Let $a,b\in\mathbb{Z} \backslash \{0\}$. Show that  $a\mid b$, $b\mid a$ $\implies a=\pm b$.  I can see why this is true, but not write it down.
If $A|B$ and $B|A$ then prove $A=\pm B$   So far I have $A|B \iff AX=B$ and $B|A \iff BY=A$ with $X,Y \in \mathbb{Z}$  Not sure how to finish, any help.
Showing $\sum\limits_{p \in P} \frac{1}{p}$ where $P$ is the set of all primes is divergent Moment of inertia of a hollow sphere wrt the centre? I've been trying to compute the moment of inertia of a uniform hollow sphere (thin walled) wrt the centre, but I'm not quite sure what was wrong with my initial attempt (I've come to the correct answer now with a different method). Ok, here was my first method:  Consider a uniform hollow sphere of radius $R$ and mass $M$. On the hollow sphere, consider a concentric ring of radius $r$ and thickness $\text{d}x$. The mass of the ring is therefore $\text{d}m = \frac{M}{4\pi R^2}\cdot 2\pi r\cdot\text{d}x$. Now, use $r^2 = R^2 - x^2:$ $$\text{d}m = \frac{M}{4\pi R^2}\cdot 2\pi \left(R^2 - x^2 \right)^{1/2}\text{d}x$$ and the moment of inertia of a ring wrt the centre is $I = MR^2$, therefore: $$\text{d}I = \text{d}m\cdot r^2 = \frac{M}{4\pi R^2}\cdot 2\pi\left(R^2 - x^2\right)^{3/2}\text{d}x $$ Integrating to get the total moment of inertia: $$I = \int_{-R}^{R} \frac{M}{4\pi R^2} \cdot 2\pi\cdot \left(R^2 - x^2\right)^{3/2}\ \text{d}x = \frac{3MR^2 \pi}{16}$$  which obviously isn't correct as the real moment of inertia wrt the centre is $\frac{2MR^2}{3}$.   What was wrong with this method? Was it how I constructed the element? Any help would be appreciated, thanks very much. How to install Eclipse? I want to install the latest version of Eclipse but the Ubuntu Software Centre contains an older version.  Is there a PPA or some other way to install latest Eclipse? Please describe the steps for full installation. The ls -al command shows the following output;  -rwxrw-r--    10    root   root 2048    Jan 13 07:11 afile.exe   What are all the fields in the preceding display? It is being surprisingly hard to achieve. I've tried so far:   Download file manager apps from Google Play. Unfortunately, "share" shares the files, not the file names. None of them had a option like copy file names Download backup apps. Unfortunately, none of them backup the file names of a folder.   Any idea to copy to the clipboard or share the file names of all the files of a folder? So I have been extruding a cube to make a low poly banana and when I go from edit to object mode half the edges don't appear as well as half is dark.      I tried looking for an solution couldn't find one and its making me feel like I am doing something wrong so I decided to come here. Determining application path in a Python EXE generated by pyInstaller Repeat each line multiple times Would like to have each line in a file repeated a fixed number of times.  e.g. have each line repeated four times:  a b c   becomes:  a a a a b b b b c c c c   I've done some searching, and there are a lot of questions and answers along the lines of doing the reverse, e.g. merging duplicate lines into single lines, and maybe a few about doubling lines by printing them again.  It would be easy to do this in C, but I wish I knew more about the native commands so I wouldn't have to resort to these kinds of one-off throw-aways all the time. Prove $\sin^2\theta + \cos^2\theta = 1$ How do you prove the following trigonometric identity: $$ \sin^2\theta+\cos^2\theta=1$$  I'm curious to know of the different ways of proving this depending on different characterizations of sine and cosine. Is the above statement true? If I have already proven that the set is linearly independent, is it automatically given that the subset is also linearly independent? If so, how would I go about showing that as a proof. Is it possible to create 3D red/cyan anaglyphs in a hand drawing? There are plenty tutorials online on how to create 3D red/cyan anaglyphs from existing photos in Photoshop or similar software. However, I'm wondering if it is possible to draw a 3D image by hand on for example a whiteboard using markers.  My plan is to tape together a red and blue marker and draw with it using the same rotation/inclination of the marker across the drawing. Will this give a 3D effect when viewed through red/cyan glasses? Or is the 3D anaglyph method more complex than just offsetting red and cyan lines? I tried to create a Windows 10 USB boot medium with WoeUSB. I formatted a USB drive with an NTFS partition, but WoeUSB complains with:  Installation failed! Exit code: 256 Log: WoeUSB v@@WOEUSB_VERSION@@ ============================== Mounting source filesystem... Error: File "/media/woeusb_source_1543626298_6098/sources/install.wim" in source image has exceed the FAT32 Filesystem 4GiB Single File Size Limitation and cannot be installed.  You must specify a different --target-filesystem. Refer: https://github.com/slacka/WoeUSB/wiki/Limitations#fat32-filesystem-4gib-single-file-size-limitation for more info. Unmounting and removing "/media/woeusb_source_1543626298_6098"... You may now safely detach the target device   I've also tried to start WoeUSB via command line, did not work.  sudo woeusb --partition Win10_1809Oct_English_x64.iso /dev/sdb   My iso is located in:     home/sawyer/Downloads/Win10_1809Oct_English_x64.iso  Thanks in advance! I've been trying to get windows installed for three nights now, so any help towards that goal is greatly appreciated. Open more than one instance of Windows 10 settings How can I open more than one instance of the Windows settings?  It wasn't a problem with the old style, but with the new one I can't seem to open multiple settings simultaneously and side by side.  Example of the new settings window: Switch command output language from native language to english How to approximate the student-t CDF at a point without the hypergeometric function? Is there a way to closely approximate the CDF of a student-t distribution at a point $x$ without involving the hypergeometric function? For example, by using a series expansion, or expressing the CDF in terms of simpler functions?  The t-distribution can   $$t = \frac{X\sqrt{n}}{Y}$$  where $X$ is normally distributed with mean 0 and variance $\sigma^2$; $\frac{Y^2}{\sigma^2}$ has a chi-squared distribution. Maybe this expression can help simplify the t CDF?  I am trying to write an algorithm to compute the student-t CDF for any real number, so using the hypergeometric function will be extremely difficult. I am also trying to avoid integrating the PDF to get the CDF. I have an unlocked North American (Canada) iPhone 5s, where can I buy a data only SIM card in Japan ? I'm not interested in renting nor in a voice plan, I'll be staying for 4 weeks.  I'll be landing in Tokyo, so preferably around there. Can I get them from the airport somewhere ? I'm assuming I won't have problems with the network/frequencies, am I correct ? I would like to install Ubuntu alongside Windows 7 on a HP G62 Notebook. Although I have installed  Ubuntu in a dual-boot many times before, I found out that this model has already four partitions.    Partitions:   SYSTEM (NTFS, 199MB, used 66.59MB) Partition without any tag: NTFS (579GM, used 129GB)  RECOVERY (NTFS 16.74 GB, used 2.42GB)  HP_TOOLS (FAT32, 103.34 MB, used 13.23MB)   Since I am not an expert with partitions I would like to get advice on how to do this.  My first idea is this one:   Free some space from /dev/sda2 (I don't know if could also free some other space) delete the HP_Tools partition (I have already created a backup) create an extended partition with the free space in #1 containing three parititions: swap (1gb); / (EXT4, 30GB);  /home (EXT4, 120GB)   Another option is to use wubi instead.  What do you think? Is there any other way to achieve this?  PS: I really think this HP policy of using 4 partitions is not a coincidence  PS: I tried using gparted from the live CD and I got a warning message saying that if I freed some space from /dev/sda2 I could create serious issues in the system Find index of nested item in python How can I get a copy of a Canadian visa refusal letter? A and B disjoint, A compact, and B closed implies there is positive distance between both sets