I want calculate remainder when $2^{1000}$ is divided by $5^{4}$ Can I calculate it using Fermat's Little Theorem?
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.
What's the fastest way to make money online? Everything in the online world is expensive, well the best things are. We all want the luxurious apartment with 10 cars garage and the best weapons but to achieve that, we need money. Which missions or activities give the most cash ? Does installing many programs really slow down Windows? Why does Stack Exchange ask for a reload on successful automatic login? I'm looking for an answer considering both technical limitations and user experience.   Does it need a reload from a technical standpoint? How to use proxy for Transmission? How do I make an Bootable Ubuntu USB containing the the latest version of Ubuntu?  For completeness, how would I be able to do this on Ubuntu, Windows, OS X or another Linux distro? What is meant by "object serialization"? Can you please explain it with some examples? What’s the point in having “www” in a URL? Other than for historical reasons, is there is reason to have “www” in a URL?  Should I create a permanent redirect from www.xyz.com to xyz.com, or from xyz.com to www.xyz.com? Which one would you suggest and why? How to use "who" vs. "that" How to both display a command line's output on console and save the output into a text file? VirtualBox 5.0.40 crashes the whole host since update to kernel 4.13.0-26-generic I'm using post thumbnails to link to a page.  Is it possible to add a class name to the post thumbnail image.  &lt;li&gt;&lt;a href="&lt;?php the_permalink(); ?&gt;" &gt;&lt;?php the_post_thumbnail(); ?&gt;&lt;/a&gt;&lt;/li&gt; JavaScript curry: what are the practical applications? I don’t think I’ve grokked currying yet. I understand what it does, and how to do it. I just can’t think of a situation I would use it.  Where are you using currying in JavaScript (or where are the main libraries using it)? DOM manipulation or general application development examples welcome.   mentions animation. Functions like slideUp, fadeIn take an element as an arguments and are normally a curried function returning the high order function with the default “animation function” built-in. Why is that better than just applying the higher-up function with some defaults?  Are there any drawbacks to using it?  As requested here are some good resources on JavaScript currying:    Crockford, Douglas (2008) JavaScript: The Good Parts  (Takes a detour into ML so skip the whole section from “A crash course in ML” and start again at “How to write curried JavaScript”)    (Mr. Resig on the money as per usual)    I’ll add more as they crop up in the comments.    So, according to the answers, currying and partial application in general are convenience techniques.  If you are frequently “refining” a high-level function by calling it with same configuration, you can curry (or use Resig’s partial) the higher-level function to create simple, concise helper methods. i Need a batch file to create a folder "test" in every removable device than i plug in my computer (pendrives and etc)  i am trying something like this  set "extDrive=" for /f %%D in ('wmic volume get driveLetter^ where drivertype=2 ) do set extDrive=%%D mkdir "%extDrive%\test"   Someone can tell me why this is not working ?? Prove that, $f'(0) \ge -\sqrt{2}$ for a function $f$ satisfying some conditions on $(-1,1)$. Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that, $f(0)=1$, $f'(x)≤0$, $f(x)≥0$ and $f''(x)≤f(x)$ for all $x≥0$. Prove that, $f'(0)≥-\sqrt2$   Progress: I was able to prove $f'(0)≥-2$. For this I have applied Cauchy MVT for $f'$ and $g$ on $[0,x]$ for any $x\in (0,1)$ where $g: [0,x]\to\mathbb{R}$ is defined by, $g(x)=\sqrt{x+1}$. But, couldn't find another approach to get $-\sqrt{2}$. Remove entire row in a file if first column is repeated Discontinuous derivative. Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-valued and defined on a bounded interval. is $y = \sqrt{x^2 + 1}− x$ a injective (one-to-one) function? I have a function  $y = \sqrt{x^2 + 1}− x$ and I need to prove if it's a Injective function (one-to-one).  The function f is injective if and only if for all a and b in A, if f(a) = f(b), then a = b  $\sqrt{a^2 + 1} − a = \sqrt{b^2 + 1} − b$  $\sqrt{a^2 + 1} + b = \sqrt{b^2 + 1} + a$  $(\sqrt{a^2 + 1} + b)^2 = (\sqrt{b^2 + 1} + a)^2$  $a^2 + 1 + b^2 + 2b\sqrt{a^2 + 1} = b^2 + 1 + a^2 + 2a\sqrt{b^2 + 1}$  $2b\sqrt{a^2 + 1} = 2a\sqrt{b^2 + 1}$  $b\sqrt{a^2 + 1} = a\sqrt{b^2 + 1}$  $(b\sqrt{a^2 + 1})^2 = (a\sqrt{b^2 + 1})^2$  $b^2(a^2 + 1) = a^2(b^2 + 1)$  $b^2a^2 + b^2 = a^2b^2 + a^2$  $b^2 = a^2$  $\sqrt{b^2} = \sqrt{a^2}$  $b = \pm a$  then the function is not injective because $b$ in not equal to $a$, but using online  the function is injective. This applies to all d20 systems, but with an example from the from the Pathfinder  (emphasis mine):  Stonecunning: Dwarves receive a +2 bonus on Perception checks to potentially notice unusual stonework, such as traps and hidden doors located in stone walls or floors. They receive a check to notice such features whenever they pass within 10 feet of them, whether or not they are actively looking.  In this case, does the GM roll the check in secret or ask the player roll the check? If the GM asks the player to roll for the check, what stops situations like:  GM: (interrupting) &quot;Gimli, Roll a perception check.&quot; Dwarf player: &quot;Oh, it's a natural one, I guess I don't see anything.&quot; Human player: &quot;Okay, I'll roll a perception check for anything unusual in this room. Nice! 19. How about 19, does 19 work for you?&quot; GM: &quot;Gimli, everything looks kosher to you. Aragon, you notice a rock coloured tarp on the ground here, covering what looks like a pit trap.&quot;  In this case, by telling the dwarf to roll his &quot;free&quot; perception check, he's telegraphed to the players that there is something to notice. How can this be fixed, without bluffing checks for nothing in particular to throw off the signal to noise ratio of similar checks? Every six months a new Ubuntu version will be released. Does anyone know at which time this will be? has anybody been successful getting pandas to work with QGIS's Python installation, on Windows? I've used the OSGeo4W installer to install QGIS2.2, which comes with Python 2.7 and Numpy 1.5.1. Any reasonably new version of pandas requires Numpy 1.6.1 or newer. So that seems like my main problem. I've tried: - Building an older version of pandas (e.g. 0.7.3 and 0.4.1) using easy_install, but that gives me a very long list of errors, at least the last of which are related to distutils\msvc9compiler.py - Using binaries of old pandas versions, but they are all built for Numpy 1.6.1 - Upgrading Numpy, which also gives me an error message. In case anybody is interested, the message ends with "File “C:\OSGeo4W\apps\Python27\lib\distutils\msvc9compiler.py”, line 299, in query_vcvarsall” newline “raise ValueERror(str(list(result.keys())))” newline “ValueError: [u’path’]"  I'm wondering if anybody else has ever been successful in getting pandas to work with a QGIS install on Windows??  It would be so easy if QGIS would ship with, say, a version of Numpy that's less than three years old. Does anybody know if such an upgrade is in the works for upcoming QGIS versions?