Suppose we are given a number $a \in \mathbb{Z}$ prove that $42|a^7-a$.  I'm not too sure how to start any ideas?
I can see that this works for any integer $n$, but I can't figure out why this works, or why the number $42$ has this property.
How to understand degrees of freedom? From , there are three interpretations of the degrees of freedom of a statistic:     In statistics, the number of degrees of freedom is the number of   values in the final calculation of a statistic that are free to vary.      Estimates of statistical parameters can be based upon different   amounts of information or data. The number of independent pieces of   information that go into the estimate of a parameter is called the   degrees of freedom (df). In general, the degrees of freedom of an   estimate of a parameter is equal to the number of independent scores   that go into the estimate minus the number of parameters used as   intermediate steps in the estimation of the parameter itself (which,   in sample variance, is one, since the sample mean is the only   intermediate step).      Mathematically, degrees of freedom is the dimension of the domain of a   random vector, or essentially the number of 'free' components: how   many components need to be known before the vector is fully   determined.   The bold words are what I don't quite understand. If possible, some mathematical formulations will help clarify the concept.  Also do the three interpretations agree with each other? Do AWS Reserved Instances need to be assigned to the instance? I have an RDS instance running of type db.t2.small.  I bought a reserved instance for 1 year of the same type, I paid the partial upfront rate.  Do I need to assign this subscription in some way to my existing instance to say 'This is the instance that is using the subscription' or does AWS just charge for the instance usage depending which instances you have running and whether an available subscription was active for it at the time which determines the running rate?  The reason that I ask is that I can't see any mention of the reduced rate on my AWS Billing area and it shows that I've used 17 hours of my reservation, but there isn't a row on my bill so far this month that indicates the new partial upfront rate that I should be charged for my instance. I get this error when trying to use apt-get:  E: Could not get lock /var/lib/dpkg/lock - open (11 Resource temporarily unavailable) E: Unable to lock the administration directory (/var/lib/dpkg/) is another process using it?     How can I fix this? Generic solution to prevent a long cron job from running in parallel? I am looking for a simple and generic solution that would allow you to execute any script or application in crontab and prevent it from running twice.  The solution should be independent on the executed command.  I assume it should look like lock &amp;&amp; (command ; unlock) where lock will return false if there was another lock.  The second part would be like if it acquired the lock, run command and unlock after command is executed, even if it returns error. Directing Domain to Hosting Directory I just purchased a domain on Dreamhost, let's say foo.com. I already have hosting, let's say at 123.4.56.789, and have setup a directory foo.com which is accessible at 123.4.56.789/foo.com/. I can set the DNS to point foo.com to that IP, but how do I route it to that folder once it's there? My hosting has a really bad cPanel that gives me almost no help. Can I use .htaccess, or something else? What's the best way to accomplish this?  Edit in response to comment(s): It's VPS hosting, so I have access to http.conf and other system files. Also, yes it's Apache. How do I disable network manager permanently? I'm using Ubuntu 18.04.1 LTS. I want to disable network manager on an Ubuntu machine, because (1) I don't need it, (2) I prefer having hardcoded configuration, and (3) network manager regularly causes issues by changing the DHCP configuration. I tried to follow :  Stop network manager sudo systemctl stop NetworkManager.service  Disable network manager (permanently) to avoid it restarting after a reboot sudo systemctl disable NetworkManager.service   Despite this, the network manager is back again every time I reboot the machine. How can I make it go away? Formatting OGR VRT file to convert CSV to KML? I think that all the zeros of the Riemann-Zeta function ${\zeta}( z ) = \frac{1}{1-2^{1-z}} \sum_{n = 0}^{\infty} \frac{1}{2^{n+1}} \sum_{k = 0}^{n} (-1)^k \binom{n}{k} (k+1)^{-z}$  have real part smaller than 1, but I don't know how to prove it. If Re(z) > 1 than  ${\zeta}( z ) = \sum_{n = 1}^{\infty} \left( \frac{1}{n} \right)^z$ Let us denote by $\chi(G)$ the chromatic number, which is the smallest number of colours needed to colour the graph $G$ with $n$ vertices. Let $\bar{G}$ be the complement of $G$. Show that  (a) $\chi(G)+\chi(\bar{G}) \leq n+1$  (b) $\chi(G)\chi(\bar{G}) \geq n$  I was able to prove (a) using induction. Any hints on proving (b)? What's the best way to break from nested loops in Javascript?  //Write the links to the page. for (var x = 0; x &lt; Args.length; x++) {    for (var Heading in Navigation.Headings)    {       for (var Item in Navigation.Headings[Heading])       {          if (Args[x] == Navigation.Headings[Heading][Item].Name)          {             document.write("&lt;a href=\""                 + Navigation.Headings[Heading][Item].URL + "\"&gt;"                 + Navigation.Headings[Heading][Item].Name + "&lt;/a&gt; : ");             break; // &lt;---HERE, I need to break out of two loops.          }       }    } } Getting from Denver Amtrak to Fort Collins The eastbound Amtrak arrives Denver 6:38 PM (18:38).  How can I get to Fort Collins from there?   The  runs  from Denver International Airport to Fort Collins, but the airport is 40 km from the Amtrak station, so that's a major detour, that would mean taking bus FX from Market Street Station 7:18 PM, arriving at the airport at 8:27 PM, then the super shuttle 9:05 PM, arriving Fort Collins CSU at 10:55 PM; almost 4 hours after departing from Denver. Greyhound Lines appears to have a bus departing 12:45 AM, arriving 02:00 AM — although probably as quick as it gets, this is not exactly a comfortable time. RTD Denver has transportation to Longmont and  from Longmont to Fort Collins, but  appears to depart 7:05 PM — so I won't make it there when arriving by Amtrak.  A taxi would cost an estimated .   Is there any other way to get to Fort Collins when arriving by the eastbound Amtrak, any way that I've missed? I created some linestring features in EPSG:32632 (WGS 84 / UTM zone 32N) exactly 10 000 m long.  QGIS calculates all line lengths (horizontal, vertical, diagonal) as 10 001.9 m. The measure tool explains "Project CRS transformation is turned on and ellipsoidal calculation is selected. The coordinates are transformed to the chosen ellipsoid (WGS84), and the distance is calculated in meters."  When I set the ellipsoid to "None / Planimetric" in project properties, I get the expected results (10 000 m).  I don't understand the phrase "coordinates are transformed to the chosen ellipsoid (WGS84)", is it a transformation to unprojected degrees (4326?)? Since 32632 already is based on WGS84, is there anything to transform? If there is a kind of "great circle" calculation involved, I would have expected lengths can only get smaller.  Is QGIS's calculation correct/meaningful, is it plain wrong or do I see rounding errors?  This are the geometries I tested:  LineString (370000 5615000, 370000 5625000) LineString (366464.46609406732022762 5616464.46609406732022762, 373535.53390593267977238 5623535.53390593267977238) LineString (365000 5620000, 375000 5620000) LineString (373535.53390593267977238 5616464.46609406638890505, 366464.46609406638890505 5623535.5339059317484498)   Edit as addition to MappaGnosis's answer: I forgot that in UTM , this seems to be the simple answer for the title question. Find the maximum value of $ f(x) = \int^{x}_{0} \sqrt{(x^2-x)^2+y^4}~dy$ Find the maximum value of $f(x)$ when $0 \leq x \leq 1$ and $$ f(x) = \int^{x}_{0} \sqrt{(x^2-x)^2+y^4}~dy$$   I'm guessing that the Fundamental Theorem of Calculus can't be used because the integrand is a function of $x$.  Any help would be greatly appreciated. Is there a formal spelling for the English letter names? Are The Lazarus Effect and Dune in the same universe? There are similar aspects between Dune, and the technology in The Lazarus Effect.   Noticeable similarities:   Lasguns - this weapon exists in both of the books, and has similar architecture in both books.   Merman and Ixians - are both technologically sophisticated in similar ways. Rely on mechanical technology as opposed to organics or similar.   Is this merely Herbert sticking to his original style? Textbooks on linear regression with least squares What is the cardinality of the set of all topologies on $\mathbb{R}$? This was  on Quora. I thought about it a little bit but didn't make much progress beyond some obvious upper and lower bounds. The answer probably depends on AC and perhaps also GCH or other axioms. A quick search also failed to provide answers. Erase Windows Vista to Install Ubuntu new here. I have an old E-machine which as Windows Vista on it and I want to erase the Windows Vista, and install Ubuntu on it instead, but I am not sure where to start. Can I first install Ubuntu, and THEN erase Windows Vista, or is there a more sure method? I appreciate your help. Use of Indefinite Articles(a,an) Consider the following sentences:  I am staying in a hotel nearby. I am staying in an hotel nearby.  I know that both are correct. But I would like to know why? I do not see any proper reason for the second one as the 'h' is not silent as in French. So, what could be the reason? I don't understand philosophy very well, and so I am wondering whether you can "prove" anything in philosophy. It always seems you can go a layer down, and find another question, almost endlessly until you get to the question of "why/how/are we here?"  Can you prove, absolutely, and 100% that something is true, in the field of philosophy?