possible to prevent DHCP service on SBS 2003 from shutting down due to rogue DHCP server? Is natural projection a natural transformation? Align  elements side by side Seeking Python script for creating .mxd files? Determining the Orbits/Orbit Space of $O(3)$ on Real 3 by 3 Traceless Symmetric Matrices Let $O(n)$ be the set of $3 \times 3$ orthogonal matrices, and $S$ be the set of traceless real symmetric $3 \times 3$ matrices.  Now  $O(n)$ will act on $S$ by conjugation. Note that by the spectral theorem $ \forall A \in S$ in the orbit of $A$ there is a diagonal matrix $D=diag(\lambda_1, \lambda_2,\lambda_3)$ with $\lambda_1\le\lambda_2 \le\lambda_3 $.  Determine the all the orbits of the action &amp; the orbit space of the action.  What I've done:  Since all elements of $S$ are diagonlizable that the orbit of any element should be the orbit of its' diagonalized form because $\forall A \in S,   \exists P \in O(3)$ st  $PAP^t=D $ with $D$ a diagonal matrix and moreover for any other  $Q\in O(3)$ we get $QDQ^t=QPAP^tQ^t=(QP)A(QP)^t=RAR^t$.   And so would that imply the orbit space is just the set of traceless diagonal matrices as $tr(QDQ^t)=tr(DQ^tQ)=tr(Q)$ (cyclic property of trace) and $Q\in S$.  Any help is appreciated. Drupal out of the box comes with a page (which is the default homepage) that is assigned to the node path.  How do I avoid users reach it, and redirect them to the actual homepage of the site? Subject vs. Object marking for whoever? I know similar questions have been asked before, but I'm having trouble reconciling the following sentence, received in an email:     Can we ask whomever is your contact there to email us a job so we can check backward compatibility?   I could understand using objective case if the sentence were "can we ask (him) to email...", but I get tripped up considering "whomever" to be the subject of "(he) is your contact there". Does borrowing from my 401(k) make sense in my specific circumstance? I have been experiencing an issue that once I have a material created and looking the way I want it, I cannot seem to export it correctly with the textures mixed.  I don't know the correct words to use, but in blender I have the material created using the nodes editor, and I have placed two textures inside of there (one for the rock texture and one for moss, my object is a rock). When I export this model and import to Unity (the game engine I am using), I find a materials folder but the material inside of it does not have the textures assigned, it's just a plain white material?  How can I export my material from blender in a way that it will allow me to import it WITH TEXTURES into Unity? PDF upload silently fails I dunno since when, but upload of PDF images fails, this hurts especially on tex.SE.  Could this be an effect of an updated /etc/ImageMagick-6/policy.xml on the imgur server?  At least on my Ubuntu 18.04 LTS machine, this file was modified two weeks ago.