$9^{123456789} \pmod{100}$  I tried to do this in a simpler way:  $$9 \equiv 9 \pmod{100}$$  $$9^1 \equiv 09 \\ 9^2 \equiv 81 \\ 9^3 \equiv 29 \\ 9^4 \equiv 61 \\ 9^5 \equiv 49 \\ 9^6 \equiv 41 \\ 9^7 \equiv 69 \\ 9^8 \equiv 21 \\ 9^9 \equiv 89 \\ 9^{10} \equiv 01$$  I noticed that the odd powers always end in 9 and the even ones in $1$. The leftmost digit follows the sequence $0,2,4,6,8,$... in this order for odd numbers and the other way around for even ones. The sequence has $5$ numbers.  $123456789$ is an odd number. $123456789 \equiv 4 \pmod 5$ so the answer should be $69,$ which is the $4$th of the sequence... but it's $89.$ Close enough, but not quite. What did I do wrong and how can I solve this in this way?
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.
I see this a lot, someone in a category of people attempts to shut down anyone with an opinion that is not from the same category. For example, people without children weighing in on parenting; the parent may say something like "well, since you don't have kids and I do, my position is of more validity than yours."  Another example: a drug user (of some kind) telling someone who has never done drugs and does not care to, "until you tried it, you have no right to judge me."  People frequently try to limit the consideration on opinions to those in their same group, while it is perfectly reasonable that the non-drug user, especially if they are a chemist or well read on the subject, may have an equally valid opinion, or even more valid. In fact some might even say that someone NOT in the category has a better chance of being objective and seeing the forest for the trees.  So what is the fallacy of trying to shut down opinions when the source of the opinion is not someone who has personally had the experience under discussion? This has happened to me many times in the last 5 years: an upgrade broke my system. Each time I end up with this situation, I have to reinstall the whole system, which is really annoying.  Is there any way to roll back the most recent upgrade to be able to have a functional system without reinstalling? If not, which is the best way to suggest this as a top priority idea?   I read this idea was described in brainstorm.ubuntu.com, but it feels it is dead... and the forums are full of examples of upgrades breaking things, that's why I feel something needs to be done about this topic. Thanks! Behavior of $\sum a_n x^n$ given $\sum |a_n - a_{n-1}| < \infty$ Let $(a_n)_{n=0}^{\infty}$  be a sequence of real numbers satisfying $\sum_{1}^{\infty} |a_n - a_{n-1}| &lt; \infty$   Which of the following conclusions must then be true?  The series $\sum a_nx^n$ converges   Nowhere on $ \mathbb{R}$ Everywhere on $  \mathbb{R}$ On some interval containing $(-1,1)$ Only on $(-1,1)$   What I have tried is Make an inequality with $|a -b| \leq |a| +|b|$ .. but i cant proceed further .. Any sort of help will be deeply appreciable . The ending I got is :      the genocide   When I reset my file, my past choices still take effect. I want to start a clean run.  I don't want to buy the game again, what do I do? Introduction to string theory I am in the last year of MSc. and would like to read string theory. I have the Zwiebach Book, but along with it what other advanced book can be followed, which can be a complimentary to Zwiebach. I would like a more mathematically rigorous book or lecture notes along with Zwiebach.   Specifically, mention whether the book discusses string theory   Rigorously? Intuitively?   What's the scope of the book? Does it cover the advanced materials, e.g. Matrix string theory, F-theory, string field theory, etc. Maybe even String Phenomenology? What Is Exponentiation? What's a word for a positive kind of "envy" without the sense of resentment? Imagine your college buddy is now running a multi-billion dollar company, while you're still stuck at the same white-collar day job 5 years after graduation. You chat with him and want to express the idea that you're happy for him and fancy having his lifestyle--something like "I wish I could buy a yacht and cruise around the world like you do; I'm glad you're doing really well!"  Basically, it's the feeling of envy when you take out the resentment from it. Is there a word for it?   (I thought about "admire", but found it too strong and broad) How do I add a custom script on a specific page? I wanted to add custom JavaScript code on certain nodes. I have written this code in the html.tpl.php template, but this doesn't work. (donate is the path alias for a node.) &lt;?php if(arg(0) == 'DONATE'): ?&gt;   &lt;script&gt;     (function() {       // ...     );   &lt;/script&gt; &lt;?php endif; ?&gt; Let $X$ be a random variable and let $g,f$ be real-valued, nondecreasing, and bounded.   Show that $\mathbb{E}[f(X)g(X)]\geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$  Having a hard time seeing where to start for some reasons, any hints?  I managed to write:  $\mathbb{E}[f(X)g(X)] = \frac{1}{4}\mathbb{E}[(f(X)+g(X))^2 - (f(X)-g(X))^2]$  But I'm not sure if this is in the right direction or not. Mac asking for password for some files when deleting This happens not for all files. When I move or delete some of the files on my external drive Finder will ask for password. Is it due to permission set on these files ? Can I use something like Sudo when running Finder to prevent these pop ups ? The reason I'm asking is that sometimes I leave the mac running to move some files from one disk to another, just to find it on the password prompt a few hours later. Thank you. Find a combinatorial argument for the following binomial identity:   $$n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} .$$   Algebraic proofs can be found at  , and a related identity at  . Java function for arrays like PHP's join()? byte[] to hex string How do I convert a byte[] to a string? Every time I attempt it, I get      System.Byte[]    instead of the value.  Also, how do I get the value in Hex instead of a decimal? Will adding more RAM speed up my computer? Apparently, you can speed up your computer by adding more RAM. Is the amount of ram to be added is directly or indirectly related to the amount of hard-disk  space, processor speed, Motherboard etc... Can the memory of a RAM be compared with Virtual Memory and Cache Memory?  Note: About my PC Configuration: Intel core i5 processor, Motherboard DP55WB 4GB RAM, 700 GB Hard-disk space. How does faster than light travel violate causality? Let's say I have two planets that are one hundred thousand lightyears away from each other. I and my immortal friend on the other planet want to communicate, with a strong laser and a tachyon communication device.  I record a message on the tachyon communication device and release the message at exactly the same time as I activate the laser, both of which are directed to the other planet which is one hundred thousand lightyears away. Say it is the year 0 for both of us at the time I did this.  If tachyons existed, then the message would arrive to my friend before the photons in the laser. It would arrive, say, a thousand years earlier. From my vantage point, that message will arrive to her at year 99,999; the same would be true for my friend's vantage point. However, she will only see the laser at year 100,000.  So since she got the message at year 99,999, she immediately sends me a reply back going through the same procedure as I did. She records a message and releases it at the same time as the laser. The tachyons will arrive 1,000 years earlier than the laser, so for me, I will receive the message at year 199,998. I will receive the laser, however, at year 199,999.  It seems to me that communication this way does not violate causality. I will still have received the message after I had sent it.   If tachyons truly violated causality, though, I realize it should arrive at year -1 for her, and so she can reply to me at year -2, which would mess me up by year 0 as I will ask her how she knew I was planning on sending her a message before I sent it. I could send her a different message, which she would end up receiving at year -1, and will end up confusing her as she would have received one message asking her out, and the other asking her how did she know I was asking her out. She then decides I am crazy and sends me a message at year -2 that she does not want to date me, and so she will have both turned me down and entertained me before I have even asked her out.  On the other hand, let's go back to year 0 and add a third device to our list: an Alcubierre drive. After I send out the message and the laser, I get impatient and do not feel like waiting 99,999 years, so I get on my Alcubierre drive spaceship and arrive on her planet at the same year 0. My friend is not in her office, so I leave a note to her also immortal secretary saying I dropped by and that she should expect a message for her in year 99,999.  I then get back on my Alcubierre drive and land back on my planet, still on year 0. Meanwhile, the tachyons and photons I sent out are still racing to arrive to her. By year 99,999, she receives the message just as I Alcubierre drive back to her, and I pick her up for dinner.  But the point of my question is, it seems to me that just going faster than light, if that alone was what you had, would not violate causality. It must be something else. I understand time dilation and that things with mass cannot travel at the speed of light, but using the Alcubierre drive, hypothetically speaking, I was still able to outpace the photons while also having mass. It still did not produce causality problems. Alcubierre drives are also valid solutions to GR.  It seems circular to me to say that what makes traveling faster than light violate causality is because it violates causality (if faster than light communication was divorced from causality problems, then the causality problem would cause itself -- thereby violating causality and, hence, we would scrap it and conclude that there is no causality problem after all).  What is it that I am missing? If someone could help me out, that would be excellent. I've been itching to ask my friend out for a few millenia now. :) How to find a whole word in a String in java Using separation of variables to solve the wave equation Can I call a constructor from another constructor (do constructor chaining) in C++? As a  developer I'm used to running through constructors:    class Test {     public Test() {         DoSomething();     }      public Test(int count) : this() {         DoSomethingWithCount(count);     }      public Test(int count, string name) : this(count) {         DoSomethingWithName(name);     } }   Is there a way to do this in C++?  I tried calling the Class name and using the 'this' keyword, but both fail. Downloading from Play Store gives error 491 When I try to download anything from the Play Store it shows error 491 and can't finish the download. I have a Samsung Galaxy S 7582. How do I fix this? Let $A =$ {$(x,y,z) \in \mathbb{R}^3 |$  $x,y,z \geq 0$ and $x+y+z \leq 1$}.   Compute the integral $$\int_{A} x^m y^n z^p (1-x-y-z)^q dx dy dz$$  using new variables $u = x+y+z$, $v=\frac{y+z}{x+y+z}$, $w=\frac{z}{y+z}$.  Also, although it was not stated in the problem, I'm assuming that $m,n,p,q \in \mathbb{Z}$.   Since $f(x,y,z)=(x+y+z, \frac{y+z}{x+y+z}, \frac{z}{y+z}) = (u,v,w)$, $f^{-1}(u,v,w) = (u(1-v), uv(1-w), uvw)$  Thus J $= \begin{vmatrix} 1-v &amp; -u &amp; 0\\  v(1-w) &amp; u(1-w) &amp; -uv\\  vw &amp; uw &amp; uv\\ \end{vmatrix} = u^2v$  Skipping the part where I find the new limits of integration, and skipping some algebraic simplification we have:  $$\int_{A} x^m y^n z^p (1-x-y-z)^q dx dy dz$$   $$= \int_{0}^1 \int_{0}^1 \int_{0}^1 u^{m+n+p+2}(1-u)^q v^{n+p+1}(1-v)^mw^p(1-w)^n du dv dw$$  However, I have no  idea where to go from here. Any help would be much appreciated.