Euler phi function proof Let $a,e$ and $f \in \Bbb{Z}^+$ and p is prime number. If $\phi(p^e) | f$ then show $a^f \equiv 1 \pmod{p^e}$ $(gcd(a,p^e)=1$
Mod of numbers with large exponents [modular order reduction] 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.
My MWE:  \documentclass{report}  \usepackage{amsmath}  \begin{document}  \begin{align} y &amp;= a \\ =&amp; -2c_1\bar{z}_1^2 + 2\bar{z}_1\bar{z}_2 + 2\bar{z}_2 \left ( -\bar{z}_1 -c_2z_2 - \frac{3\epsilon}{2}\bar{z}_2 -\sigma_{2,1}\cos(x_1)\Delta x_1 \notag \right. \\ \left. &amp;- \sigma_{2,2}\Delta x_2 -\sigma_{2,3} \Delta u +c_2 \chi_2 \vphantom{\frac{1}{1}} \right ) - 2\gamma \sigma_{2,1}^2\cos^2(x_1) \Delta x_1^2 - \gamma \sigma_{2,2}^2 \Delta x_2^2   - 2\gamma \sigma_{2,3}^2 \Delta u^2 \end{align}  \end{document}   The error I get:  Extra }, or forgotten \right.   The expected result: What to do when you find a bad link in an answer? If a link has gone stale in an answer, what is the correct way to report this in the SO universe?  e.g. Jon's first link in    I have added a comment but not found the correct link yet. Remove space around tikz How can I remove ALL space around TiKZ. So given MWE below, I want no space between fbox and the node box (so essentially I want to see 1 line). How to realize this?   Is this space in tikzpicture or in \fbox?   How to remove space between \fbox and node    MWE  \documentclass{article} \usepackage{tikz}  \begin{document} \fbox{%   \begin{tikzpicture}[node distance = 0.8cm and 0.2cm]%     \node[draw] (center) {CENTER}; %   \end{tikzpicture}% } \end{document} How do I find the address of last non-blank cell in a row (while skipping columns) Running out of inodes This winter I am planning to track my downhill skiing/snowboarding using a GPS.  Most of my riding will occur at the same resort.  I would like to be able to create a sort of "heatmap" that shows the amount of runs that I have made in a given area.  As I add more and more GPS traces to my database, my goal would be to see a sort of linear heatmap of the most traveled areas.  Given the nature of downhill skiing, you would expect the uphill chairlift lines will be the "hottest" because it will be the only places visited over-and-over again.  Given that 1) my track will not be the same every time and 2) the area covered by following one "run" may be a few hundred feet wide, what might be the best way to analyze this "linear" data to create a sort of heatmap?  My thoughts were to buffer the lines, then intersect the polys to get a sort of Venn Diagram thing going.  My preference is to use open-source technologies.  I've got QGIS and PostGIS loaded and available.  UPDATE: In regards to @blah238's response, I was thinking of something that might be able to "collect" the number of passes ("runs") through an area, and then symbolize by the count.  Conceptually, this would be similar to ArcGIS "" (but for lines, not points) or  (but for multiple line in roughly the same area).  A more visual example of a similar concept might be a traffic-flow map, where highly-congested areas would equate to "highly-traveled" ski runs/areas:    I've read the following questions that may give some ideas, but they don't really address what I am trying to accomplish: I realised, reading another Phys.SE question about  that I don't really understand how  works. Particularly concerning, for a SCUBA diver.  The top answers to that question seem to claim that balloons get their "sense of down" from a pressure differential. They continue: when a car accelerates, the air at the back of the car becomes more dense, and at the front less dense, changing the plane of the pressure differential and so also, the balloon's sense of up. I find that extremely hard to credit. However, I realised that I don't really know why less dense things float in more dense things.  I'm fairly sure it's something to do with displacement of heavier things by lighter things, and I think pressure acting on the lighter thing's surface has something to do with it, but that's about it. New messages flagged blue bubble does not go away When someone flags a message in chat all 10k+ users in chat get a little blue bubble to vote on. It shows up in the upper left hand corner of your gravatar (which is in the bottom left of the page).  When someone flags a message and it shows up, if I wait for a second to click on it, and then click on it, and it has already been handled, it shows the message  "There are no flags to display."  In the past, after showing this message, the blue (1) would disappear. However, lately it has not been going away and requires a page refresh in order to remove. Can this be fixed?  This issue is addressed here:  from September 12th 2012. But since it is still occurring 10 months later, I believe the cause is different. Why is this sentence correct? “She suggested that he go to the cinema.” Why is this sentence correct?     She suggested that he go to the cinema.   I would definitely use goes instead of go. Why don't the White Walkers take the sea route to Westeros Comment submission broken for comments that end with @username "Try and" versus "Try to"? Uninstall apps locked by phone vendor Running own executable on Android shell I need a specific command line tool and I have made a C program in my Linux shell. I have compiled the program with an ARM cross-compiler. I have then moved the program into the Android file system and tried to run it.  The output is permission denied.  What do I have to do, in order to run my own compiled programs in Android file system? PHP variable variables I want to store courseworks marks of n courseworks into n variables, such as cw1 and cw2 etc. Using variable variables how can I come with cw1, cw2 etc.   How can I dynamically create variables? Can't find wifi drivers for 0bda:f179 Realtek Semiconductor Corp I'm trying to install wifi drivers for a particular mini wifi card i bought. The output of lsusb is:  0bda:f179 Realtek Semiconductor Corp   I've searched everywhere for drivers, but could not find anything. Any help?  Thanks How can I disable Amazon search results in the Dash? How to prove that the ideal $I=(x^3-y^2)$ in $k[x,y]$ is prime?    I have constructed a map from $k[x,y]$ to $k[t]$, which maps $x$ to $t^2$, and $y$ to $t^3$. Then, I want to show that the kernel is exactly $I$. But the difficulty is that how to prove the kernel is contained in $I$.  Could you help me? Thanks a lot! I've created a script that runs every night on my Linux server that uses mysqldump to back up each of my MySQL databases to .sql files and packages them together as a compressed .tar file.  The next step I want to accomplish is to send that tar file through email to a remote email server for safekeeping.  I've been able to send the raw script in the body an email by piping the backup text file to mailx like so:  $ cat mysqldbbackup.sql | mailx backup@email.com   cat echoes the backup file's text which is piped into the mailx program with the recipient's email address passed as an argument.  While this accomplishes what I need, I think it could be one step better, Is there any way, using shell scripts or otherwise, to send the compressed .tar file to an outgoing email message as an attachment?  This would beat having to deal with very long email messages which contain header data and often have word-wrapping issues etc. I'd like to have Ubuntu installed on a USB stick, that I could plug in (almost) any computer and boot from it, and that I could use (almost) normally, as if it was installed on that computer. I'd also like to have a data partition on it that could be used by (almost) any computer, mostly to store some portable apps, just in case.    I've been told that if I installed Ubuntu in BIOS mode, I wouldn't be able to boot from a computer in UEFI mode, and vice versa.  Since I'd like to be able to boot on any computer, it's a big problem.  What can I do to be able to boot on computers in BIOS or UEFI modes in my case?  I've been searching for solutions but most of them are for specific needs that are different than mine, so it's difficult to extract the minimum requirements from those solutions to apply them to my problem.  Is there a simple, "clean" and quite generic solution to this problem?  I'd appreciate some explainations about the reasons why a specific solution is good or a specific step is needed.    My current setup:   USB stick:  OS: Xubuntu 16.04 i386 installation: full install (from a Live USB built with Rufus) drivers: none proprietary file systems:    partition table:    2: , /boot/efi, 128 Mio, FAT32, [boot,esp] 3: , 128 Mio, HFS+ 4: , 2 Mio, [bios_grub] 5: swap, 512 Mio 6: , /, 24 Gio, ext4, defaults,errors=remount-ro,lazytime,commit=60 1: , /media/data, ~3.6 Gio, FAT32, defaults,lazytime,check=strict  RAM:   /tmp, tmpfs, defaults,nosuid,nodev,noexec,mode=1777   swap:    vm.swappiness=5 swap partition: 512 Mio, for emergency zRAM swap files:    /var/swap/hib.swp: same size as RAM, for hibernation, generated by a script at startup /var/swap/opt.swp: if needed, generated by a script   overprovisioning: ~5.6%   built-in: ~1.2 Gio (apparently) added: 512 Mio  scheduler: deadline