I wonder about an equation given as a solution to the following task:     Calculate the multiplicative inverse $$5^{−1} \pmod {13}$$   The solution ends in this equation: $$     5^{11} \equiv 5^{10} \cdot 5 \equiv −1\cdot 5 \equiv 8 \pmod {13} $$ and the check:     $$     5 \cdot 8 \equiv 40 \equiv 1 \pmod{13} $$      1) What is the inverse here now?      2) Why is $5^{10} \equiv -1 \pmod {13}$?   As far as I understood Fermat the rule is given as: $$     x^{(p−1)} \equiv 1 \pmod p $$ which won't come into account here, since in this case  $p = 13$ and not ideal for calculating $5^{11}$.  If I would use $x^{p−1}$ it would be $5^{12}$ in this case, so I'd still need to divide by $5^1$ to get to the result of $5^{11}$.  I am so confused although it should be really easy.
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.
How to make a logo where a letter is made of particles of varying sizes? How can I make a logo in Adobe Illustrator like the AntiParticle logo below, from ?   It is a large number circles of different, varying sizes, which form the shape of a text character. Can you enter UK later than intended date of travel on the visa application? Restricting a user to a certain directory on SSH login I am having trouble restricting a user to a fixed directory when they connect with SSH  I want the user restricted to /var/www/home  however I connected myself and they are free to move about the system  I set the home directory to /var/www/home so they go there upon login however this doesnt restrict them.  I have tried a few things from suggestions in Google but nothing so far seems to work / do what I want  For example I have changed the chmod of www to 755 and root owned but still no effect I want to upgrade my disk from MBR to GPT, and I tried using Windows' MBR2GPT.EXE tool to do the job. I used Windows' DISKPART tool to identify the disk I want to convert. However, this happened:  C:\WINDOWS\system32&gt;MBR2GPT.EXE /convert /disk:0 /allowfullos  MBR2GPT will now attempt to convert disk 0. If conversion is successful the disk can only be booted in GPT mode. These changes cannot be undone!  MBR2GPT: Attempting to convert disk 0 MBR2GPT: Retrieving layout of disk MBR2GPT: Validating layout, disk sector size is: 512 bytes Disk layout validation failed for disk 0 MBR2GPT: Conversion failed   and the conversion failed. Using Diskpart again, I get the following output:  DISKPART&gt; list disk    Disk ###  Status         Size     Free     Dyn  Gpt   --------  -------------  -------  -------  ---  --- * Disk 0    Online          465 GB      0 B   Disk 1    Online         3841 MB   960 KB  DISKPART&gt; select disk 0  Disk 0 is now the selected disk.  DISKPART&gt; list partition    Partition ###  Type              Size     Offset   -------------  ----------------  -------  -------   Partition 1    Primary            100 MB  1024 KB   Partition 2    Primary            464 GB   101 MB   Partition 3    Recovery           813 MB   464 GB   Partition 4    Recovery           511 MB   465 GB   Looking at the partitions on disk 0, there are 2 primary partitions and 2 hidden ones, I read  that the partitions might have something to do with the conversion failing, so I believe this is the culprit. If anyone could offer help converting my disk, that would be greatly appreciated. I stupidly ran sudo chown -R carlos:carlos / and carlos is my user. This changed all files to be Carlos and I could not boot up again (because the root files were now owned by #1000) and I cant get sudo acess in live cd (because the owner of /usr was changed to my user). Can anyone help me? FYI I have mounted my hard drive. My question is a little silly I know, but I'm curious to know if a particle of light can be theoretically trapped between two reflective screens.   For example once the particle of light has left it's source, and is reflected against a mirror (assuming there is no loss of energy) and you immediately replace the source of light with another mirror. Would the light beam bounce in between the sheets endlessly? When will my device get the Android 5.1 update (Lollipop)? Android 5.1 (Lollipop) was officially announced on March 9th, 2015.  You can review the high-level changes at the announcement on the official .  Of course, not all devices will get this update right away and others won't get it at all. Each manufacturer and carrier often choose to add their custom modifications which tend to delay the official release.  When will each device get Android 5.1, either officially or unofficially, if ever?  See also: What is ubuntu 14.04 doing while showing "stopping early crypto disks"? I created a virtual machine to install ubuntu 14.04 on VirtualBox 4.2.12.*. I created virtual CD to install it with fixed-size virtual disk of 200GB. After successful installation, it restarted as usual. But after showing message of below, it hangs. Actually, I created virtual machines to install ubuntu at Virtual Box successfully several times. It's very easy. The installed ubuntu versions ranged from 9.10, 13.04 and even 14.04 before. I don't know why this time it hangs at restarting after successful installation. The only difference between this failure restart and previous successful ones is that the virtual disk is much larger, 200GB. It failed three times.  * Deactivating swap... [OK] * Stopping remaining crypto disks... [OK] * Stopping early drypto disks... [OK] Is the cube root of a prime number rational? The vote fraud script breaks legitimate polling questions - A serial downvoter's lament I've downvoted several of the answers on these Meta discussions:       Both questions are typical self evaluations, Stack Exchange runs them on all young Beta sites and as you may have noticed they are essentially polls, a series of example questions from the site that we are supposed to vote on.  Today I noticed that almost all of my votes have disappeared on both evaluations, only an upvote remains in the later one. My votes triggered the serial voting script, something that :      Heh, it appears I just serially downvoted you (I downvoted a lot of the answers here), hopefully the script is smart enough to not reverse my votes. Will check back in 24h, and if it does, I'll rant about it on MSO ;)    My voting behaviour on both evaluations was very simple:   I had the evaluation open in the left monitor, Opened all the example questions in their own tab, in the right monitor, Scanned each question quickly, and voted on the Meta answer (most of my votes were downvotes), Repeat for the next one, and so on...   I didn't spend more than a few seconds on any question, as I'm fairly regular on the site and I've already read all of them while casually browsing the site. In fact, if my memory isn't failing me, I had voted (up/down) and/or commented, and/or voted to close on each of the 20 example questions. I might have spend a minute or two re-reading a couple, but I mostly scanned through them. In any case I think my voting pattern is quite normal, and the script should be somehow tweaked to take serial posts, posted by the same user in quick succession, into consideration, especially on Metas.   Since I had anticipated that the script would kick in, I could have tried to fool it by pausing a bit between votes or trying to look for other Meta posts to vote on, in between voting on the evaluation answers. However I'm a very active voter on Meta Workplace, there aren't many posts I haven't already voted on, and the few that I haven't is probably because I neither agree or disagree with (my opinion on them is simply "meh"). In any case I think my voting behaviour was natural, and although the serial voting script is awesome, this time it caught and reversed a perfectly legitimate behaviour.  Animuson, in , mentions that the script isn't really optimal on Area51 as well:      Yes, please implement this on Area 51 proposals too! Oftentimes a single user would suggest a plethora of "example" questions to their own proposal which are just terrible, yet all of everyone's votes get reversed and the question just sits there at 0. I've noticed several times that my votes have gotten reversed.    I haven't noticed it myself, but it certainly seems plausible, especially early in a proposal's life.  Mysticial found a related discussion, , but it's focused on voting patterns on main sites, where reputation is also a factor. I can understand jumping through hoops to satisfy the script on main sites where, unfortunately, there are a lot of people trying to get some cheap rep or "punish" others with their downvotes, but not on Metas, and especially not on these evaluation posts.   Voting fraud is a problem, and more often than not the script works wonderfully, but in my case it was extremely counterproductive. Critical evaluations are essential for a young Beta site, and their answers typically only get a handful of votes. I wonder how many of these votes have silently gone away...  Fix this please. I have two directories that should contain the same files and have the same directory structure.  I think that something is missing in one of these directories.  Using the bash shell, is there a way to compare my directories and see if one of them is missing files that are present in the other? Open-source alternative to Adobe InDesign There is some open-source software, similar to Adobe InDesign? It would be awesome if it do exist...  I know all the others like Inkscape, Gimp, etc... but for InDesign i have none.  Could you give me one direction? Why is propagation of uncertainties quadratic rather than linear? How can I execute a file without giving myself execute permissions (with chmod u+x) for it?   If I try and tick the 'Allow executing file as program' checkmark, the checkmark is immediately removed. What's the best way of replacing a plug or switch in a house with aluminum wiring? Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero. Let $f$ be an entire function. Suppose $f(z)$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero. How to prove? Totally I have no idea... Please give the solution in detail since I do not understand. Third grade spiral in TikZ - dimension too large I'm trying to draw a smooth third grade spiral (r^3 = a^3\cdot \varphi) over $60\times 60$mm grid and I'm getting the "Dimension too large"-error if I set $a &gt; 320$ but I need to fill the grid area completely:  \begin{tikzpicture}[x=1mm, y=1mm]      \draw[step=1, line width=0.1mm, opacity=0.3] (-30, -30) grid (30, 30);      \draw [domain=0:50, variable=\t, samples=1000]          plot ({\t r}:{pow(320*\t, 1/3)}); \end{tikzpicture}     Is there any way to avoid this error? DataView.Sort - more than just asc/desc (need custom sort) How to add a script to Startup Applications from the command line? I have a shell script. I want to configure my script to run automatically during startup. I know how to do this from GUI. But I wanted to do this from terminal. How can I do this?  On a primary research I found that the file needs to be moved to /etc/int.d/ directory. But this operation needs sudo permission. I wanted to do this without super user permissions.   I also found that there are files under ~/.config/autostart/ which are having some settings regarding the startup applications. But I don't know how to edit them to achieve this.   Can someone tell me the exact procedure to achieve this? Explain reflection laws at the atomic level The "equal angles" law of refection on a flat mirror is a macroscopic phenomenon. To put it in anthropomorphic terms, how do individual photons know the orientation of the mirror so as to bounce off in the correct direction?