I am having difficulty in solving following types of problem:  Sometimes we are given a number in terms of $n$ and we have to check whether it is divisible by a particular composite number.  For example, I am posting a question here      suppose $k= n^5- n$ then prove that $k$ is divisible by $30$.   And this was my approach:      $$n^5-n= n (n^4-1) =n(n^2+1) (n+1) (n-1)$$   Since $k$ is a product of $n^2+1$ and three consecutive integers, it must be a multiple of $2$ and $3$. So it gives $k= 6m (n^2+1)$. But how can I prove that it's also a multiple of 5? And this is where I  get confused.   Now, suggest some alternate way to prove above problem or some corrections in my method.
How can I prove that prove $n^5 - n$ is divisible by $30$? I took $n^5 - n$ and got $n(n-1)(n+1)(n^2+1)$ Now, $n(n-1)(n+1)$ is divisible by $6$. Next I need to show that $n(n-1)(n+1)(n^2+1)$ is divisible by $5$. My guess is using Fermat's little theorem but I don't know how.
Activating 'missing' standard GRASS utilities in QGIS Processing toolbox Random test data: How to do it right? If I roll two fair dice, the probability that I would get at least one 6 would be.... What's the difference between mobile and desktop processors? Sign of Work and potential energy in electrostatics Conceptual question:  Suppose we have a configuration of point charges. If the potential of the energy of the system is negative, this means work is positive.   I'm kind of rusty with my mechanics, does the fact that the work is positive mean that to arrange the configuration you have to do work against the electric field? I.e. supply energy via some external force?  In terms of electrostatics, is there a general way of interpreting the sign of potential energy and work? Admob SMART_BANNER size I want to better understand why the construction something + [adjective] makes grammatical sense. Background: I was recently reading a Washington Post article and came across the following sentence.  So if you fall in that category, [the movie] feels more like magic than repetitive.  (Note: I've copied the relevant spoiler-free sentence above; following the link to the article will open you up to movie spoilers! ) The phrase more like magic than repetitive struck me as odd, presumably because it was non-parallel (more like [noun] than [adjective]). I would have expected &quot;...more like magic than something repetitive.&quot; (Actually, that needs to be rewritten to flow better, but the point is that at least it appears to be grammatical.) Thus, one would have the noun phrase &quot;something repetitive,&quot; of the form &quot;[noun] [adjective].&quot; This got me to thinking about the use of adjectives after nouns. There is a great Stack Exchange article  that involves the characteristics of the adjective that indicate it should come after the noun. But I haven't been able to find examples of how the word &quot;something&quot; requires a postponed adjective. Consider the following sentences:  I want to buy something red. I am looking for something sweet to eat.  You cannot say:  *I want to buy red something. *I am looking for sweet something to eat.  But you can say:  I want to buy something (that is) red. I am looking for something (that is) sweet to eat.  So, my question:  Do the above examples mean that the &quot;something [adjective]&quot; construction is an example of the ellipsis of &quot;that is&quot;?  Is there something special about the word &quot;something&quot; that requires the postponed adjective? Note that I am NOT looking for details about adjectives that are postponed; I'm interested in the word that precedes the adjective (either &quot;something&quot; or similar words).   By the way, I am not interested in the following constructions, which are different:  I want to paint the house red. I tried to build the brick house taller.  Thanks in advance! Ubuntu server shows me that I  use allmost all disk:  Usage of /:   95.5% of 118.12GB   And I try to find big folders and files, run ncdu:  ncdu 1.8 ~ Use the arrow keys to navigate, press ? for help                                                                                                                                                  --- / ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------     5.5GiB [##########] /root                                                                                                                                                                                    2.3GiB [####      ] /var   628.6MiB [#         ] /usr   209.9MiB [          ] /lib    28.2MiB [          ] /boot     8.6MiB [          ] /bin     7.7MiB [          ] /sbin     6.6MiB [          ] /etc   208.0KiB [          ] /run   112.0KiB [          ] /tmp    48.0KiB [          ] /opt e  16.0KiB [          ] /lost+found     8.0KiB [          ] /dev     8.0KiB [          ] /media     4.0KiB [          ] /lib64 e   4.0KiB [          ] /srv e   4.0KiB [          ] /selinux e   4.0KiB [          ] /mnt e   4.0KiB [          ] /home     0.0  B [          ] /proc     0.0  B [          ] /sys @   0.0  B [          ]  initrd.img @   0.0  B [          ]  vmlinuz   According to ncdu I use about 10 GiB of 128 GiB - it is about 10 %. Contradiction.  How to clean my ubutntu server without rebooting?  I thought that ncdu lies and used another apps to find big files and folders. All of them shows the same result as ncdu.  And df -h command shows that disk is full.  # df -h Filesystem      Size  Used Avail Use% Mounted on /dev/xvda       119G  113G     0 100% / udev            2.0G  8.0K  2.0G   1% /dev tmpfs           788M  212K  788M   1% /run none            5.0M     0  5.0M   0% /run/lock none            2.0G     0  2.0G   0% /run/shm   Update  sudo du -sch /* result:  /# sudo du -sch /* 8.7M    /bin 29M /boot 8.0K    /dev 6.6M    /etc 4.0K    /home 0   /initrd.img 210M    /lib 4.0K    /lib64 16K /lost+found 8.0K    /media 4.0K    /mnt 48K /opt du: cannot access `/proc/4470/task/4470/fd/4': No such file or directory du: cannot access `/proc/4470/task/4470/fdinfo/4': No such file or directory du: cannot access `/proc/4470/fd/4': No such file or directory du: cannot access `/proc/4470/fdinfo/4': No such file or directory 0   /proc 5.0G    /root 212K    /run 7.8M    /sbin 4.0K    /selinux 4.0K    /srv 0   /sys 112K    /tmp 629M    /usr 2.3G    /var 0   /vmlinuz 8.1G    total   8.1G  total as usual. But I see cannot access rows, maybe problem because of them.  Then I checked the biggest folder in /. It is /root :  /# sudo du -sch /root/* 96K /root/Downloads 2.5G    /root/Dropbox 36K /root/nohup.out 4.0K    /root/npm-debug.log 4.0K    /root/readonly 980K    /root/redis-2.6.16.tar.gz 228M    /root/tmp 2.7G    total The series $\sum_{n=2}^{\infty }\frac{1}{\log (n!)}$ I'm trying to find out whether this Series $\sum_{n=2}^{\infty } a_{n}$ converges or not when $$a_{n}=\frac{1}{\log (n!)}$$  I tried couple of methods, among them: d'Alembert $\frac{a_{n+1}}{a_{n}}$,   Cauchy condensation test $\sum_{n=2}^{\infty } 2^{n}a_{2^n}$, and they both didn't work for me.  Edit: I can't use stirling, and integral.  Thank you Visiting Russia as an Arab. Am I going to be hassled? I am teaching myself information about groups, but don't really understand how to work through this problem. Here is what I have been thinking so far (please note that I do not need to work through showing any closure): $\mathbb{Z}^*$ denotes the nonzero integers. Prove or disprove: $(\mathbb{Z}^*, \cdot)$ and/or $(\mathbb{Z}^*, \div)$ is a group.  For $(\mathbb{Z}^*, \cdot)$:  Group. Associative Law: Let $a,b,c \in \mathbb{Z}^*$. Then $(a\cdot b)\cdot c=a\cdot (b \cdot c)$, and the associative law holds. Existence of Identity: Let $a \in \mathbb{Z}^*$, then $a \cdot 1 = 1 \cdot a = a$, and the identity holds. Existence of Inverses: Let $a \in \mathbb{Z}^*$. Then the inverse of $a^{-1}=-a$, and the inverse holds. Note that this only works because we are ommitting zero.  Hopefully this looks like I am on the right track. I don't know to what extent I can explain logic vs showing it with mathematical operations, so I would appreciate guidance in this respect.  For $(\mathbb{Z}^*, \div)$: Not a group. Disprove by counterexample.  Yeah, I dunno about this one. I don't think it's true, because we will be introduction rational expressions that won't allow this to hold, but I'm not sure where I can use this information to disprove the statement. My Ubuntu 14.04 is stuck when trying to login my desktop. The screen goes black and soon after that the login screen comes back.   I followed all steps in  but the issue was not solved. Changing content of footline in beamer How is it possible to set a specific text in the footline of the slides in beamer? In my case, author names and the title are shown. Instead I'd like to show the conference name and country.  \documentclass{beamer} \usetheme{CambridgeUS}  \author{Elena Piper, Amina Shahid, April Fiorese} \title{Evaluating the Change of Performance Metrics for Key ABS}  \begin{document} \maketitle  \end{document} Unidentified address in Gmail Many times, when I copy&amp;paste an email address into the "To" field in a Gmail message, it refuses to send the email, saying that “the address has not been identified”. This is very frustrating. Is there any solution other than just manually typing the address? A triangle with vertices on the sides of a square, with one at a midpoint, cannot be equilateral Why is asymmetric cryptography bad for huge data? I've been told that asymmetric cryptography requires that the message to be encrypted be smaller than its key length. Why is this?  I know about hybrid encryption, which . But I still want to know why public-key cryptography needs the data to be shorter than the key length. Activate physics by keyframe I'd like to have a group of debris fall from a ceiling onto the floor, but not have them drop all at the same time. Is there a way to set a keyframe that dictates when the object enters the simulation? For example, object 1 starts falling on frame 0, object 2 on frame 5, object 3 on frame 24, etc. What is the lifespan of an SSD drive? Are SSDs comparable to HDDs for reliability and is there published research on this?  Also are all SSD manufacturers at a similar 'reliability level'? When I need to generate unique id's based on some information hashing is typical choice. However, sometimes that id needs to be of a particular size.  I've seen a lot of schemes (HMAC-MD5-96 in SSH, CGA in SeND for IPv6) that use a portion of a hash, so I'm thinking it might be alright to use it that way.    Which properties of a hash are preserved and which do not apply anymore?  Obviously, with less bits, the chance of collision goes up, but can I still rely on the partial hash for uniform character distribution?  What about the ?  I'm guessing that if a hash modifies the entire hash upon the smallest change to the input, that would imply that a portion of that hash would be equally well changed.  Am I thinking about this correctly?  Also, would these behaviors be different for different hashing algorithms? My 5 year- old daughter is constantly defiant! My 5-year old daughter is very defiant.  It started getting bad about 6 months ago. She doesn't do what she is told, and any time she hears the word "no" she throws terrible, dramatic fits, screaming ridiculously loud, and in some cases hitting.  She refuses to make eye contact when I warn her to stop doing something even when I am down at her eye level and it takes multiple orders to finally get her to look at me. She constantly tells me no.  I feel like I constantly tell her no and that she feels this gives her the right to tell her father and I no.  I have tried time outs which are not effective at all; she only throws a fit the whole time, never seeming to run out of wind. Spankings are only effective until the shock of the action itself wears off and then she returns to the same behavior.  My husband and I recently took away ALL of her toys/ possessions being she is a pretty materialistic child (she loves getting toys and always wants something new). We told her that if she was good and listened and stopped telling us no and throwing fits, that after three days we would let her choose three boxes of toys to bring back in her room. This worked but as soon as she got those three boxes unpacked, we packed them back up because she went right back to the no's and fit-throwing within a few hours.  I am mainly the one disciplining her and I feel she is definitely the most defiant with me. I believe in explaining to her why things are not OK and why I tell her no, but I really don't feel it helps and I never feel as if she listens to what I tell her.  I am at my wits' end and feel like a failure of a mother. She is my first and only child, and the only grandchild on both sides of our family. She is an angel at school but home is completely different.  I need help figuring this out but nobody I talk to really has any advice for me. I am at the point where I am wondering if she or I need counseling because I feel out of options.