I have convinced myself that this true, however I'm at a loss of where I should start with this proof. Looking at a similar proof with 3 instead of 9, I saw the use of the basis representation theorem, but I'm not 100% comfortable with that, so it was hard for me to follow. Is there a way to do this with mod?
Suppose we have a natural number $N$ with decimal representation $A_kA_{k-1}\ldots A_0$. How do I prove that if the $\sum\limits_{i=0}^kA_i$ is divisible by $9$ then $N$ is divisible by $9$ too?
How to install the latest versions of NodeJS and NPM? I have a layout similar to this:  &lt;div id="..."&gt;&lt;img src="..."&gt;&lt;/div&gt;   and would like to use a jQuery selector to select the child img inside the div on click.  To get the div, I've got this selector:  $(this)   How can I get the child img using a selector? Good Number Theory books to start with? I'm wanting to parse a string into a nullable int in C#.  ie. I want to get back either the int value of the string or null if it can't be parsed.  I was kind of hoping that this would work  int? val = stringVal as int?;   But that won't work, so the way I'm doing it now is I've written this extension method  public static int? ParseNullableInt(this string value) {     if (value == null || value.Trim() == string.Empty)     {         return null;     }     else     {         try         {             return int.Parse(value);         }         catch         {             return null;         }     } }      Is there a better way of doing this?  EDIT:  Thanks for the TryParse suggestions, I did know about that, but it worked out about the same.  I'm more interested in knowing if there is a built-in framework method that will parse directly into a nullable int? Given a compact Hausdorff space $X$, if $a \in X$ is a $G_\delta$ singleton (i.e., $\{ a \}$ is a $G_\delta$ set), then there is a countable local base at $a$.   The point $a$ can be written as countable intersection of open sets let's call $U_{n}$.  How can I use the compactness? What is the next step for proof? Could you give me hint please? I have a problem with using "too" in a sentence. In what situation would you use "too" instead of "so" or "very"? Please include an example to illustrate the difference. Lasso regression feature selection I have been reading many articles on LASSO regression, and everyone claims that LASSO address multicollinearity showing contour plots of cost function touching the corner of the diamond(x1+x2). In the first place, I didn't understand how does this phenomenon translate to addressing multicollinearity when there are many groups of correlated variables in the data.  Verbatim from one of the research paper: "• If there are grouped variables (highly correlated between each other) LASSO tends to select one variable from each group ignoring the others". C++ template typedef Prove $|\arctan x - \arctan y|\leq |x-y|$ Class not found loading JDBC org.postgresql.Driver Are there any examples of Humans and Aliens producing offspring in Star Wars? The title is pretty self explanatory, although to expand on the question further I am looking for any examples of inter-species breeding in the Star Wars universe, not just inter-species (ahem) "relations". I'm working on a python script that passes file locations to an scp subprocess. That's all fine, but I'm in a situation where I may end up concatenating a path with a filename such that there's a double '/ in the path. I know that bash doesn't care if you have multiple file separators, but I'm wondering how exactly that is rectified. Is it bash that strips extra /s or does it really not matter ever?  I ask because it will save me several lines of code to check for extra /s while concatenating. I know it's not a big deal, but I'm curious as well. I have a bash script that has the line cd //usr (instead of cd /usr), which seems to imply there might be a significance to using multiple /s in a path Supersymmetry in Quantum Field Theory I have an idea of supersymmetry in quantum mechanics, can you suggest a book on "supersymmetry in quantum field theory", which has sufficient mathematical rigour like "Peskin and Schroeder" What is the difference between the words which and that?  For example I have the following sentences:   “I have a car which is blue.” “I have a car that is blue.”   Are there any rules specifying usage of which and that? Is there a way to set transparency of windows in GNOME? Sometimes ago I used KDE 3.5 I think, and really liked the transparency option for windows, which gave all my text editor a really killer look and feel, really nice/ethereal environment for coding.  I haven't managed to find such a possibility with GNOME.  Any idea if it is possible and how? Sponsor Gallery, Page, or Something Like That? Is save data from previous Dragon Age games used in Inquisition? Does Dragon Age Inquisition make use of save data from previous games in the series?  If so, what benefits does this provide? Why does Ubuntu create a file ending in ~ for every file I create with a text editor like GNU Emacs? How did Jaqen H'ghar get caught? Following a recent 9gag post:    Did Jaqen H'ghar end up in that cage because he knew that he would meet and "recruit" Arya, or was it some strange coincidence? Is it described somewhere in the books, or left unexplained? Rotating symbols per attribute value in QGIS? I am trying to apply a rule to rotate symbols based on the value of an attribute.  I tried to copy , which is essentially my question, however it does not work for me.   I have a layer with two attributes, the slope and aspect.  I want to colour the symbols by the slope (this part works) and rotate by the aspect.  The steps I take are:   Layer Properties -> Style -> Graduated -> Symbol -> Change Select Simple Marker (arrow) and press the button to the right of angle -> field type (double) -> Aspect-sel    Applying these commands changes the symbol to an arrow, and I can easily apply the colour gradient, but the symbols do not rotate. The attribute Aspect-sel varies from 2.87 to 334.23.    ETA: I was using Version 2.8.3 - on suggestion I updated to 2.14.10, which solved the problem.