By brute force I'm noticing that for all $a ∈ 𝔽_7$, we have $a^{19}\bmod 7 \equiv a $. Now this looks very much like Fermat's Little Theorem, but this theorem should hold if the primes are the same, whereas 19 and 7 are clearly not. Is this just a coincidence?
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.
A fleet of nine taxis is dispatched to three airports, in such a way that three go to airport A, five go to airport B and one goes to airport C.  If exactly three taxis are in need of repair. What is the probability that every airport receives one of the taxis requiring repairs.  My method was total number of ways is (9!)/(3!*5!*1!)= 504 (this is correct)  Now you have three groups and of every group one position is already filled. So: ....R, ..R, R So the numbers of ways to distribute the rest is: (6!)/(2!*4!)=15  But according to answer model of my professor, it should be 90. So I assume he says the number of ways of distrubiting these 3 broken cabs over the 3 spots is 3*2*1. But to me this seems wrong? The broken taxis don't differ so R(1)R(2)R(3) is same as R(2)R(1)R(3). If you standing at the airport A, it doesn't matter which of the broken cabs stands there? Do immature fruits perform photosynthesis? Most immature fruits are green: peppers, pine cones, plums, lots of them. I want to know if the green is from chlorophyll in the cells. Do the fruit cells perform photosynthesis? When you cover a green stem or leaf, it will turn pale and stretch. That is because the stems have little need for chlorophyll in the dark, which is why they are pale. They stretch because the auxins in the stems are not destroyed by the photons, and the stems stretch out and topple over. If a green fruit is covered, will it turn pale and stretch like that? I used to do a full factory wipe then flashed a stock firmware with Odin, that helped me get back to having access to my mobile network then I flashed to Resurrection Remix 5.5.3 and lost access to my mobile network (Emergency Calls Only) I checked my IMEI and it appears to be valid. I didn't backup my /efs I didn't do anything to it. What seems to be the solution? I have a Samsung Galaxy S3 (I-9300) Some packages, have a Provides: header. For example, the package postfix on Debian Wheezy has Provides: mail-transport-agent.   The package mail-transport-agent which doesn't exist physically is considered as a "virtual" package.   How can I know, on a Debian based system, if a given virtual package is "installed" (or "provided")? Can I list every "provided" virtual package?  Hint: to list every existing virtual package, installed or not, do: aptitude search "~v" From Windows Vista to 7, can I get rid of folder Windows.old? I've just changed my laptop OS from Windows Vista to Windows 7.  I didn't choose to have both OS together, I just want to have Windows 7.   I was specting Windows 7 instalation program to format/clean my HD and leave me most of its capacity for a fresh start. But I've found, once the OS has been isntalled, that all files and programs from the Vista era are still in my HD, in a folder called Windows.old   I don't need these programs and files, I want to have as much HD space as possible and Windows 7 as clean as possible, what should I do?  Can I just delete Windows.old folder? Would that affect the how Windows 7 works?  Maybe is because I ended up angry with Vista, but I don't like the fact that there wasn't any cleaning/formatting ..   Thanks a lot overbrace/underbrace but with an arrow instead Is there a similar command to \overbrace{foo}^{bar} but instead of braces, an arrow pointing up/down?  EDIT: Here's a little demonstration of how I'd expect something like this to work:    And possible syntax would be \overarrow{a=b}^{\text{theorem X}} Do the SSH or FTP protocols tell the server to which domain I am trying to connect? When using the ssh or ftp commands from the Bash shell, does the server that I am connecting to learn of the domain name used? I understand that the domain name is locally translated into an IP address via DNS. In HTTP, after that happens, the server is told the original domain name as well in order to serve the correct page, or to present the correct TLS cert (SNI).  host serverfault.com GET /   Does a similar phenomenon happen when connecting to ssh or ftp?  I ask because I am  trying to ssh into a server (GoDaddy webhosting) which expects a domain name, but is not letting me in when I try to connect via user@IPaddress as the DNS is not yet moved to the GoDaddy IP address. We have set up GA on several websites to gather information about traffic to the site. We know about the organic keywords that are displayed under the Acquisition section of the traffic report. But we have been asked to find a way to display "not provided" keywords typed in by the end user in our Google Analytics report.  Is it possible to accomplish this? Sum over multiple indices Reinstall software-center in Ubuntu 16.04 Following the guidelines suggested in , I am going to post a proposed proof as an answer to the theorem below. I believe the proof works, but would appreciate any needed corrections.  Theorem If a series $\sum_{n=1}^{\infty}a_n$ of real numbers converges then $\lim_{n \to \infty}a_n = 0$ Syntactic characterisation of the intersection of CTL and LTL "Easy" vs. "simple" Are those two words equivalent or is there a difference? When would I say something is easy rather than simple, or vice versa? Should the user be forced to sign in for a wish list on an ecommerce site? I believe that signing into a site can be a big barrier to users so prefere e-commerce sites that offer guest access.  The question is should wishlists be a guest feature or do the business advantages enforcing the user to sign up outweigh the potential increases in profits.   Does anyone have any research or views on this? I realize this isn't possible, but I can't see why not, especially if you change the model a little bit so that the balls simply travel through a tube of water on the way up, rather than exactly this model.  Please be clear and detailed.  I've heard explanations like "the balls wouldn't move" but that doesn't do it for me - I really do not see why the balls on the right would not be pulled/pushed up, and the rest of the chain wouldn't continue on. What's the difference between running a program as a daemon and forking it into background with '&'? What are the practical differences from a sysadmin point of view when deploying services on a unix based system? About 'start' and 'begin' Are 'begin' and 'start' interchangeable? Both &quot;This is a new start&quot; and &quot;This is a new beginning&quot; work, right? Then during games, they say &quot;Let's start&quot;, but I never heard &quot;Let's begin&quot;. So is there supposed to be a difference? And then one guy said this &quot;They could not begin to start a conversation&quot;. What is this supposed to mean? Is it correct? Prove every element of $G$ has finite order. How can I get distribution name and version number in a simple shell script? I'm working on a simple bash script that should be able to run on Ubuntu and CentOS distributions (support for Debian and Fedora/RHEL would be a plus) and I need to know the name and version of the distribution the script is running (in order to trigger specific actions, for instance the creation of repositories). So far what I've got is this:  OS=$(awk '/DISTRIB_ID=/' /etc/*-release | sed 's/DISTRIB_ID=//' | tr '[:upper:]' '[:lower:]') ARCH=$(uname -m | sed 's/x86_//;s/i[3-6]86/32/') VERSION=$(awk '/DISTRIB_RELEASE=/' /etc/*-release | sed 's/DISTRIB_RELEASE=//' | sed 's/[.]0/./')  if [ -z "$OS" ]; then     OS=$(awk '{print $1}' /etc/*-release | tr '[:upper:]' '[:lower:]') fi  if [ -z "$VERSION" ]; then     VERSION=$(awk '{print $3}' /etc/*-release) fi  echo $OS echo $ARCH echo $VERSION   This seems to work, returning ubuntu or centos (I haven't tried others) as the release name. However, I have a feeling that there must be an easier, more reliable way of finding this out -- is that true?  It doesn't work for RedHat. /etc/redhat-release contains : Redhat Linux Entreprise release 5.5  So, the version is not the third word, you'd better use :  OS_MAJOR_VERSION=`sed -rn 's/.*([0-9])\.[0-9].*/\1/p' /etc/redhat-release` OS_MINOR_VERSION=`sed -rn 's/.*[0-9].([0-9]).*/\1/p' /etc/redhat-release` echo "RedHat/CentOS $OS_MAJOR_VERSION.$OS_MINOR_VERSION" Will inverse functions, and functions always meet at the line $y=x$? If I have a function, the inverse function, by definition will be a reflection of the original function in the line $y=x$, so if I wanted to find the point of intersection, instead of solving it with equating both of the functions to equaling each other, could I assume that the point of intersection, between the two functions, will always be at the point $y=x$? This would enable me to solve problems much easier, instead of having to solve quartic equations, for example.