We have $m,n \in \mathbb{N}$ and $gcd(m,n) = 1.$   How to prove that $gcd(m+n,m-n) = 1$ or $2$?
I want to show that for $\gcd(a,b) = 1$ $a,b \in Z$   $\gcd(a+b, a-b) = 1$ or $\gcd(a+b, a-b) = 2$ holds.      I think the first step should look something like this:      $d = \gcd(a+b, a-b) = \gcd(2a, a-b)$   From here I tried to proceed with two cases. 1: $a-b$ is even, which leads to $\gcd(a+b, a-b) = 2$ 2: $a-b$ is odd, which leads to $\gcd(a+b, a-b) = 1$  My main problem I think is, that I do not know how I should include $\gcd(a,b) = 1$ in the proof.  Any help is appreciated. Thx in advance. Cherio Woltan
svmono template and `\mathcal` Why, with Springer's  svmono template, do the letters written with the \mathcal font change? It seems that \mathcal becomes \mathscr.  Edit: Look at this image:    With the svmono template \mathcal returns the first alphabet and not the (classical) latter.  Here the packages I use in my document:  \documentclass[envcountsame,envcountchap]{svmono}  \usepackage{subfiles} \usepackage[utf8]{inputenc}  \usepackage{mathptmx} \usepackage[bottom]{footmisc} \usepackage{microtype} \usepackage{stmaryrd}    \usepackage{amsmath} \usepackage{courier} \usepackage{mathrsfs}  \usepackage{amssymb}  \usepackage[all,cmtip]{xy} \usepackage{faktor} \usepackage{graphicx} \usepackage{mathdots} \usepackage[colorlinks={true},linkcolor={blue},citecolor={blue}]{hyperref} \usepackage[frame,width=15truecm,height=23.5truecm,center]{crop}   With this settings I have the above strange behavior of \mathcal. I have installed Ubuntu on my laptop and now I want to remove it and the Grub Loader. How I can do it without the Windows installation disk (I cannot use the Recovery Console). In The Matrix Revolutions, when Neo and Trinity are trying to get to the Machine City, when the Machines launch weapons against them, Neo is able to destroy some of the incoming weapons. This is in the real world at this point, not in the Matrix.  How does Neo have any powers against the machines outside of the Matrix? Addendum: I'm looking for in universe answers only.  I would think that would be obvious, but I'm adding this after an answer that is essentially, &quot;It's a movie...&quot; window-switching keyboard shortcuts interfere with emacs and cannot be changed in the main keyboard system preferences How to break multi digit number into separate digits in VHDL? I found the method .But I do not know how to perform this in VHDL. Let a fixed point number (12 downto -19) like 3456.478396 I need break this number entirely into separate numbers 3456.478396 --> 3,  4, 5,  6,  (dot) ,  4,  7,  8,  3,  9,  6...  into 11 integer type variables. How to perform this in VHDL?  signal example : sfixed(4 downto -4); --'example' contains 9.75 then it is storage as "01001.1100". --simply example ="10011100". --10011100 --&gt; integer 156   How to get back original 9.75 number from 156 to separate into single digits?  I USED,  library ieee; library ieee_proposed; use ieee.std_logic_1164.all; use ieee.numeric_std.all; use ieee_proposed.float_pkg.ALL; use ieee_proposed.fixed_pkg.ALL; use ieee_proposed.fixed_float_types.ALL; How to get string with pattern from std::regex in VC++ 2010 I try to resolve a problem, how to implement a xor gate with nand's gate.  A xor B = A'B + AB'  So, this becomes :  A xor B = A'B + AB' + AA' + BB' = A(A' + B') + B(A' + B') = (A + B)(A' + B') =   (with De Morgan apllied on the second term)  = (A + B)(AB)' = ..... ?  In this point i am blocked. If someone can help me, please. Thank you. Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$ Electric circuits in TeX, LaTeX, and Friends What are some ways to draw electric circuits in TeX systems?  I'm making this community wiki since I haven't found a question related to this.  I was thinking along the lines of . What do you think of that? Other examples are more than welcome, of course. Process Builder - Breaking Production Deployments? I have a Process Builder flow set up and I am running some tests around the InvocableMethod Apex code. The tests pass fine in my staging and production environments. Yet, when I try to deploy to production, I get a strange error.  CANNOT_EXECUTE_FLOW_TRIGGER, The record couldn’t be saved because it failed to trigger a flow.  Thankfully I received a somewhat more informative error email.  UPSERT FAILED ---  ERRORS :  (CANNOT_EXECUTE_FLOW_TRIGGER) The record couldn’t be saved because it failed to trigger a flow. common.exception.AssertionException: ORA-20070: Cannot read from apex_class and apex_trigger while running tests ---  for SFDC record with ID : null,  Unfortunately, the deployment has nothing to do with the flow and it is unclear why it is failing. What is the root cause of this error and how can I fix it? EDIT: I found out this is a . The other claimed workaround is to deploy via the Metadata API. UPDATE: Because I was subscribed to the Known Issue, I received the following confirmation this issue is resolved.  We are pleased to notify you that the following Known Issue has been resolved: Executing an InvocableMethod during an apex test run from a change set can cause Errors. Thank you for being patient while we worked to address this issue. Please feel free to reach out to Customer Support if you have further questions. Sincerely, Salesforce Success Team How do I write a correct micro-benchmark in Java? How do you write (and run) a correct micro-benchmark in Java?  I'm looking for some code samples and comments illustrating various things to think about.  Example: Should the benchmark measure time/iteration or iterations/time, and why?  Related: There are now errors when updating and I cannot install most software due to a corrupted  /etc/apt/sources.list   file. Is there a copy I can download to replace it?  The file would be for Ubuntu 12.04 (Final Beta) in the United States. Why would you put Identity Increment other than 1 in SQL? Tikz, Tree without overlaps I'm trying to get a nice tree without the overlaps, can anyone help me to improve my code?  \documentclass{standalone} \usepackage{amssymb} \usepackage{amsmath} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{dsfont} \usepackage{tikz} \usetikzlibrary{shapes} \usetikzlibrary{graphs}  \begin{document}       \begin{tikzpicture}[baseline,level/.style={level distance=20mm, sibling distance =35mm/#1},scale=1]         \node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}             child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                 child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                     child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                     child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                         child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         }                     }                 }                    child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                     child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                         child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             }                         }                     }                     child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                         child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             }                         }                     }                 }             };     \end{tikzpicture} \end{document} How can i make the grass bend when this cube hits it?  Here's the picture: I saw this movie a while back. I don't remember how it starts, but from where I remember watching it, it may have not been too far from the start. Apparently, Earth has been conquered by some alien empire or something and humanity are basically cavemen being used as slaves. From what I remember, the main antagonist alien is seen taking one of the slaves to a room where a machine is used on him to make him smart; this is apparently a big no-no so the main antagonist and his sidekick have to keep it quiet. The idea was that by making one of the slaves smart, it could increase the work output of the others. Apparently the antagonist is using the slaves on Earth to gather gold bullion  which is apparently still very valuable to the aliens but he's doing it illegally (not the slave labor; that's apparently ok). The protagonist who had the machine used on him becomes the intermediary between the slaves and the aliens and begins teaching the other slaves. I remember him teaching them mathematics and I remember him saying that it was universally accepted (I think he may have been referring to algebra or trigonometry). At some point he becomes aware that they are slaves and organizes a plan to free not only his fellow slaves but the Earth. Some escape and find an old museum which has old flight simulators, and shows that humanity wasn't always a slave race and that it may have been modern times in which Earth was invaded (i.e. no lasers or starships). I think they also find Fort Knox as there is a huge collection of gold bricks and weapons. The protagonist comes up with a plan to destroy some sort of teleporter the aliens use to transport stuff to and from Earth, however that alone wont be enough as they could fight back, which will be bad for humanity, which is still undeveloped. So the plan also includes using the teleporter before destroying it to to transport and detonate a nuclear warhead on the alien home world which apparently cripples any attempt to immediately send a fleet but also as a show of strength. After executing their plan, the main antagonist is captured by the humans and as an insult is locked in a cage surrounded by the gold he sought. If I recall correctly the place is set to detonate as to destroy the precocious gold and that will be used against the aliens should they mobilize a fleet in the future (i.e. attack us and we destroy Earth's supply of gold). The antagonist's side kick for some reason I remember as a big black man from whom the antagonist had removed his hand at some point and also apparently the antagonist was cruel to him. After the antagonist is locked up, he is trying to appeal to his sidekick's loyalty, however the sidekick takes the humans' side, I'm also getting an image that apart from treating him kindly the sidekick's missing hand was restored. Number of solutions of $x_1+2x_2+\cdots+kx_k=n$? Let $F$ be a field, let $\omega$ be a primitive $n$th root of unity in an algebraic closure of $F$. If $a$ in $F$ is not an $m$th power in $F(\omega)$ for any $m\gt 1$ that divides $n$, how to show that $x^n -a$ is irreducible over $F$? Mandatory excerpt for custom post type Equivalence of the definition of the Subbasis of a Topology I came to know about 2 definitions of a subbasis of a topology. 1. A collection of subsets of X viz. S is said to be a subbasis of a topology T on X if S covers X and the finite intersections of elements of the subbasis form the basis elements (H.L.Royden). 2. If we consider the all the topologies on X containing S, then their intersection is also a topology on X and it is the smallest topology containing S. Then S is said to be a subbasis of the smallest topology, and the topology is said to be generated by S.  Now I want to prove their equivalency, i.e. S is a subbasis of a topology T iff T is generated by S. I have tried in the way: First let T is generated by S. Then by 2nd definition T is a topology containing S. Let B be the set of all finite intersections of elements of S.Then S is a subset of B and B is a subset of T (as T is a topology). LET T1 be the collection of possible unions of elements of B, we will show that T = T1. After this I am not able to proceed. Also I am unable to sketch the converse part. If something is wrong, please let me know, and also I hope to get help. Thank You.