Question:   Which of the numbers $0,1,2, ..., 60$ is $10 ^ {60 \cdot 2 ^ {100} +2}$ congruent with modulo $61$?  How should you think about this? I know that you should think think that you should break it out a little more like $x ^ {a * b} = (x ^ b) ^ a$ use modolo in the parentheses, so as to always get smaller parts of the number as far as possible, etc, etc. .   But be a little thoughtful if that's how you should do it. also have the idea that you can put all numbers in a list, up to mod 60, and see which numbers are congruent.   But since 61 is the quite high number, it feels excluded.   Also $10 ^ {60 * 2 ^ {100} +2}$ is also a large number.   So need a little push
I've read about Fermat's little theorem and generally how congruence works. But I can't figure out how to work out these two:   $13^{100} \bmod 7$ $7^{100} \bmod 13$   I've also heard of this formula:  $$a \equiv b\pmod n \Rightarrow a^k \equiv b^k \pmod n $$  But I don't see how exactly to use that here, because from $13^1 \bmod 7$ I get 6, and $13^2 \bmod 7$ is 1. I'm unclear as to which one to raise to the kth power here (I'm assuming k = 100?)  Any hints or pointers in the right direction would be great.
Problem storing arrays with update_user_meta I am writing a function that adds an ID number to an array and puts the array in the usermeta. $_GET['auction'] is the post_id.  Below is the function:  $reminders = get_user_meta( $current_user-&gt;ID, "reminders" ); print_r( $reminders ); if( in_array( $_GET['auction'], $reminders ) ) {     echo "Failed: Auction already in list"; } else {     array_push( $reminders, intval( $_GET['auction'] ) );     if ( update_user_meta( $current_user-&gt;ID, "reminders", $reminders ) ) {         echo "Success";     } else {         echo "Failed: Could not update user meta";     } } print_r( $reminders );   Here is the output after adding one auction:  Array ( )  Success Array ( [0] =&gt; 7 )    Here is the output after adding two auctions:  Array ( [0] =&gt; Array ( [0] =&gt; 7 ) )  Success Array ( [0] =&gt; Array ( [0] =&gt; 7 ) [1] =&gt; 73 )   And here is the output after adding three auctions:  Array ( [0] =&gt; Array ( [0] =&gt; Array ( [0] =&gt; 7 ) [1] =&gt; 73 ) )  Success Array ( [0] =&gt; Array ( [0] =&gt; Array ( [0] =&gt; 7 ) [1] =&gt; 73 ) [1] =&gt; 0 )   Notice that adding a new element to the array using array_push works fine. But when the array is stored in the usermeta then retrieved again, the last element in the array has been put into an array of its own, creating an infinitely dimensional array. I'd prefer to keep this array one dimensional.  Is there a way I can run update_user_meta and get_user_meta without it changing the structure of my array? Persist variables between page loads Finding domain/range of a specific shader node input/output, inside the Blender UI? Is there anywhere within the Blender UI that displays the dimensionality &amp; bounding sets (e.g., all floats, floats in [0,1], floats in [0,255], etc) of arbitrary shader node inputs &amp; outputs?  I consistently find myself digging through stackexchange/the Blender docs/experimenting with less-than-equal-to to try to figure out what I need to divide by to normalize shader nodes' input/output values to within a specific range, or to center them at 0, etc. This seems like functionality that Blender's very rich UI might already have built into it, though, so I am wondering if I am just overlooking it somewhere.  As an example, I can do something like this, varying the threshold between 0, 0.5, and 1, and observing that in rendered view my sphere switches from all black to mixed black/white to all white, to convince myself that Blender's (Perlin) Noise Texture's Fac output naturally outputs floats in [0,1]:    But it seems odd that I should have to experiment, or search through docs or stackexchanges, to discover something like this. Is this information maybe already neatly displayed somewhere inside the Blender UI, and I just don't know where to look? Why can't to-infinitive be used as subject in "Not to learn French is my biggest regret."? E.g. 1 &quot;Not learning French is my biggest regret.&quot; E.g. 2 &quot;Not to learn French is my biggest regret.&quot; I know that e.g. 1 is correct and e.g. 2 is wrong, but what is the grammar rule making e.g. 2 incorrect? (At first I thought the rule was that you shouldn't use to-infinitive as the subject, but then I realized it couldn't be the rule as to-infinitive is sometimes used as the subject, as in &quot;To tell the truth is always right.&quot;) XPath select node with namespace How can you find out which process is listening on a TCP or UDP port on Windows? find() function in python2.7.5 Can polymorphing monsters spam their ability to effectively give themselves a massive health pool? Some creatures have an ability that lets them polymorph, such as the Couatl's Change Shape action. I am wondering if this ability could be used again whenever the form is dropped to give the creature a massive effective health pool.  For example, if the Couatl used their action to Change Shape and become a Veteran, then was hit for enough damage to kill the polymorphed form, could it then re-use the Change Shape ability to transform into a fully healed Veteran once more? I've seen the full proof of the Triangle Inequality  \begin{equation*} |x+y|\le|x|+|y|.  \end{equation*} However, I haven't seen the proof of the reverse triangle inequality:  \begin{equation*} ||x|-|y||\le|x-y|. \end{equation*} Would you please prove this using only the Triangle Inequality above?  Thank you very much. Explain what is a transistor in plain English! Long-distance movement arrow in gloss I have a question on drawing an arrow using gb4e. Currently, I am trying to use the method described in the link below. The problem takes place when I draw the long distance arrow like (5). As you can see the picture below, the line cross over the gloss. I would like to draw the horizontal line like the example (1). How can I evacuate the crossing?    \documentclass[11pt, dvipdfmx]{article} \usepackage{tikz} \usetikzlibrary{calc} \newcommand\tikzmark[1]{\tikz[remember picture, baseline=(#1.base)] \node[anchor=base,inner sep=0pt, outer sep=0pt] (#1) {#1};}  % This code from http://tex.stackexchange.com/q/55068/2693 \tikzset{     ncbar angle/.initial=90,     ncbar/.style={         to path=(\tikztostart)         -- ($(\tikztostart)!#1!\pgfkeysvalueof{/tikz/ncbar angle}:(\tikztotarget)$)         -- ($(\tikztotarget)!($(\tikztostart)!#1!\pgfkeysvalueof{/tikz/ncbar angle}:(\tikztotarget)$)!\pgfkeysvalueof{/tikz/ncbar angle}:(\tikztostart)$)         -- (\tikztotarget)     },     ncbar/.default=0.5cm, }  % Thanks to Paul Gessler adn Percusse for code improvement here \newcommand{\arrow}[2]{\begin{tikzpicture}[remember picture,overlay] \draw[-&gt;,shorten &gt;=3pt,shorten &lt;=3pt] (#1.base) to [ncbar=\arrowht] (#2.base); \end{tikzpicture} \setlength{\arrowht}{0ex} } \usepackage{gb4e} % The following code modified from  % http://permalink.gmane.org/gmane.comp.tex.linguistics/1036 % This adds some extra space after the first line \newlength{\arrowht} \setlength{\arrowht}{0ex} \newcommand*\cgdepthstrut{{\vrule height 0pt depth \arrowht width 0pt}} \renewcommand\eachwordone{\cgdepthstrut\rmfamily} \renewcommand\glt{\vskip -\topsep} \let\trans=\glt \newcommand\arrowex{\setlength{\arrowht}{2.5ex}\ex}   \begin{document} \begin{exe} \ex\label{scramble-det} \begin{xlist}         \arrowex          \gll          \ldots dat Jan \tikzmark{een} auto gisteren \tikzmark{t}  gekregen heeft.\\         \ldots that John a car  yesterday t gotten has \\         \ldots that John a car yesterday.         \arrow{t}{een} \end{xlist} \end{exe}  \begin{exe} \ex\label{scramble-dets} \begin{xlist}         \ex          \gll          \ldots dat Jan een auto gisteren t gekregen heeft.\\         \ldots that John a car  yesterday t gotten has \\         \ldots that John a car yesterday. \end{xlist} \end{exe}  \begin{exe} \ex[]{     \gll     John-wa totetumonaiukurai baka-da.    \\     John-Top very stupid-cop \\     `John is very studip.'     }  \ex[]{     \gll     John-wa totetumonaiukurai dousiyoumonai tondemonai sukuigatai baka-da.    \\     John-Top very very very very stupid-cop \\     `John is very studip.'     }  \arrowex[]{     \gll     \tikzmark{John}-wa totetumonaiukurai dousiyoumonai tondemonai sukuigatai  \tikzmark{t} baka-da.    \\     John-Top very very very very stupid-cop \\     `John is very stupid.'         \arrow{t}{John} }  \end{exe}  \end{document} Where to or how to ask "Best of" Questions for what software tools to use I am wondering where is a good place or how is a good way to ask a question on what software tools to use for a specific task.  A lot of the time when searching for information on these types of problems stackoverflow comes up with the top and actually useful discussions and answers, but the question is locked.  I completely understand the point of discouraging "best of" or list questions as discussed in this question () and agree they often aren't constructive enough to justify the effort in moderating them.  What I am wondering is where is the right place to have a discussion like this? Stackoverflow has a good and active community so perhaps there is another section or related site I can ask less specific questions. And I do find some value in reading people's opinions, even if subjective on issues like this.  Here are some examples of the kind of questions I mean:    and   First results in google when searching for those subjects and some useful discussion.  Perhaps asking a question like this but with a strict and detailed specification it might be considered a valid and reasonable question to ask on stackoverflow; perhaps it's just a case of re-thinking how to ask a question. Or perhaps there's a more relevant place for this kind of discussion. Please let me know what you think. QGIS 3.0 and Qt 5.7 time schedule When will QGIS 3.0 with QT5.7 (possibly) be released as a stable and long term support version? We're planning to upgrade an existing farm from 2007 to 2013. I learned that the best way and maybe the only way to achieve this is by migrating in two steps. 2007 to 2010 and 2010 to 2013. The actual farm is a sql-server(2008r2 64 bit) and a sperate sharepoint-server(web-frontend, etc....) I want to keep the sql-server as it is, if possible and we want to install a new 64bit host for sharepoint 2013. We will design a new branding for 2013 afterwards, so we need to keep the content, but not the layout. There's so much about migration available on the web, but Can I find something like best practises? A "non-believer" of C# was asking me what the purpose to extension methods was. I explained that you could then add new methods to objects that were already defined, especially when you don't own/control the source to the original object.  He brought up "Why not just add a method to your own class?" We've been going round and round (in a good way). My general response is that it is another tool in the toolbelt, and his response is it is a useless waste of a tool... but I thought I'd get a more "enlightened" answer.  What are some scenarios that you've used extension methods that you couldn't have (or shouldn't have) used a method added on to your own class? What do you call something that is not first in a sequence? Is there a word to describe something that is not the first element in a sequence, but can be in any other position? A synonym of "not first", in fact.     This element is __ in this sequence. How can I record my screen on Ubuntu?   The app I'm looking for has ideally all of these features:   Can record in a format that can be played back easily on any platform and/or accepted by YouTube or another popular video site Can record just a window (instead of the whole screen), possibly selecting it with a mouse click Can start recording after a configurable delay (e.g., I launch the app and have time to do arrangements to my desktop/window before actual recording starts) Would you ever stop rolling the die? Combo box in Java - nullpointer Below is an attempt at a proof of the following problem. Any feedback would be greatly appreciated. Thx!  Let $E$ be a set and $O_n = \{x: d(x, E) &lt; \frac{1}{n}\}$. Show  If $E$ is compact then $m(E) = lim_{n \rightarrow \infty} m(O_n)$.  This is false for $E$ closed and unbounded or $E$ open and bounded.   Note that all sets are real and measurable refers to Lebesgue measurable.  Suppose $E$ is compact. Then $m(E) &lt; \infty$. Also, since each $O_n$ is an open set of $\mathbb{R}^d$, $m(O_n) &lt; \infty$. If either of the following is true, we have that $lim_{n \rightarrow \infty}m(O_n)=m(E)$: $O_n \nearrow E$ $O_n \searrow E$ Now  $\bigcap_{n=1}^{\infty} O_n = \{ x:d(x,E) = 0\}$. That is, $x \in \bigcap_{n=1}^{\infty}O_n \iff \exists n&gt; \frac{1}{\epsilon},\ n \in \mathbb{N} \iff x\in \{x \mid d(x,E) =0\}$. Then we have that $\bigcap_{n=1}^{\infty}O_n = E$ and $O_n \searrow E$. Thus $ \ m(E) = lim_{n \rightarrow \infty} m(O_n)$. Suppose $E$ is closed and unbounded. Let $E = \{(x,y): y = 2 \}$ and $O_n = \{(x,y): d(x,E) &lt; \frac{1}{n}\}$. Since $E$ is a line in $\mathbb{R}^2$, $m(E) = 0$. Also, since the measure of a rectangle in $\mathbb{R}^d$ is its volume, $m(O_n) = \left| O_n \right| = \infty$, since the rectangle $O_n$ is unbounded. It is therefore apparent that $m(E) \not = lim_{n \rightarrow \infty} m(O_n)$. Suppose $E$ is open and bounded. Let $E=(0,1)$ and $O_n = \{x \mid d(x,E) &lt; \frac{1}{n}\}$. $\mathbb{R}-E$ is closed and $O_n \in \mathbb{R}-E$. Since $\mathbb{R}-E$ contains all its limit points $lim_{n \rightarrow \infty}O_n = O \in \mathbb{R}-E$. Thus $lim_{n \rightarrow \infty} m(O_n) \neq m(E)$.