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Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them.   Let $a$ be the primitive root then I know other primitive roots will be among $\{a,a^2,a^3 \cdots\cdots a^{\phi(n)} \}$ because any other number will be congruent modulo $n$ to one of these. Then I figured as the answer is $\phi(\phi(n))$ so for $a^k$ to be primitive root, $k$ must be coprime with $\phi(n)$.But I don't know the reason.Am I missing any silly point? Maximum area of isosceles triangle with perimeter 6 cm Problem. What is the side lengths of an isosceles triangle with $6cm$ perimeter which has the largest area? I solved this problem as below and I would like to learn  that if there is anything wrong: Let us say that the side lengths are $a$ and the bottom length is $6-2a$. By using pythagorean theorem we can conclude that the height $h$ is equal to $\sqrt{6a-9}$. Then, the area of the triangle can be given as $f(a)=(3-a)(\sqrt{6a-9})$. So $f'(a)=-\sqrt{6a-9}+\cfrac{3}{\sqrt{6a-9}}$. Finally, $f'(a)=\cfrac{12-6a}{\sqrt{6a-9}}$ and $f'(a)=0 \iff a =2$. How to achieve this smudge effect in photoshop? I came across this photo by Jacob Sutton and wanted to know if someone knew how to get this effect in photoshop?    (source: ) Can I have two Schengen visas in one passport? Can "firstly" be used in the same meaning as "at first/initially"? I've come across many sources claiming that the words firstly and first (as an adverb) can be used interchangeably. They, however, only seemed to be comparing their meanings with regard to enumerations - first(ly) I want to say ..., second(ly) ..., last(ly) ...  What bothers me more is if the word firstly is also usable within the meaning of at first or initially like in the sentence:     Muons (subatomic particles) were firstly observed in cosmic rays.   The word first fits there very well, but is that the case for firstly too? I have not found a dictionary definition or example saying so. What is the relation between linear purity and von Neumann entropy of a state? Given a density matrix $\rho$, its linear purity is $\mathrm{Tr} \rho^2$, its von Neumann entropy is $-\mathrm{Tr} \rho \log \rho$. Knowing one how do I calculate the other?  Edit:   My thought is that the linear purity and the von Neumann entropy are a parallel set of benchmarks for mixedness. Their inter-conversion should be like unit conversion.  See it this way, as a state goes from a pure state to a maximally mixed state, $\mathrm{Tr} \rho^2$ goes from $1$ to $0$, and $-\mathrm{Tr} \rho \log \rho$ goes from $0$ to $\log d$, like two rulers. A point on one should correspond to a point on the other. How can I copy files from A to B, only including files that exist in B? I'm on Windows 7 Professional, and this is my scenario :  Folder "Asd" (C:\)              Folder "Asd" (D:\) File 1                          File 1  File 2                          File 3  File 3                          File 4 File 4 File 5   I'd like to select the folder on C:\ (without selecting each single file, of course) and copy it over the same folder on D:. Usually, it will copy all files. But what I'm looking for is to copy only files (from C:) that are contained on folder d:.  So in my case I'd like to replace only File 1, File 3 and File 4.  File 2 and File 5 must be ignored.  Is it possible? What are the most downvoted questions and answers on Stack Overflow, Server Fault, Meta, and Super User history to date?  Similarly, what are the most upvoted questions and answers on SO, SF, Meta, and SU to date? iPhone Apps in Waiting mode for the last 4 days I tried to download the "Dictionary Free" app from the App Store four days ago, but it's been stuck in "Waiting" ever since. I've tried everything: rebooting the phone, restoring from an iTunes backup, side-loading the app from iTunes, but nothing has worked.  When I turn Airplane Mode on and then off again, I get an alert popup that says "Unable to download application. It will be available for download when you log in to the iTunes Store on your computer".  I've tried what it recommends, but nothing seems to work. When should one use "should" and when should one use "must"? I tend to use should when it's a suggestion I don't have a strong opinion on, i.e. it could be done in many other ways than the one I'm suggesting and it can still happen.     You should stop by that gas station to go to the bathroom.   On the other hand, I usually use must for things that have to happen in a certain way; any other way would make that thing not to happen. The problem is that some people find this usage harsh, as they receive it like if I'm giving them an order.      You must cross the street to get to that store.   To me, it's just a depiction of reality, not an order. But I'm neither a native speaker nor a regular person. :-)  So, what's the proper use for must and should? Does this use differ among the English dialects? Repeatedly rolling a six sided die four times and summing the highest three results gives you a distribution with what mean and standard deviation?  I've only taken AP statistics, but I would like to learn how to do this. Let $r$ be the smallest positive integer such that $a^r \equiv 1 \bmod{n}$. Prove $r \mid \phi(n)$. Let $a \in \mathbb {Z}$, $n &gt; 1$ a natural number with $\gcd(a, n) = 1$, and let $r$ be the smallest positive integer such that $a^r \equiv 1 \bmod{n}$. Prove that $r \mid \phi(n)$.  Euler's generalization of Fermat's Little Theorem: If $n \in \mathbb {N}$ and $m \in \mathbb{Z}$ such that $\gcd(m, n) = 1$, then $m^{\phi(n)} \equiv 1\bmod(n)$.  By Euler's generalization I would have $a^{\phi(n)} \equiv 1\bmod{n}$.  Then I could say $a^r \equiv a^{\phi(n)} \equiv 1 \bmod{n}$. At this point I can see that $r$ clearly must be some multiple of $\phi(n)$.  If I want to show that $r \mid \phi(n)$, then I can show that $\phi(n) = rk$, for some $k \in \mathbb {Z}$.  However, I am a bit stuck about how to make this last step showing that  $\phi(n) = rk$, where $k \in \mathbb {Z}$. How to delineate watersheds in QGIS? I am using qgis 2.14.17 with ubuntu16. Having as input data a DEM of the region and a river network. In the manual of qgis they suggest to use a toll of qgis: Catchment area (Parallel); I dont have this in my toolbox, anybody made the same experience and could tell me the reason for it and how to get it. It seems to offer the best solution for me.  Tools which are available are from GRASS7 and TauDEM, the latter need much more data it seems. And I am not sure if I am able to create these just out of a DEM? The GRASS7 has one tool r.water.outlet: but here one has to put the outlet coordinates, which clearly I could get out of the river network, but the idea is that they are generated by delineating the watersheds.  Somebody has an idea how to solve it?  I managed now to get SAGA run in the toolbox, nevertheless would be great if somebody could help me with the qgis tool, since it should appear?! How to download Ubuntu Server backups automatically? I'm thinking of an investment in a new root server running ubuntu. Due to the fact, that the provider won't have extra backup-space, I need an other solution.  Is it possible to download the server backups automatically when my desktop machine is connected to the internet and in idle state? How do I move to a new tab in gedit using the keyboard? If I have multiple tabs open, how do I move from one to another? (Ubuntu 11.10; gedit 3.2.1) Flexslider view preview working but not on actual page My block is not working for the flexslider: The images are not appearing, but the view preview does work.  I don't know if it is relevant, but a similar problem would happen with just a slideshow view, although one image would appear and it would not cycle.  Could jQuery be a problem?  The following is the actual page with the block that is not showing any images.        The following is the view preview, which works. I have upgraded from Ubuntu 16.04 LTS to Ubuntu 18.04 LTS. I was surprised to see that the context menu entry "Make Link" (keyboard shortcut Ctrl+M) in Nautilus doesn't seem to exist anymore. It used to appear upon right mouse click on a file or folder and allowed the user to create a symbolic link. The Software Center has a section of For Purchase programs &mdash;see screenshot below&mdash;, which is annonying me every time I want to install something.  Is there any way to remove or hide these programs from the Software Center?     Links to answers:                     Please leave a comment or send a mail to jmendeth@gmail.com   if it's not working for you or you're having trouble! Find the infinite sum $\sum_{k=1}^{\infty}\sqrt k - 2\sqrt {k + 1} + \sqrt {k + 2} $ I am facing a problem, where I have to find the partial sum of a sequence/sum and with that, the infinite sum of the sequence.  $\sum_{k=1}^{\infty}\sqrt k - 2\sqrt {k + 1} + \sqrt {k + 2} $   The problem here is that I don't know how to proceed. I would be thankful if someone would steer me in the right direction. Can a monk make a bonus unarmed attack after using shocking grasp? I'm playing a 5e Wizard/ Monk build. My DM and I are questioning the viability of the cantrip Shocking Grasp. As a monk I can do another unarmed strike as a bonus action after I attack with a melee attack. Being a touch spell, can I punch (touch) with Shocking Grasp and then do my monk's bonus action?