Having trouble trying to understand how to tackle this question.  $19^{16} \mod{20413}$. When approaching this question I understand the steps involved such as $19^{16} / 20413 = 1.413028039\cdot10^{16}$. However not sure what to subtract this answer to achieve a $0$ before the decimal point in order to multiply by $20413$ to get the answer. From the answer sheet the intended answer is $11546$.  Appreciate the help.
I came upon an interesting way to relatively quickly compute modular exponentiation with large numbers. However, I do not fully understand it and was hoping for a better explanation.  The method basically states if you have $x^y \bmod N$, it can be computed by repeatedly multiplying by $x$ modulo $N$:  $$x \bmod N \rightarrow x^2 \bmod N \rightarrow x^4 \bmod N \rightarrow x^8 \bmod N \rightarrow \ldots \rightarrow x^{2[\log y]} \bmod N$$  This is supposed to calculate in $\mathcal{O}(\log_2 N)$ time.  An example is given:  $$x^{25} = x^{11001_2} = x^{10000_2} * x^{1000_2} * x^{1_2} = x^{16} * x^8 * x^1$$  I don't really understand how this is faster. Aren't you just doing the same thing? You're just breaking down the exponents into representations of base $2$. Also where does $x^{2[\log y]}$ come from? If you're breaking it down into exponents of two where does a logarithm come from? (Aren't logs representations of numbers in exponents of $10$?)  I'm sorry if this appears to be a really stupid question - I have minimal knowledge in math.  Also, I wasn't sure if this belongs here or on stack exchange, so give me the word and I'll move it.
If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative Check if table exists in SQL Server So a buddy and I were playing a game of Magic that turned out to be a really quick game because he got  out on the battlefield early. He was tapping it without attacking because of its obviously awesome ability but I don't know if you can tap a creature for its Inspired (untap-triggered) ability without attacking for it to then be tapped. Are you allowed to tap a creature whenever you want? Are there well known HTTP-only sites? Captive WIFI portals suck.  So often when I open in a browser (Desktop Chrome or Mobile Chrome) a HTTP site, I get the captive portal, but with auto-completion and so quickly I connect again to WIFI.  The problem is that after the captive portal redirects, I'll have also a HTTPS redirect and Chrome remember the certificate and to use only HTTPS.  So I cannot use the same site twice (in a session).  A well-known public HTTP only site will resolve this.  Well-known sites usually work, causing some less debugging of the WIFI connection. Does anyone know of a simple Image Editor, with functionality comparable to ,  for Ubuntu?  I've always found GIMP to be overkill and too complicated for what I want to do. Is it consistent with ZF for there to exist a set $S$ such that the power set $P(S)$ is countable? If so, what is the weakest form of the axiom of choice needed to prove that no such set exists? how to create a roster sheet in sharepoint 2010 This is about training management, when an employee requests for training, it goes to approval workflow and once approved, it is like the employee is enrolled.  Once it is approved I want a roster sheet to be created for that course for all the employees registered and who got approved.   The report I need to generate has to be of the enrolled users of one course and one particular session  If I give copy listitem as approved then different courses would come together in the list  The format for the report Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$   I proved $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ by interpreting $AB$ as a composition of linear maps, observing that $\operatorname{ker}(B) \subseteq \operatorname{ker}(AB)$ and using the kernel-image dimension formula. This also provides, in my opinion, a nice interpretation: if non-stable, under subsequent compositions the kernel can only get bigger, and the image can only get smaller, in a sort of loss of information.  How do you manage $\operatorname{rank}(AB) \leq \operatorname{rank}(A)$? Is there a nice interpretation like the previous one? I would like to transform 5V 500mA (from USB power) into 12V with a few cheap components.  Is this possible with the help of some generic parts (and not with the help of some expensive transformer, etc.)?  Also, how many milliamps would this provide? Both static_cast and reinterpret_cast seem to work fine for casting void* to another pointer type. Is there a good reason to favor one over the other? Why do DNGs only show in low resolution on Google Photos? Through my NAS at home I store and backup all my photos on Google Drive. These photos also show up on Google Photos. I use Lightroom and convert all photos to DNG when copying to my NAS.  When I look at the photos in Google Photos the image is rather small. Clicking the (i) info button tells me the image has a resolution of 1024x680, file size 9.8 MB and a MP of 0.7. This is probably just a preview or thumbnail of some sort, but how do I view the image in larger resolutions in Google Photos? As a non-native speaker of English, it can often be hard to spot some of the errors that a native speaker would find. For a long time, I've been thinking about getting a native speaker to help me with proof-reading my texts (articles, theses, application, etc), but I don't quite know where to look. What are some good ways to get proofreading for your texts?  For example, are there any websites on the Internet where one can exchange proofreading services between different languages? (Say, if you help me proofread my text in language x, I'll help you with yours in language y.) We have some new hardware in our office which runs its own customized Linux OS.   How do I go about figuring which distro it's based on? As far as we know the gaze of the basilisk will instantly kill someone or eventually petrify him/her in case the gaze is somehow softened or reflected.   But why would it melt the film in Colin's camera? The film is just plastics with some specific chemicals.   A bit broader question would be - is the some information that the basilisk's gaze is somehow destructive to non-alive objects too and under what circumstances could it happen? Difference between confidence intervals and prediction intervals For a prediction interval in linear regression you still use $\hat{E}[Y|x] = \hat{\beta_0}+\hat{\beta}_{1}x$ to generate the interval. You also use this to generate a confidence interval of $E[Y|x_0]$. What's the difference between the two? Calling C/C++ from Python? What would be the quickest way to construct a Python binding to a C or C++ library?  (I am using Windows if this matters.) How to replace the_post_thumbnail template tag and show the first inside the post image instead leaving oil on a frying pan for use later I live alone, so I cook for myself. And often times I cook steaks/burgers on a pan, and later in the same day would return to use the same pan/oil without washing/using new oil.  Is this safe, or should one wash and use a new pan before cooking every steak/burger patties? I've actually done this for more than a day once (cooked burgers in the morning, steaks later that afternoon, and then burgers for the next morning). Display bare HTML from hook_menu() What would be the cleanest way to route URLs to functions and return only the HTML output from those functions (a.k.a. the menu callback)?  If I use hook_menu() the string that's returned will be the HTML within the rest of the site. I want the page to display only the string returned from the menu callback.  Should I use drupal_set_header() in the same way drupal_json() does? Is this not inefficient having fully bootstrapped Drupal, and then changing the header at the last minute?  I want the fastest code path possible since this is going to be for widgets on external sites.  I could start with a standalone .php file and bootstrap what I need but the hook_menu routing is nice.    How should you do this? How many constructor arguments is too many? Let's say you have a class called Customer, which contains the following fields:   UserName Email First Name Last Name   Let's also say that according to your business logic, all Customer objects must have these four properties defined.  Now, we can do this pretty easily by forcing the constructor to specify each of these properties.  But it's pretty easy to see how this can spiral out of control when you are forced to add more required fields to the Customer object.    I've seen classes that take in 20+ arguments into their constructor and it's just a pain to use them.  But, alternatively, if you don't require these fields you run into the risk of having undefined information, or worse, object referencing errors if you rely on the calling code to specify these properties.  Are there any alternatives to this or do you you just have to decide whether X amount of constructor arguments is too many for you to live with?