How to calculate   ((13)^15)^17 (mod 17)   using fermat's little theorem?
How do I efficiently compute $a^b\bmod c$:  When $b$ is huge, for instance $5^{844325}\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$? When $(a,c)\ne1$, for instance $6^{103}\bmod 14$?  Are there any other tricks for evaluating exponents in modular arithmetic?  This is being asked in an effort to cut down on duplicates, see  and .
Windows 8.1 64-bit: High memory usage I have 8GB of memory on a 64 bit windows 8.1 machine, and windows is telling me I'm using 91% of my memory, yet its simultaneously telling me I'm only using 3GB.  I literally just restarted my machine a few minutes ago because it was being incredibly rediculously slow. I'm currently only running chrome (with a ton of tabs, but doesn't add up to anywhere near 7.5GB). What could be going on here?   Here's a bunch of my current info       I took a look at this:  but I wasn's able to use xperf and the rest didn't lead me to a solution.  UPDATE - RamMap:   UPDATE 2: I checked Resource Monitor at Wouter's suggestion and found an "Image" called lxdecoms.exe with over 6GB of Commt (KB). I tried killing the process, but while it lowered my Commit Charge, it didn't affect "Used Physical Memory". So I restarted hoping to capture it again in full, but its not back yet and yet the problem persists.  I should note that I do have a Centos virtual box virtual machine running on my machine, not sure if that has any weird memory impacts that aren't reflected in the numbers in these tools.   I got xperf working, so here's my resource monitor and a wpa analysis of my xperf output (from xperf -on ReferenceSet -BufferSize 1024 -MaxFile 512 -FileMode Circular  &amp;&amp; timeout 5 &amp;&amp; xperf -d MemUsage.etl):     What is N/A that's taking up 2.7GB of memory? My VM? Using the properties panel I can add, with "+"  button, only eight UV maps. Is it possible to have more than 8 of them? I have been trying to teach myself QGIS by using this forum and others to help. Most of my questions can be answered by downloading plugins that are available through 3rd party repositories but for some reason I am not able to connect with these. Anyone have any ideas why this is the case? I just edited an answer on Unix&amp;Linux and when scrolling down, the semi-transparent right sidebar boxes "How to edit" and "How to format" kept their fixed positioning and overlapped with the page footer.     While posting this, I have seen several (not) duplicate questions about this issue, and although from my personal taste, I would prefer the boxes to stop scrolling before the footer is reached, I can accept the development teams decision to let it overlap.   However, on all those other posts, the boxes had an opaque background, so that the footer was hidden and the box content was readable. Here the boxes are transparent though, so the texts get mixed and are no longer well readable. How to compress entire drive C in Windows including locked files? $\mathbb R = X^2$ as a Cartesian product I wonder if it is possible to consider $\mathbb R$ as a Cartesian product $X\times X$ for some set $X$. From the point of view of the dimensionality, there are spaces with a Hausdorff dimension $1/2$ (sort of Cantor sets), but I guess there are other problems in this construction  unrelated to the dimension.  Edited: replying on the comment by Asaf. I want that for all $r\in\mathbb R$ there exists a unique representation $r=\langle x,y\rangle$ where $x\in X$ and $y\in X$. Also, what if we want to have $X$ to be a topological space and $h:X\times X\to\mathbb R$ to be a homeomorphism? Is it a weakness to use a single shared secret for protecting messages using a Encrypt-then-MAC scheme?    Assuming a system is using AES-256-CBC and a SHA1-HMAC and the same secret key for both operations.  Upon intercepting one of these messages and attacker could brute force this secret key by trying to solve the HMAC or by successfully decrypting the first block of the ciphertext.  Performing  a decryption on a single block of AES-256 should be a lighter operation than performing a SHA1-HMAC on a potentially large message.  Therefore AES-256 is ideal for a massively parallel brute-force attack and the use of the SHA1-HMAC doesn't hurt the system.  Is this approach sound, or is there an attack that has not been taken into consideration? No matter where I look, I just can't find gnome number [insert number here]. Where can I find every gnome in the game? Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem. Pure significance of line integrals of vector fields I can understand how the line integral of a scalar function has pure-mathematical significance, representing an area beneath a curve. But is there a pure significance to the line integral of a vector field? The only understanding I have of this is the physical example of work done on a moving particle. Thanks for any insight. Okay the scenario I'm thinking of is as follows:   I control a ,  and 3 untapped lands (2 swamps, 1 Island to be perdantic). In my opponents turn he casts a burn spell to kill my duskmantle.   I activate his first ability in response.   The way I am hoping the following interactions will go is:   Duskmantles ability will hit the stack at the top and resolve The burn spell resolves dealing its damage to Duskmantle The burn spell card hits the graveyard and triggers the infinite loop for duskmantle and mindcrank killing my opponent Duskmantle perishes   Is this the way that the stack would pan out? Do I win in this situation regardless of the stack being correct or not above? (I am not fully familiar with how the stack really works in magic) You can see the cards from my deck . Using subcaption package but subfigures end up stacked rather than side by side I'm trying to put two jpegs side by side using the subcaption package, but the images are consistently stacking on top of each other rather than side by side. All the solutions I've found simply state that the pictures have to be small enough to actually be side by side but I've tried making them smaller, too small even and they stay stacked. My code looks like this:  \begin{figure}[H] \centering \begin{subfigure}{0.5\textwidth} \centering     \includegraphics[width=0.3\linewidth]{inclusion.jpg}     \caption{inclusions}     \label{fig:inclu} \end{subfigure}  \begin{subfigure}{0.5\textwidth} \centering     \includegraphics[width=0.3\linewidth]{deform}     \caption{grain deformation}     \label{fig:deform} \end{subfigure} \caption{filler text} \label{fig:manmade} \end{figure}   Help would be greatly appreciated. Prove that the power set of an $n$-element set contains $2^n$ elements Theorem. Let $X$ denote an arbitrary set such that $|X|=n$. Then $|\mathcal P(X)|=2^n$.      Proof. The proof is by induction on the numbers of elements of $X$.      For the base case, suppose $|X|=0$. Clearly, $X=\emptyset$. But the empty set is the only subset of itself, so $|\mathcal P(X)|=1=2^0$.      Now the induction step. Suppose $|X|=n$; by the induction hypotesis, we know that $|\mathcal P(X)|=2^n$. Let $Y$ be a set with $n+1$ elements, namely $Y=X\cup\{a\}$. There are two kinds of subsets of $Y$: those that include $a$ and those that don't. The first are exactly the subsets of $X$, and there are $2^n$ of them. The latter are sets of the form $Z\cup\{a\}$, where $Z\in\mathcal P(X)$; since there are $2^n$ possible choices for $Z$, there must be exactly $2^n$ subsets of $Y$ of which $a$ is an element. Therefore $|\mathcal P(Y)|=2^n+2^n=2^{n+1}$. $\square$      From the above explanation, I don't understand why the set that contains $\{a\}$ will contain $2^{|n|}$ elements when it should clearly be $2^{|1|}$.  The construction of a new set $S$ is the union of the old set with cardinality $n$ and an new element $\{a\}$, therefore the set that do not contain $\{a\}$ still has cardinality $n$ and the set that contains $\{a\}$ is just $\{a\}$, one element...  Can someone please elucidate? What's the difference between mobile and desktop processors? I just read about the new Samsung Galaxy Note Edge having a 2.7&nbsp;GHz quad-core processor and 3GB of RAM.  The laptop I bought last year by HP is 4&nbsp;GB of RAM and 2.3&nbsp;GHz quad-core and my iMac is even older and is 2.5&nbsp;GHz i5.  Does this mean that the new Samsung gadget is more powerful than my desktop?  Is the 2.7&nbsp;GHz the same kind of GHz as non-mobile devices (is it scaled up, or compared, etc.)?  Why, in terms of power, do modern computers not have two of those Samsung quad-core processors running in parallel pushing out 5.4&nbsp;GHz processing power for the amount of electrical energy as two Galaxy Note batteries? Do not show ellipsis when not needed in search results When doing a search, the results page shows an ellipsis when it is not needed. For instance:    The message of that answer is shorter than the text overflow limit, which means the ellipsis (...) should not be shown. Let $D \subset \mathbb{C}$ be the open unit disk, $f:D \to \mathbb{C}$ a holomorphic function and $d=\sup \{|f(z)-f(w)|:z,w\in D \}$. How can I show that $|f'(0)|\leq \frac{d}{2}$?  I tried using Schwarz's lemma on the function $g:D \to \mathbb{C}$ defined by $g(z)=\frac{f(z)-f(0)}{d}$, but I am only getting $|f'(0)|\leq d$. My terminal on osx 10.11.6 will not accept `[` character Every time I type it the terminal flashes but doesn't insert it. If I paste in a command that has a [ it will paste everything except the ['s  This has been happening before I updated to 10.11, but has become a massive issue, I can't use jq at all because of it.  I also have the same issue with iTerm2. Does anyone have a Terminal command-line recipe for making a new bootable USB flash drive for one of the new MacBook Air's?  Is there a way to dd copy the USB boot device that comes in the box and make the new copy bootable?  I'd like to make a new backup copy of the USB rescue device for my MBA, preferably on a much larger flash drive with room for a backup of some of my working directories.  Added:  And is it possible to make, not just a rescue drive, but a bootable USB drive that will boot a fully updated version of the OS (say 10.6.8 or later)?  If so how on a MacBook Air 11, and how big a USB stick will this require? Appearing on the second page (under the section Digression: Size worries) of the following paper about the Yoneda Lemma:     It says that $a$ $priori$ $[\mathcal C^{op},Set](H_A,X))$ is a class. I don't understand why this is the case.  My understanding is that $a$ $priori$ each member of $[\mathcal C^{op},Set](H_A,X))$ is a class.  So it seems to me that $[\mathcal C^{op},Set](H_A,X))$ could be a collection of proper classes.  I've looked at this , but it hasn't helped me much.  Any help is appreciated -Thanks I'm looking for a 90s (or early 2000s) scifi (mini)series where a boy and old man chase a black guy in metal suit. I think they were going through different dimensions. I remember part of it, but I don't think I ever saw how it ended. I remember my friend telling me there was a twist at the end. I think they were trying to find something and he was always ahead of them..