Problem:  If $n&gt;4$ and is composite, then $n$ divides $(n-1)!$.   My Attempt: I have tried to solve this problem using the approach described by @Michael Hardy but have not succeeded as of now. I wish to know how one can solve this problem using his  approach. For reference, Michael's answer is given below:      Suppose $p$ is prime and $p\mid n$.  Let $k$ be the multiplicity of $p$ as a factor of $n$, so that $p^k\mid n$.   $$ \underbrace{(n-1)! = 1\cdot2\cdot3\cdot4\cdot5\cdots\cdots p\cdots\cdots p \cdots\cdots p\cdots\cdots(n-1)}_{\text{Does $p$ show up at least $k$ times here?}} $$If $p$ appears at least $k$ times in the usual factorization of $(n-1)!$ displayed above, then $p^k\mid(n-1)!$.  If this works for every prime factor of $n$, then they're all there, so $n\mid(n-1)!\,{}$.   I guess, the question really boils down to proving the fact that the prime number $p$ shows up at least $k$ times in $(n-1)!$.
I have a proof and need some feedback. It seems really obvious that the statement is true but it is always the obvious ones that are a little trickier to prove. So I would appreciate any feedback. Thank you!  Here is what I am asked to prove:     If $n$ is composite then $(n-1)! \equiv 0 \pmod n$.   Proof:  $n$ is composite $\implies n=ab$ where $a,b \in \mathbb{Z}$ and $0&lt;a,b&lt;n$.  Case 1: If $a=b$ then $n=a^{2}$. Now $n \mid (n-1)! \implies a \mid (n-1)!$, so $$\begin{aligned} (n-1)! &amp;\equiv 1\times 2\times \dotsb \times a \times\dotsb\times (n-a)\times\dotsb\times (n-1) \\ &amp;\equiv 1\times 2\times \dotsb\times a \times\dotsb\times -a\times\dotsb\times -1 \\ &amp;\equiv 0 \pmod n \end{aligned}$$  Case 2: $0&lt;a&lt;b&lt;n$.  Then, since $a \mid n$, $b \mid n$ and $n \mid (n-1)!$ we have that $a \mid (n-1)!$ and $b \mid (n-1)!$.  So this implies $(n-1)! \equiv 1\times 2\times \dotsb\times a \times\dotsb\times b\times\dotsb\times (n-1) \equiv 0 \pmod n$, Q.E.D.
How to splice together several instances of InterpolatingFunction? So I was thinking about something for the past while  Consider a large spherical foam-ball with homogeneous density. Where a foam ball is defined as an object that can absorb matter with 0 friction (example: a gravitational well without an object inside). This is a purely theoretical construct.  If the foam-ball has a radius R and a charge Q. What charge must the foam-ball have, such that there is well defined sphere or horizon such that any object with a negative charge (even if it is the slightest bit) must travel faster than C in order to escape the field of the ball.  Aka what charge would turn this into a black-hole for all objects with opposite sign on their charge?  Once again I assume the foam-ball is a single particle, that doesn't repel itself... its just a very large homogenous 'thing' with charge. What is the unisex form of a word like fisherman? Do you have to use fisherman and fisherwoman separately, or is fisherperson acceptable? I couldn’t find a dictionary with the word …  In general, what do you do when a word does not have a unisex form? 1) Alex's house      2) Alex' house     When the noun ends with the letter 's' or 'x', do I need to put 's' after an apostrophe or not?  I remember I read some rules related to this in my school grammar book, but now I've forgotten it. Does it look good to edit the question you're answering? I have a question about editing practices:  Does it look good to edit a question you have just answered?  For example: say I had just answered a question, then saw that the question had some minor formatting issues. Should I edit it? Does it turn up voters off when they see that I am answering a question I changed, possibly to better suit my answer? In a table, how can I draw a horizontal line that goes only through some of the cells? I mean a line like the one in the Foo and in the Bar area of this table:  +----+-----+ |Foo |1 |2 | |    |1 |2 | |    +--+--+ |    |1 |2 | |    |1 |2 | +----+-----+ |Bar |1 |2 | |    |1 |2 | |    +--+--+ |    |1 |2 | |    |1 |2 | +----+--+--+ I have an iPhone 6 (with proper iTunes backup) where all of my contacts where, but I bought a new iPhone 7 and had to initialize it as new iPhone, due to several time-related constraints, so now I have a few but new contacts on my new phone, but none of the older ones, but would be the best practice oriented, for just recovering the old contacts from the iTunes backups?  PS: I still have the old iPhone 6, so it could be used as "temporary storage" if needed. Salesforce query on profile not working on user language change I have a query on profile like  profilename = [Select name from profile where id=:userinfo.getUserprofileid ].name  Now I have a check in  VF page  like  profilename  == 'System Administrator' . this works fine if the logged in user language is english. If the user language is different then profile name will be  'System admin' in that language . So how to prevent this beahviour?   Also  is there any particular field in profile which I can use  to identify the profile is system admin? so that I can have check on that field instead of checking for name which is causing the problem, for example I think modifyall filed is only checked for system admin. Duplicate id in list My service is working fine for single record insert when i send bulk records at that time it showing soapenv:ClientSystem.ListException: Duplicate id in list: 01ti0000002gUP1AAM Class.Sample.behaviourOfWebService: line 235, column 1  the error line is Database.upsert(pList)     p.productCode = pCode;    p.name =pName;    pList.add(p);  } How to make HTML code inactive with comments More than 1000 gallons of paint is/are sold each day Possible Duplicate:             Can anyone help me determine the correct verb in this sentence? I am not sure what to do. If it were not such a complex introductory phrase, it would be more obvious. The general consensus of my friends who are not professional writers is that the verb should be are.   To me the question is whether or not the subject is singular (i.e., “a large collective volume of paint”, perhaps in a tank) or plural (a lot of the individual gallon containers of paint).  If simplified to other options, it would be like these:   Paint is sold. Gallons of paint are sold. More paint is sold. More gallons of paint are sold. More than 1000 gallons of paint is sold. [emphasis on total volume] More than 1000 gallons of paint are sold. [emphasis on individual containers] What Wording Did Witches and Wizards in Other Cultures Use for Spells? The spells that we know about in the Harry Potter world all seem to be either in Latin, derived from Latin, or resembling Latin.  This makes sense for the European World, but there were likely witches and wizards in Africa, Asia, and the Americas.  These cultures would have developed independently, and the magical people in them would likely have still found ways to cast their spells.  Do cultures with languages that did not derive from Latin use the same words and phrases that are related to Latin for their spells?  Or did they have other phrases based on the languages in their cultures? Can not move polygons but only vertices of an object by bmesh! I tried control+shift+power button as well as command+shift+power button. Neither seem to work. Bash while read loop breaking early I've been banging my head against for wall for a while with this one.  I want to SSH into a set of machines and check whether they are available (accepting connections and not being used). I have created a small script, tssh, which does just that:  #!/bin/bash  host=$1 timeout=${2:-1}  ssh -qo "ConnectTimeout $timeout" $host "[ \`who | cut -f1 | wc -l \` -eq 0 ] &amp;&amp; exit 0 || exit 1"   This script works correctly. Returning 255 if there was a connection problem, 1 if the machine is busy and 0 if everything is good. If anyone knows a better way to do this please let me know.  So next I try and call tssh on my set of machines using a while read loop, and this is where it all goes wrong. The loop exits as soon as tssh returns 0 and never completes the full set.   while read nu ; do tssh "MYBOXES$nu" ; done &lt; &lt;(ruby -e '(0..20).each { |i| puts i }')   At first I thought this was a subshell problem but apparently not. Any help, along with comments on style/content, would be much appreciated! I know I'm going to kick myself when I find out why... What is "my guy syndrome" and how do I handle it? Select Permissions Across Multiple Schemas Within the Same Database I have been asked by a client to implement some views for reporting purposes, the views will be accessed via PowerBI, Excel and SSRS. The designated user will only have access to the view and no underlying tables must be available to the user.  The problem I have encountered in that the SQL within the view touches 3 different schemas (All within the same database):   Pupil Provider  Security   The view is:  CREATE VIEW dbo.vTestPermissions AS SELECT a.Column1,        b.Column1,        c.Column1 FROM Pupil.Table1 a JOIN Provider.Table2 b ON a.Column1 = b.Column1 JOIN Security.Table3 c ON a.Column1 = c.Column1   The owners of the Tables/View are as follows:   Pupil.Table1 - Owner Pupil Provider.Table2 - Owner Provider Security.Table3 - Owner Security vTestPermissions - Owner dbo   When I select from the view I get an error:     The SELECT permission was denied on the object 'table3', database   'TEST', schema 'Security'   I have tried giving SELECT permission with and without the GRANT option to the schema, and the tables, but that makes the underlying tables available to the user.  Any help on this will be greatly appreciated. Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$ Let $F$ be the field $\Bbb Q(\alpha_1,...,\alpha_k)$ where  $\alpha_i^2\in\Bbb Q\ \forall i$  I want to show that $\sqrt[3]{2}\notin F$  I thought if $\sqrt[3]{2}\in F$ then $\sqrt[3]{2}\in\Bbb Q\cup(\alpha_1,...,\alpha_k)$ but this is not a disjoint union so I'm not sure what to do... taking the square of $\beta:=\sqrt[3]{2}$ gives $2^{2/3}=\sqrt[3]{4}\notin\Bbb Q$ but that doesn't really advance things  So I'm looking for a hint Can the Eclipse Java formatter be used stand-alone Is there a way to use the formatter that comes with eclipse, outside of eclipse?  I would like to format some java files using my formatter.xml file that I have configured using eclipse.  Does anyone have any code examples that would allow me to do this?  I would also like to use this standalone, so the specific jars that are used would be nice. MCP4131: How to know what value resistors to use on the multiplexed SPI IO? There are two aspects here. The first question is related to R1 below:    The datasheet says: "R1 must be sized to ensure VIL and VIH of the devices are met" but gives no formula. So first question:  (1) How does one determine value of R1?  The datasheet gives maximum SPI SCK frequency for this device as 10MHz for 5V supply as given in Tables 1-1 and 1-2. However, for the read operation on the device where the SDI/SDO are multiplexed, the idea is different because:    The point 2 under the Table above says that the max SCK can be increased by using a pull up resistor. About these things I am wondering the following:  (2) Why is max SCK frequency lower for multiplexed SDI/SDO read operation only? (3) If I connect a pull-up resistor it will certainly form a voltage divider with the R1 in the first picture. How do I determine the value of both of these resistors then?  Finally, since the Table 1-1 and 1-2 give SCK Input Frequency for voltage 1.8V to 2.7V as 1MHz rather than 10MHz which applies to 2.7V to 5.5V.   (4) Why is the max SCK lowered at lower supply voltage?