$$\newcommand{\gcd}{\text{gcd}}$$     Prove: if $d=\gcd(m,n)$ so $\gcd\left(\frac{m}{d},\frac{n}{d}\right)=1$   Intuitively it is obvious, but I am having a hard time to formalize the proof, what I have came to this:  $d=\gcd(m,n)$ so $d|m$ and $d|n$ therefore $m=dx$ and $n=dy$ now if $\gcd\left(\frac{m}{d},\frac{n}{d}\right)\neq 1$ that mean that $m$ and $n$ have a common factor, after the division in $d$ which is the greatest common divisor is contradiction.
How would you go about proving that $$\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$$  for any two integers $a$ and $b$?  Intuitively it is true because when you divide $a$ and $b$ by $\gcd(a,b)$ you cancel out any common factors between them resulting in them becoming coprime. However, how would you prove this rigorously and mathematically?
I'm new to Ubuntu. I tried installing it with Windows 8.1, but failed I lost all my files.  I have successfully formatted my PC and installed Ubuntu, but how do I install codecs to play audio and videos files ? How to get the Runtime-Type of an Object dynamically (for Primitive Data Types and SObjects) Is it possible to get the Type of an Object (it's not an SObject)?  String s = 'test'; doSomething(s);  public static void doSomething(o Object) {     // need to find out if o is a string or an integer     type = IsThereAMethodToGetTheTypeOf(o) // ?     if(type='String') {         // do something for strings here ....     } else {         // do something else ....     } }   I think the Type class can't help me, because it has no way to get the type of an existing object instance, right? Any there any other way?  Since there is most likely no perfect solution possible within the current platform limitations, I posted an Idea on IdeaExchange here:   For SObject I have already this  public static string getType(SObject obj) {     if(obj==null) return '';     return obj.getSObjectType().getDescribe().getName()+''; } better Curve/Text to Mesh generation? Using $\text{n}^{\text{th}}$ root of unity  $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$  Prove that   $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$ Prove that the normalisation of $A=k[X,Y]/(Y^2-X^2-X^3)$ is $k[t]$ where $t=Y/X$ (Reid, Exercise 4.5) This is a problem about finding the normalisation of a quotient polynomial ring. So I have to find the integral closure of the ring in its field of fractions. The problem statement is as follows:     Let $A=k[X,Y]/(Y^2-X^2-X^3)$. Prove that the normalisation of $A$ is $k[t]$ where $t=Y/X$.   Can I do this by showing that the field of fractions $\text{Frac} A$ of $A$ is equal to $k[t]$, and subsequently showing that the field of fractions is normal? (This could be done by showing that $k[t]$ is a UFD?)  I am lost at calculating/determining $\text{Frac}A$, or similarly proving that $k[t]=\text{Frac}A$. Also, how do I show that it is normal?  I hope you can help! It seems that there are many ways to prove the Sylow theorems. I'd like to see a collection of them. Please write down or share links to any you know. How to install the latest stable version of Chromium? When electric dipole placed in non uniform electric field, what is the approach to calculate torque acting on it? Can it be zero? How to add a new kernel to grub2 How can we add a kernel to grub2? i knew the stuff we could do with the former versions, but it seems to have changed. What files should we edit? Why do I get TypeError: can't multiply sequence by non-int of type 'float'? Quick question about ideals of field $F$ in $F[x]$ are principal. Just got a question about ideals of $F[x]$ being principal with $F$ being a field.  The proof basically starts assuming $g(x) \in N \subset F[x]$ where $N$ is an ideal and $g(x)$ is a minimal degree element. The rest is just deducing the what the division algorithm says.  But my question is that shouldn't we let $g(x)$ be arbitrary? Why are we choosing the minimal element? Hi can you help me solve this exercise? Thanks. Let $f: [0;+\infty) \to \mathbb{R}$ be nonnegative and continuous function. Suppose $\forall a &gt; 0$  $\sum_{n=1}^{\infty} f(na)$ is convergent. Prove that $\int_{0}^{\infty}f(x) dx$ is convergent. I tried to solve it by using the Riemann sum, but for fixed a it doesn't work. I have no other ideas. Split polygons into equal areas within each region I've a set of regions within a city. I'd like to split each region into five subregions and then get the centroid of each one of them.  I know how to make a vector grid, but the problem I have is that the generated points are not taking into account the regions, they're only related to the general map.  I have this:    And I want to split each region in five equal size areas. Something like this:    I'd like to build a visualization similar to this one: . In order to accomplish that, I need to split each region, because the original ones are pretty big. Which one is correct to say: "It's me" or "It's I"? Most of the recipes I'm using (mainly Modernist; stocks, caramelized vegetables, confit) suggest cooking at 15psi. My pressure cooker has two settings 12psi and 7psi. I've always had success just using 12psi in recipes where it says 15psi. However recently the 12psi mode has broken on my PC. I can now only cook at 7psi. How can I convert my recipes to use this lower pressure?  For example I would cook a chicken stock at 12psi for 90mins, I presume at 7psi the cooking time would need to be longer but by how much? How to make two people play at the same time? Installing TTF fonts in LaTeX One of the annoying aspects of LaTeX is the limited number of fonts that come by default, and the pain involved in making new fonts 'LaTeX' ready. I have a collection of truetype fonts that I'd like to prepare for use, and I definitely want to make sure I have vector versions of these fonts (i.e not type 3/bitmapped versions). Is there a relatively painless way to do this ? Is there an easy way to remove overlapping faces? In the example remove doubles doesn't work  You might be able to do the following: A to select all faces. X &gt; Only Faces. F to create faces, but after deleting the faces I'm left with something like this.  If I press A to select all and then press F to create faces then there are still overlapping. Blend file How to get vertex normals from a shape-key in Python? I'm trying to create an exporter for a format, which includes shape key support. This format requires both vertex coordinates and normals to be specified. This should be logical, because the vertex selected on the image below won't have the same normals:     Comparing it to this one:    But I actually haven't found any way to export shape key normals. Both Bmesh Vertex and ShapeKey Vertex only store the coordinates.  Is there any way to get them? Reputation changes not visible in the achievement box (again) Just like , which was marked as . But today, it's not working again. I'm still notified of new badges:      Update: it looks like new events are now updated; reputation events from earlier this UTC afternoon are still missing: