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Letter O with vertical line: operator null Let $D$ be a , and let $a,b,x \in D \setminus \{0\}$. Then is it true that $\gcd (ax,bx)=x \cdot \gcd (a,b)$ ?    Let $c=\gcd (a,b)$ and $d=\gcd(ax,bx)$, then as $cx|ax$ and $cx|bx$ so $cx|d$. We would be done if we could show $d|cx$, but I am unable to show that. Please help me to solve this problem. Resizing Mac OS partition - macOS High Sierra Why are so many forces explainable using inverse squares when space is three dimensional? What does $\lim\limits_{x \to \infty} f(x) = 1$ say about $\lim\limits_{x \to \infty} f'(x)$? Given that $f$ is differentiable, what does $\lim\limits_{x \to \infty} f(x) = 1$ say about $\lim\limits_{x \to \infty} f^\prime(x)$ ? Intuitively I feel that it's $0$.  I attempted to solve this by trying to evaluate $$\lim_{x \to \infty} \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$ by trying to interchange the limits after showing that $\frac{f(x + h) - f(x)}{h}$ converges uniformly as $x \to \infty$. But I couldn't proceed further.  Trying to work backwards, as a specific example, $f(x) = \arctan(x)$ came to my mind. It's derivative certainly goes to $0$ as $x \to \infty$. Doesn't this show that $\frac{f(x + h) - f(x)}{h}$ converges uniformly to $0$ as $x \to \infty$ ?  I'm totally confused! I would really appreciate if anyone told me what I am doing wrong. Inserting a term into a custom taxonomy Which is the correct sentence, if there is a correct one?     I use to be a hitman.        I used to be a hitman.   I've read the second sentence recently in a book, but I was sure it should be I use to be a hitman. What does kill 0 do actually? In the man page, it says:  kill [ -s signal | -p ] [ -a ] [ -- ] pid ...  pid... Specify the list of processes that kill should signal.  Each pid can be one of five things:           0      All processes in the current process group are signaled   And I tried like this in bash:  $ man kill &amp; [1] 15247 $ [1]+  Stopped                 man kill $ kill 0 $ ps 15247 pts/41   00:00:00 man   Here 0 is used as pid. As I understood, kill 0 will kill all processes in the current process, which includes pid15247. However, it didn't do anything in this example. Does anyone have ideas about how to use it? understanding "mount" as a concept in the OS I want to understand what mounting is. It is used in different contexts and situations and I can't find resources which:   Describe the mount concept Explain the actions taken by the computer/OS/utility when a mount is performed How and in which situations mount is used Which features in the Linux mount command are of frequent use and some examples   ( I hear mount applied to diverse entities directories, flash drives, network card, etc ) Pugh, "Mathematical Analysis", exercise 2.17:     Assume $f : M \to N$ is a map from one metric space to another which satisfies the following condition: for every convergent sequence $(a_n) \subset M$, the sequence $(f(a_n)) \subset N$ converges. Prove that $f$ is continuous.   (I know that if $f$ commutes with limits then the answer is affirmative, but I can't prove it with that hypothesis). Is "mens" a valid word? I've been living in Ireland for almost a year now and I start noticing they use the word "mens" a lot. I can see it used in:   Shops, to denote the area where you can find men's clothes In sport, when they talk about "mens team".    My guess is just that they are lazy about the use of quotes, so that mens should actually be men's. However, there may be some rule I'm not aware of. So, is "mens" only limited to Irish English? When I'm allowed to use it? How to use the symbol \angle at the section title Dynamic Paint not showing in rendered image Find the angle ADE of the given triangle. Given :  Angle ABE = 50, Angle DAC = 20, Angle BAD = 60, Angle ACB = 20 and Side CD = Side DA.Unit of angle values given is degree.  Find : Angle ADE.    I'm stuck and I need help here. Immutability, Association and typing What's the best way to convert a string to an enumeration value in C#? I have an HTML select tag containing the values of an enumeration. When the page is posted, I want to pick up the value (which will be in the form of a string) and convert it to the corresponding enumeration value. In an ideal world, I could do something like this: StatusEnum MyStatus = StatusEnum.Parse(&quot;Active&quot;);  but that isn't a valid code. Disable automatic activation of Gnome Shell activities on mouse over Show that the series $\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$ converges This came to my mind after I learned the following from :  where T : struct   We, C# developers, all know the basics of C#. I mean declarations, conditionals, loops, operators, etc.  Some of us even mastered the stuff like , , , , ...  But what are the most hidden features or tricks of C# that even C# fans, addicts, experts barely know?  Here are the revealed features so far:    Keywords    by  var by  using() statement by  readonly by  as by  as / is by  as / is (improved) by  default by  global:: by  using() blocks by  volatile by  extern alias by    Attributes    by   by   by   and  by   by   by   by    Syntax    (coalesce nulls) operator by  Number flaggings by  where T:new by  Implicit generics by  One-parameter lambdas by  Auto properties by  Namespace aliases by  Verbatim string literals with @ by  enum values by  @variablenames by  event operators by  Format string brackets by  Property accessor accessibility modifiers by  Conditional (ternary) operator (?:) by  checked and unchecked operators by  implicit and explicit operators by    Language Features   Nullable types by  Anonymous types by  __makeref __reftype __refvalue by  Object initializers by  Format strings by  Extension Methods by  partial methods by  Preprocessor directives by  DEBUG pre-processor directive by  Operator overloading by  Type inferrence by  Boolean operators  by  Pass value-type variable as interface without boxing by  Programmatically determine declared variable type by  Static Constructors by  Easier-on-the-eyes / condensed ORM-mapping using LINQ by  __arglist by    Visual Studio Features   Select block of text in editor by  Snippets by      Framework   TransactionScope by  DependantTransaction by  Nullable&lt;T&gt; by  Mutex by  System.IO.Path by  WeakReference by    Methods and Properties   String.IsNullOrEmpty() method by  List.ForEach() method by  BeginInvoke(), EndInvoke() methods by  Nullable&lt;T&gt;.HasValue and Nullable&lt;T&gt;.Value properties by  GetValueOrDefault method by    Tips &amp; Tricks   Nice method for event handlers by  Uppercase comparisons by  Access anonymous types without reflection by  A quick way to lazily instantiate collection properties by  JavaScript-like anonymous inline-functions by    Other   netmodules by     by     by Calculation of $\lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$