I start with knowing that two numbers are coprime if:  $n*k + m*j = 1$  So, setting $k = a$ and j = $a+1$ I can solve as follows:  $n*a + m*a + a$  Then,  $a(n+m) + m = 1$  Where can I go from here?
I have a question.  I have been given this proof: "For any $n$ in the integers where $n&gt;2$, show there are at least $2$ elements in $U(n)$ that satisfy $x^2=1$."  I have gone through and actually proved this, (that the numbers are $1$ and $n-1$) but i didn't' know how to prove that $n-1$ is in fact in the set $U(n)$. Is it because two consecutive numbers are always relatively prime?
Desktop doesn't remember brightness settings after a reboot Every time I reboot my machine the brightness goes back to 100% in Gnome. I wish it would keep the last setting. Is there anyway? I have a dual boot of Windows 10 and Ubuntu 16.04. Recently I upgraded it to 18.04. As soon as I did it, it started showing DISK IS LIKELY TO FAIL SOON (37° C / 99° F).    I checked the status of my disk via Windows and it shows it’s perfectly healthy. I assumed the error occurred because my partition size of Ubuntu was 64GB (which I now increased to 86GB). Please help me out with this.    Edit: The results of smartctl - attributes Is there a way to see my previous performances with respect to winter-bash? May be something like leagues for winter-bash? I understand that hats are different each year, but at least it would be nice to see the number of hats collected. Loopy integral $\int xe^x\sin(x)$ Can a mangaka continue another mangaka's manga? There are plenty of manga that never had an ending for many reasons. For example, High School of the Dead's mangaka, Daisuke Satou, . Some mangaka also cannot continue their work because of health problems, as well as other reasons. Say the original mangaka either prepared or did not prepare an ending himself/herself, can another mangaka continue their work? Has this ever happened before? What suggestions should I make when reviewing a poorly written paper? Since I need a break from the reviewing...  Often in my field I review papers from researchers in Asia (reviews are not double blind) which are poorly written. By poorly written, I don't mean that the prose is not pleasing to the eye, that there are problems with how the writing flows, or that there are innocuous typos or spelling mistakes. Instead, the grammar used by the authors differs seriously from normal English, and often produces significant changes in the meaning of sentences.  Usually such papers are ultimately readable, but one has to go back and re-read sentences or paragraphs several times to infer the meaning (and this is with strong knowledge of the research area!), so I generally won't reject a paper solely for this reason. However, I would feel remiss in my duties as a reviewer if I didn't instruct the authors to improve this aspect of the paper. I'd like to provide helpful advice for the authors without just giving them a laundry list of the errors.   Typically I highlight a few places where I had significant trouble understanding what they actually meant to say, and suggest seeking a professional editor before publication.   Are these reasonable things to suggest?   What other suggestions can I make? How to establish this inequality: $(1-a)(1-b)(1-c) \geq 8abc$ for $a+b+c=1$? Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c = 1$. Then how to establish the following inequality? $$ (1-a)(1-b)(1-c) \geq 8abc.$$  My effort:   Since $a+b+c =1$, we can write $$  (1-a)(1-b)(1-c) = 1 - (a+b+c) +(ab+bc+ca) - abc = 1 - 1  +(ab+bc+ca) - abc =  (ab+bc+ca) - abc = abc(a^{-1} + b^{-1} + c^{-1} ) - abc = 3abc (\frac{a^{-1} + b^{-1} + c^{-1}}{3}) - abc.$$ What next? How do I get a Cron like scheduler in Python? I'm looking for a library in Python which will provide at and cron like functionality.  I'd quite like have a pure Python solution, rather than relying on tools installed on the box; this way I run on machines with no cron.  For those unfamiliar with cron: you can schedule tasks based upon an expression like:    0 2 * * 7 /usr/bin/run-backup # run the backups at 0200 on Every Sunday  0 9-17/2 * * 1-5 /usr/bin/purge-temps # run the purge temps command, every 2 hours between 9am and 5pm on Mondays to Fridays.   The cron time expression syntax is less important, but I would like to have something with this sort of flexibility.   If there isn't something that does this for me out-the-box, any suggestions for the building blocks to make something like this would be gratefully received.  Edit I'm not interested in launching processes, just "jobs" also written in Python - python functions. By necessity I think this would be a different thread, but not in a different process.  To this end, I'm looking for the expressivity of the cron time expression, but in Python.   Cron has been around for years, but I'm trying to be as portable as possible. I cannot rely on its presence. Rendering an animation the way 3D view looks Is there some way to render exactly whats in the camera view in the 3D view? By that I mean just like a screenshot really, all paths, lamps, and cameras, just like things look in the camera view. The reason I can't just take a screenshot is that I need an animation and it would be too cumbersome to take 1000+ screenshots! Check all dependencies of a Record Type in the Organisation before deactivating it I am looking forward to delete a Record Type in my Salesforce organisation. However before I go ahead and do the same I would love to know all of its dependencies in the entire org.  Please help me in finding a tool or a method to easily get all the list of dependencies associated with that particular record type.This may include ,  a)      Workflows/Process Flows b)      Validation Rules c)       Apex Triggers/Classes d)      Any other dependencies if we deactivate the record type.(including Picklist and Page layouts)  Thanks in advance. 16.04 new installation gives grub-efi-amd64-signed failed installation /target/ ubuntu 16.04 at the end I am installing Ubuntu 16.04 in an Intel NUC computer with UEFI enabled. I am using an SSD from another computer. During the installation I chose Erase the disk and make a new full install  At the end of the process I see:  grub-efi-amd64-signed failed installation /target/ Ubuntu 16.04    And the system does not boot.  I tried to use boot-repair but apparently it does not solve the problem. Foolproof way to ensure Google news pulls the correct image for its thumbnails? Google news results have an acompanying thumbnail next to articles that show up in the results.    If Google's crawler can't find a thumbnail to pull from our site, it uses its next best guess from another site, therefore linking the image to another site but still uses our headline.  Example:  Headline from Reuters, Image from Livemint:   Our pages absolutely have images, they are not massive in file-size or dimensions, yet we are not having them pulled / crawled correctly.  We have read up on the suggestions from Google, and from others around the web and nothing is panning out.  Has anyone had any experience where they can ensure Google news will pull a thumbnail of our choosing? it can not access popup.js file after creating a chrome extension Reputation has been changed but not reflected on tooltip. PHP __PHP_Incomplete_Class Object with my $_SESSION data Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$ Trying to put a footnote in subsection title:  \subsection{Effect of Lateral Confinement on SW Band Structure\footnote{Parts of this sub--section are based on \citet{Venkat2013}}}   produces the following error:  Argument of \@sect has an extra } ...section are based on \citet{Venkat2013}}}   I am using:   \documentclass[paper=A4, fontsize=12, toc=listof, toc=index, toc=bib]{scrreprt}   Please advise. How do I go about evaluating the following by contour integration?  $$ \int^1_0 \frac{dx}{(x^{2} - x^{3})^{1/3}} $$  The question does not fit in the standard form of : $\int^{2\pi}_0$ or $\int^\infty_{-\infty}$  etc.  I tried substituting $ x = cos(\theta) $ but got stuck. How can I copy a file from another directory to the current one? I have this problem as part of a school assignment:  NOTE: In case it's hard to read in the image above, here's the directory tree: $ tree . `-- sample_dir1     `-- sample_dir         |-- admin         |-- cambridge         |   |-- cafeteria         |   |-- library         |   `-- security         |       |-- annex         |       |-- building         |       `-- parking         |-- faculty         |-- history.exe         |-- markham         |   |-- annex         |   |-- building1         |   `-- parking         |-- oxford         |   |-- outline.doc         |   |-- programming         |   |   `-- report.pdf         |   `-- security         `-- stenton             |-- gen_ed             `-- lib_arts                 |-- english.txt                 `-- match.doc  15 directories, 11 files  I tried doing the command: $ cp ../cambridge/security/parking ./parking2  ...but it isn't working. The question in the image above states that the current directory is stenton, and that we need to make a copy of the file named parking (from the security directory), and to name the new file parking2, and place it in the current directory, stenton, using relative pathnames. What am I doing wrong? My cp command seems to be correct. When I play Black Ops 2, it seems that my xbox controller is inverted. Example: if I press up, my character looks down and vice versa. It was not like this before. Does anyone know how to reset the controller back to its original settings?