Prove $ n \equiv s(n)\ ($mod$\ 3)$ using the fact that $\ [10^n] = [1]$. Prove $ n \equiv s(n)\ (mod\ 3)$ using the fact that $\ [10^n] = [1]$. Let $n = (a_k \times 10^k) + (a_{k-1} \times 10^{k-1}) + \cdots +(a_1 \times 10^1)+ (a_0 \times 10^0)$ and $s(n)=(a_k + a_{k-1}+ \cdots +a_1+a_0)$.   When trying to solve this question, I combined the information given and found that $$n-s(n) = (a_k \times 10^k) + (a_{k-1} \times 10^{k-1}) + \cdots +(a_1 \times 10^1)+ (a_0 \times 10^0)-(a_k + a_{k-1}+ \cdots +a_1+a_0)$$ $$=(a_k \times 10^k-a_k ) + (a_{k-1} \times 10^{k-1}-a_{k-1}) + \cdots +(a_1 \times 10^1-a_1)+ (a_0 \times 10^0-a_0)$$  $$=a_k (10^k-1) + a_{k-1} (10^{k-1}-1) + \cdots +a_1 (10^1-1)+ a_0 (1-1)$$ $$=a_k (10^k-1) + a_{k-1} (10^{k-1}-1) + \cdots +a_1 (9)$$ I'm not sure where to go from here, I thought maybe I could deduce that since, for the case of $\ [10^n] = [1]$ -- since that means that $10^n \equiv 1 (mod \ 3)$ -- I could say that since there exists an integer, call it p, such that $10^n - 1=3p$, and then put that "statement" in the parentheses in the last "=" line that I had above. I have a feeling it doesn't make sense though, and it would be incorrect.
Why $9$ & $11$ are special in divisibility tests using decimal digit sums? (casting out nines & elevens) I don't know if this is a well-known fact, but I have observed that every number, no matter how large, that is equally divided by $9$, will equal $9$ if you add all the numbers it is made from until there is $1$ digit. A quick example of what I mean:  $9*99 = 891$ $8+9+1 = 18$ $1+8 = 9$  This works even with really long numbers like $4376331$ Why is that? This doesn't work with any other number. Similarly for $11$ and alternating digits sums.
java.lang.NullPointerException at org.glassfish.hk2.classmodel.reflect.impl.TypesImpl.getType Double dual mappings $\newcommand{\Hom}{\operatorname{Hom}}$Let $U,V$ be vector spaces. Denote the dual space of $U$ with $U^\intercal$ and the dual mapping of a linear map $\Phi$ be $\Phi^\intercal$. Define the double dual embedding on $U$ to be $\iota_U:U \to U^{\intercal\intercal}$. ($u \in U, f \in U^\intercal$) $$ \iota_U(u)(f) = f(u) $$     If $\Phi \in \Hom(V,U)$, then $\Phi^{\intercal\intercal} \circ \iota_V = \iota_U \circ \Phi:V \to U^{\intercal\intercal}$   Proof: Let $v \in V, f \in U^\intercal$. Then $$ \iota_U \circ \Phi(v)(f) = f(\Phi(v)) $$ but how do I show that $$ \Phi^{\intercal\intercal} \circ \iota_V(v)(f) = f(\Phi(v)) $$ ? How to Preload Tabs/Set Multiple Websites as Default in Firefox, IE, etc Any website can set as the homepage of Firefox which loads spontaneously. Is this for multiple tabs? Then BBC Health, CNN, Yahoo Health, msn.com would all extemporarily materialise (in four separate tabs); I then wouldn't have to open each website manually. Irreducibility check for polynomials not satisfying Eisenstein Criterion. My Question is to check Irreducibility for polynomials not satisfying Eisenstein Criterion.  As an Illustration, to check whether $x^{p-1}+.....+x+1$ for p a prime is irreducible or not, we replaced $x$ by $x+1$ and by using Eisenstein's Criterion for the resulting polynomial we conclude that resulting polynomial is irreducible and so is the original polynomial.  In general, for a given irreducible polynomial $f(x)$ with coefficients in a known U.F.D, is there some element $a$ such that we can apply Eisenstein's criterion to $f(x+a)$?  I am sure there would be no general structure for this but I expect there to be at least some special cases.   Any Reference/suggestion would be appreciated.  Thank You. Now that Unity is gone, the usual way of setting a Compose Key (via Keyboard Preferences > Layout) no longer works (because in Ubuntu 18.04, Devices > Keyboard has no Layout tab).  How do we set a Compose Key in Ubuntu 18.04 Bionic Beaver? Highlight a word with jQuery I'm trying to align a legend inside a table caption, so that the colons are vertically aligned. I can do it with the align ambient outside of the caption, but if I use the same code inside the caption the pdf won't compile (I use LuaTeX to compile). I'm using the package caption.  EDIT: i want something like this:    for which i've written this code:  \begin{align} \begin{split} a : {} &amp; the~first~letter~of~the~alphabet \\ b : {} &amp; the~second~one \\ and : {} &amp; so~on \end{split} \end{align}   There might be a more elegant way... but i don't know it.  Anyway, I can't do it inside the caption, that's why i'm stuck. I have found a strange behavior in TikZ. When using the \draw ... to ... syntax and trying to place nodes along the line with the node[...]{...} syntax, the nodes do not turn up at the expected location.  \documentclass{standalone} \usepackage{tikz} \begin{document} \begin{tikzpicture}   \draw(0,0) to[out=60,in=120] (1,0) node[midway]{x}; \end{tikzpicture} \end{document}     How is this done properly? Underworld Connections: Tapping the Land for mana and ability? Prime ideal in a polynomial ring over an integrally closed domain Integral Representation of the Zeta Function: $\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$ How does one get from this $$\zeta(s)=\sum_{k=1}^{\infty}\frac1{k^s}$$ to the integral representation $$\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$ of the Riemann Zeta function?  I can see that it can be rewritten as  $$\Gamma(s)\zeta(s)=\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$ and the Gamma function as an integral yields  $$\zeta(s)\int_{0}^\infty \frac{x^{s-1}}{e^x}dx=\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$  But this approach does not work as the right integral does not converge.  So how does one go from the summation to the integral representation? Example: I type man ls, than I want to get man only.   By using !! I can get man ls but how do I get man? Visit the UK with Schengen tourist short-stay visa I'm Russian, and I have touristic  Schengen short stay visa. I want to know, can I stay for some period in UK without getting UK visa, or it is not possible (I heard about 36 hours free of visa staying or something like that) table of contents no word separation Is it possible to have in the toc no word separation but have a word separation in the chaptername:  For example:   Contents  1. max Mustermann   Chapter  Max Muster- mann Switching between windows with scroll wheel on Ubuntu Dock I read a story once about a man who escapes from his pursuers by doing an uncontrolled jump into hyperspace. If I'm not mixing my stories up he was blind but could see through the eyes of other sighted people/animals nearby. He was able to see through his pet but it dies and he thinks he's finished - he's blind and lost in hyperspace but there's a rat on board which he eventually catches by turning off gravity and then he can see through the rat's eyes. He works out how hyperspace operates and can hop back to where he came from.   Can anyone remember the name of the story and the author? I enjoyed it when I was a kid and would like to read it again. What equipment and settings do I use for panning shots of moving objects? How do you go about getting great panning shots of things like cars and bicycles so that the background is nice and blurred but the target is sharp as can be. I have tried on many occasion to do panning shots and I just can't get the setup right.  What lens setup, shutter speed would be a good place to start? When should static_cast, dynamic_cast, const_cast and reinterpret_cast be used? What are the proper uses of:   static_cast dynamic_cast const_cast reinterpret_cast C-style cast (type)value Function-style cast type(value)   How does one decide which to use in which specific cases? At the moment, I have created a code that retrieves data from the database and displays it. However, for some reason, I cannot see the files I want to retrieve on my page. My goal is that the data gets retrieved from the database, and is displayed on the webpage. I do not need to make a connection with the database since Wordpress does that automatically.  My code:  &lt;?php  global $wpdb; // this adds the prefix which is set by the user upon instillation of wordpress $table_name = $wpdb-&gt;prefix . "wpex_programma"; // this will get the data from your table $retrieve_data = $wpdb-&gt;get_results( "SELECT * FROM wpex_programma; " ); print_r($retrieve_data); ?&gt; &lt;ul&gt; &lt;?php foreach ($retrieve_data as $retrieved_data){ ?&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;column_name;?&gt;&lt;/li&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;another_column_name;?&gt;&lt;/li&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;as_many_columns_as_you_have;?&gt;&lt;/li&gt; &lt;?php  } ?&gt; &lt;/ul&gt;   My question: the data is not shown and I believe it is not retrieved. How can I fix that? The table name is correct, and I have data in my database tabel. The print_r($retrieve_data)returns the following: Array ( [0] =&gt; stdClass Object ( [id] =&gt; 35 [naam] =&gt; /public_html/wp/wp-content/plugins/AbonneerProgrammas/FilesUpload/Country Kickin 1.mp3 ) )  My query log:    My database structure:   New:  &lt;?php  global $wpdb; // this adds the prefix which is set by the user upon instillation of wordpress $table_name = $wpdb-&gt;prefix . "wpex_programma"; // this will get the data from your table $retrieve_data = $wpdb-&gt;get_results( "SELECT * FROM wpex_programma; " ); print_r($retrieve_data); ?&gt; &lt;ul&gt; &lt;?php foreach ($retrieve_data as $retrieved_data){ ?&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;id;?&gt;&lt;/li&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;naam;?&gt;&lt;/li&gt; &lt;?php  } ?&gt; &lt;/ul&gt; Watched screen recording but it's not saved to the desktop I recently recorded a session with my dad. I stopped the recording after the session ended (with the setting set to save to Desktop), clicked the icon that appeared on the lower right corner to open the file, and played back a few minutes to make sure it worked. I then clicked Share to send it via AirDrop to my computer, which did not work; the status remained &quot;cancelled&quot; despite my efforts. I assumed I'd be able to access it through my phone by sending it through a third app, so I pressed cancel on the AirDrop pop-up window. The entire screen recording closed and (apparently) vanished forever after the action. I'm extremely frustrated because I can't seem to locate this video.