Let $𝑛$, $𝑎$ and $𝑏$ be positive integers such that $𝑎𝑏 = 𝑛^2$. If $gcd(𝑎, 𝑏) = 1$, prove that there exist positive integers $𝑐$ and $𝑑$ such that $𝑎 = 𝑐^2$ and $𝑏 = 𝑑^2$.    I'm a bit lost. I know that $1 = ac + bd$, and I can simplify $ab = n^2$ to $a = \frac{n^2}{b}$ and $b = \frac{n^2}{a}$, but I don't know where to go from there. I try substituting $a$ and $b$ but cannot figure out how to get that $a = c^2$ and $b = d^2$.
Problem:  If $\gcd(a, b) = 1$ and If $ab = x^2$ ,prove that $a$, $b$ must also be perfect squares; where $a$,$b$,$x$ are in the set of natural numbers  I've come to the conclusion that $a \ne b$ and $a \ne x$ and $b \ne x$ but I guess that won't really help me.. I understand that if the $\gcd$ between two numbers if $1$ then they obviously have no common divisors but where do I go from this point?  Any tips at tackling this would be great. It looks quite easy though I'm still trying to get my hand around these proofs! Any pointers in the right direction would be great.  Thank you in advance,
Prerequisites for Algebraic Geometry Is it possible to get into algebraic geometry by just knowing calculus and linear algebra or is this too far of a stretch?  If not could anyone give me a list of book/lecture notes recommendations in chronological order to dive into algebraic geometry with my state of knowledge?  Thanks New mobile chat doesn't follow the keyboard height The Japanese keyboard on iOS dynamically changes it's height to show suggestions and hides the input box of new mobile chat.        Additionally, third-party keyboards make the input box completely hidden. The chat seems to treat third-party keyboards as having no height.        iOS 9.2.1 / Mobile Safari  There is no problem in old (stable) design but it doesn't fix the input box to bottom! Word for a temporary project name In software development, and I imagine in many other fields, we often make up temporary nicknames for new projects or features. These are not meant to be the real names for the end product and are often funny, playful, or sarcastic in nature. These are also not intended to be "codenames" that hide the nature of the project or feature. Is there a word for such nicknames? We have various ways of invoking apex. Triggers, Inbound Webservices, @InvocableMethod,@RemoteAction and what not. Often we call some or other utility methods from these execution contexts. I have some generic utility method,that should behave differently from the context it is called?  I know about Trigger.isExecuting but it only tells if its running in trigger context or not.  I have a utility method that creates lead from Webservices, VF button click and via Lightning . Whenever we create a lead I want some special operations to be done via the source it was originated.(Its not just updating a simple field leadSource, I have some methods that creates some childs and update other records of other type) . I can achieve this by passing an extra parameter to utility method to tell about source, but i have then to edit hundred of other class that uses this utility method. If i could dynamically find out via which context this piece of code is running,then i would have to change only one class to satisfy my requirement.    Is there something in APEX that tells in what context the apex method was called? How do I install and manage GNOME Shell extensions? When using wish to replicate a spell, what school of magic does it count as? Issue in using longtable within table environment I have an issue making a long list with wrapped text. The table can be several page long. I use longtable and assign the width for each column but as soon as I put it in the \begin{table} \end{table}, it does not work. It clips the table. I want to make a label for the table so I put it in the table environment.  And is there a way to automatically adjust the width so that I don't need to assign the value manually?  Thank you.   Tung  \begin{table} \centering \begin{longtable}{|p{6.5cm}|p{2.5cm}|p{2.5cm}|p{2.5cm}|} \hline Name                                      &amp; Company     &amp;  Catalog Number   &amp;   Comments    \\ \hline Small DNA FRAG Extract Kit-100PR          &amp; VWR         &amp;   97060-558 &amp; \\ \hline Acrylamide 40\% solution 500 mL           &amp; VWR         &amp;   97064-522 &amp;  \\ \hline Bis-acrylamide 2\% (w/v) solution 500 mL  &amp; VWR         &amp;   97063-948 &amp;  \\ \hline GeneRuler 100 bp DNA Ladder, 100-1,000 bp &amp; Fermentas   &amp;   SM0241  &amp;    \\ \hline Mini Vertical PAGE System                 &amp; VWR         &amp;   89032-300  &amp;  \\ \hline \end{longtable} \label{tab:list} \end{table} How to prove that exponential grows faster than polynomial? In other words, how to prove:  For all real constants $a$ and $b$ such that $a &gt; 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$  I know the definition of limit but I feel that it's not enough to prove this theorem. What can an app do with the "USE ACCOUNTS ON THE DEVICE" permission? Can such an app freely read my emails/calendar/message/docs/etc.?  The "USE ACCOUNTS ON THE DEVICE" permission is also known for developers as .  Example of such apps: , . I just want to write \Sha without ruining everything I realize similar questions have been asked before but I am not satisfied with any of the answers I've seen. I would like to use the Cyrillic letter Ш to denote a particular group in mathematics. Most solutions involve using OT2 or T2A encoding; for example, at . However, this makes the other text in the document look bad.   For example, using the solution above, I get    while I would like to have     The first seems to be shaded strangely and the characters don't seem to align right.  Does anyone know how I can write the character Ш (in math mode) without changing the default font for the rest of the text? What does the notation like 8.6e-28 mean? What is the 'e' for? I have a problem with the interpretation of a test result in which the p-value is 8.6e-28. How should it be interpreted? What is the e for?  Specifically, I used White's Test for the homoscedasticity assumption of a linear regression:  White's general test statistic :  150.0535  Chi-sq( 9)  P-value =  8.6e-28 Looking for old book, SF, artificial world, different zones for different species that visitors "inhabit" I read this book at least ten years ago.  Any help identifying it would be appreciated.  Characters are teleported to an artificial world where each take over the mind of a different species.  I remember a moving sentient plant, a faun-like character, and a wasp-like character.  The planet is subdivided into different zones designed for each species.  One of the characters had long sharp fingernails with poison in them.  Thanks for any help, steve What are the most important things I need to do to encourage Google Sitelinks? My website used to have sitelinks and now it doesn't. It's very possible that it's due to changing the website to a sidebar design instead of having an "interstitial" type landing page which limited the number of choices, but I'm not sure.  Here is how sitelinks might look for a site:    What are some things that I can do to improve my chances of getting sitelinks? Is there any other way to do a clean install withot formatting /home? I think my Ubuntu is a bit broken from messing around with it.  Unfortunately during installation, I did not make a separate partition for /home. Now I need to re-install but want to keep my music and films.  Any ideas... Does anybody know how to easily undo a git rebase?  The only way that comes to mind is to go at it manually:   git checkout the commit parent to both of the branches  then create a temp branch from there cherry-pick all commits by hand replace the branch in which I rebased by the manually-created branch   In my current situation this is gonna work because I can easily spot commits from both branches (one was my stuff, the other was my colleague's stuff).  However my approach strikes me as suboptimal and error-prone (let's say I had just rebased with 2 of my own branches).  Any ideas?  Clarification: I'm talking about a rebase during which a bunch of commits were replayed. Not only one. The reference to entity "foo" must end with the ';' delimiter I am trying to pass a reference to a two-dimensional array to a function in C++. I know the size of both dimensions at compile time. Here is what I have right now:  const int board_width = 80; const int board_height = 80; void do_something(int[board_width][board_height]&amp; array);  //function prototype   But this doesn't work. I get this error from g++:  error: expected ‘,’ or ‘...’ before ‘*’ token   What does this error mean, and how can I fix it? Writing test code for wrapper class Number of vertices in the Klein bottle triangulation given in "Topology and geometry for physicists" by Nash and Sen I stumbled upon the following triangulation of the Klein Bottle in page 76 of the book Topology and geometry for physicists by Charles Nash and Siddhartha Sen.    Below the figure, it says "the triangulation $K$ consists of eight $0$-simplexes[...]".  I think this is not an eight-vertex triangulation, but rather a nine-vertex triangulation of the Klein bottle. I also know, thanks to the great comments under  question, that an $8$-vertex triangulation of the Klein bottle is a minimal one, hence the one portrayed in the book isn't minimal.  Am I correct? When is the past perfect exactly needed?