Find three natural numbers $a,b,c$ such : $\begin{cases}ab=1(\mod c)\\ac=1(\mod b)\\bc=1(\mod a)\end{cases}$
If $p,q,r$ are all primes,and $p|qr-1$,$q|pr-1$ and $r|pq-1$,find all possible values of $pqr$.
IndentationError: unindent does not match any outer indentation level Is my sugar daddy scamming me What is the simplest way to prove the Earth is round? Assume you've come in contact with a tribe of people cut off from the rest of the world, or you've gone back in time several thousand years, or (more likely) you've got a numbskull cousin.  How would you prove that the Earth is, in fact, round? Terence Tao, Analysis I, Ex. 5.5.2: Entry point needed Terence Tao, Analysis I, 3e, Exercise 5.5.2:     Let $E$ be a non-empty subset of $\mathbb{R}$, let $n \ge 1$ be an   integer, and let $L &lt; K$ be integers. Suppose that $K/n$ is an upper   bound for $E$, but that $L/n$ is not an upper bound for $E$. Without   using Theorem 5.5.9, show that there exists an integer $L &lt; m \le K$   such that $m/n$ is an upper bound for $E$, but that $(m-1)/n$ is not   an upper bound for $E$. (Hint: prove by contradiction, and use   induction. It may also help to draw a picture of the situation.)    does not do the trick for me, since I'm looking for the proof by contradiction, equipped only with the set of propositions and theorems covered so far.  What I'm actually looking for is an entry point to the proof, since I find it quite hard to negate the assumption.  My attempt is to induct over $n$. Assume I have been able to prove the statement for $n = 1$, is the following a valid negation of the assumption?  For all integers in the range $(L, K]$, at least one of the following statements holds:   $(m-1)/(n+1)$ is not an upper bound and $m/(n+1)$ is not an upper bound. $(m-1)/(n+1)$ is an upper bound and $m/(n+1)$ is not an upper bound. $(m-1)/(n+1)$ is an upper bound and $m/(n+1)$ is an upper bound.   If I were able to prove that there is an integer in the given range such that none of the above holds, am I done?       Theorem 5.5.9 (Existence of least upper bound). Let $E$ be a non-empty subset of $\mathbb{R}$. If $E$ has an upper bound, (i.e.,   $E$ has some upper bound $M$), then it must have exactly one least   upper bound.     I went through another loop considering  of Steve Kass, but I ended up wondering why the proof needs induction at all:  Assume for all integers $m$ where $L &lt; m \le K$ we have that $m/n$ and $(m-1)/n$ are upper bounds of $E$. Then $(m-1)/n = L/n$ is also an upper bound. But $L/n$ is no upper bound, by assumption. Assume $m/n$ and $(m-1)/n$ are no upper bounds. Then $m/n = K/n$ is also no upper bound. But $K/n$ is an upper bound, by assumption.   Hence, there is an integer in the range $(L, K]$ s.t. $m/n$ is an upper bound for $E$, and $(m-1)/n$ is not. TP-LINK TL-WN823N Unable to connect to network I have purchased a TP-LINK TL-WN823N (EU) V2.0 device, which is a USB device that should work as a Network Interface Card, allowing me to connect my computer to wireless networks. Unfortunately, I am unable to find any wireless networks using this device using either wicd or network-manager (no networks appear in the list of networks supplied by either application).  A CD came with the device claiming "Linux support". The CD contained a URL to a corrupted archive file for downloading installation resources. I try to mask my immense frustration.  Here is the additional dmesg output I receive when I connect the device to my computer (similar content is present when I boot the computer with the device connected):  [17096.009912] usb 2-1.2: new high-speed USB device number 4 using ehci-pci [17096.103501] usb 2-1.2: New USB device found, idVendor=2357, idProduct=0109 [17096.103513] usb 2-1.2: New USB device strings: Mfr=1, Product=2, SerialNumber=3 [17096.103527] usb 2-1.2: Product: 802.11n NIC [17096.103529] usb 2-1.2: Manufacturer: Realtek [17096.103530] usb 2-1.2: SerialNumber: 00e04c000001   This is my lsusb, where the offending device corresponds to Bus 002 Device 004:  Bus 002 Device 003: ID 0a5c:5800 Broadcom Corp. BCM5880 Secure Applications Processor Bus 002 Device 004: ID 2357:0109   Bus 002 Device 002: ID 8087:0024 Intel Corp. Integrated Rate Matching Hub Bus 002 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 001 Device 004: ID 0c45:6433 Microdia  Bus 001 Device 002: ID 8087:0024 Intel Corp. Integrated Rate Matching Hub Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub   As lsusb -v:   It seems the device is working to me, but no new interface is found in 'ifconfig':  docker0   Link encap:Ethernet  HWaddr 02:42:6e:bf:92:16             inet addr:172.17.0.1  Bcast:0.0.0.0  Mask:255.255.0.0           UP BROADCAST MULTICAST  MTU:1500  Metric:1           RX packets:0 errors:0 dropped:0 overruns:0 frame:0           TX packets:0 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:0            RX bytes:0 (0.0 B)  TX bytes:0 (0.0 B)  enp0s31f6 Link encap:Ethernet  HWaddr 30:5a:3a:e1:96:b1             UP BROADCAST MULTICAST  MTU:1500  Metric:1           RX packets:0 errors:0 dropped:0 overruns:0 frame:0           TX packets:0 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1000            RX bytes:0 (0.0 B)  TX bytes:0 (0.0 B)           Interrupt:16 Memory:df200000-df220000   lo        Link encap:Local Loopback             inet addr:127.0.0.1  Mask:255.0.0.0           inet6 addr: ::1/128 Scope:Host           UP LOOPBACK RUNNING  MTU:65536  Metric:1           RX packets:412 errors:0 dropped:0 overruns:0 frame:0           TX packets:412 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1           RX bytes:31876 (31.8 KB)  TX bytes:31876 (31.8 KB)   Running 'lshw -class network' as root contains only information about the 'enp0s31f6' interface.  Additionally for sanity, my 'uname -a':  Linux home-z170a-ubuntu 4.4.0-22-generic #39=Ubuntu SMP Thu May 5 16:53:32 UTC 2016 x86_64 x86_64 x86_64 GNU/Linux   I am using Ubuntu 16.04.  My questions are:   Why are no networks present in the list for wicd or network-manager? Why is there no additional network interface created when I plug in the USB device? Is it possible to create this network interface?   Thanks for reading. DNS subdomain (child) NS records If we have example.com we would setup things something like this: $ORIGIN example.com. @  1D  IN  SOA ns1.example.com. hostmaster.example.com. (                   2002022401 ; serial                   3H ; refresh                   15 ; retry                   1w ; expire                   3h ; nxdomain ttl                  )           IN  NS     ns1.example.com.           IN  NS     ns2.example.com.  since in the parent zone .com. there is already NS record for the example.com. child zone and A glue record for the ns1.example.com. for what purpose do we state NS records again in the child zone? and also, do we need to state A record for the name servers in the child zone itself? Extracting numbers from vectors of strings How do I bypass "this connection is not private" warnings in safari? What is the /boot partition really for? I'm reading a relatively old text on Linux partitions and file systems (the LPIC 1 Certification Bible). It says:     Some versions of the Linux boot loaders cannot access a kernel that is   outside the first 1024 cylinders on a disk. By putting the /boot   partition at the beginning of the drive you can be assured of not   having a problem when accessing the kernel at boot. This problem shows   itself most often in cases of dual booting Linux along with another   operating system that is on the first partition.   Why would a boot loader have "no access to the kernel outside the first 1024 cylinders on a disk"?  Also, what does "putting the /boot partition at the beginning of the drive" mean? \textcite macro produces "Author (year)" correctly but colours first paren with hyperref I am using a custom \textcite macro that I found somewhere. It works nicely except that it makes the opening paren of "Author (year)" part of the link (also colouring it blue) which is not desired.  This is likely just a minor change to the definition of the macro, but the solution eludes me (I don't get where the opening paren is added at all).    Opening paranthesis should not be part of link and should not be coloured.  (M)WE:  \documentclass{scrartcl}  \usepackage{csquotes}  \usepackage[backend=biber,             bibencoding=utf8,             hyperref=true,             natbib=true,             minnames=1,     %% prevent truncation to year-only             backref=true, %% TODO REMOVE before final             bibstyle=apa,             citestyle=authoryear,             ]{biblatex} \DeclareLanguageMapping{english}{english-apa} \addbibresource{bib.bib}   \DeclareCiteCommand{\textcite}   {\boolfalse{cbx:parens}}   {\usebibmacro{citeindex}%    \printtext[bibhyperref]{\usebibmacro{textcite}}}   {\ifbool{cbx:parens}      {\bibcloseparen\global\boolfalse{cbx:parens}}      {}%    \multicitedelim}   {\usebibmacro{textcite:postnote}}   \usepackage[table,svgnames]{xcolor}  \definecolor{solblue}{HTML}{268BD2}  \usepackage[pdftex]{hyperref}  \hypersetup{    pdfhighlight = /P,    colorlinks=true,    urlcolor=solblue,    filecolor=solblue,    linkcolor=solblue,    citecolor=solblue,    raiselinks=true,    breaklinks,    unicode }   \begin{document}  foobar \textcite{thesis_pockrandt} foobar  \end{document}   bib.bib:  @phdthesis{thesis_pockrandt,     author = {Christopher Maximilian Pockrandt},     title = {{Approximate String Matching: Improving Data Structures and Algorithms}},     year = {2019} } Why didn't Skynet kill Kyle Reese? How do I align the viewport to a face normal? I would like to align the view straight on to a face, perpendicular to the normal. How do I align the 3D viewport to the face of a mesh? Disable password access through SSH? I'm setting up a server for backing up my desktop box remotely. I want to prevent remote password access to the server, permitting only users with private keys (AKA my desktop box) to access it. How can I block remote password access while allowing physical password access to the server? Why can in-class initializers only use = or {}? The main effect will be non-significant if the interaction is significant? Does moving leave the object in a usable state? When should I use "a" vs "an"? Convert .bbl file to .bib file How can I convert a .bbl file to a .bib file. I  lost my .bib file but I still have my .bbl file. Now , I need to add more references to my .bbl file according to a specific format which I need to produce using Bibtex, so how can I convert my .bbl file to a .bib file so I can add the new references according to the format.  Thanks for any insight. We're building a feature for a website and trying to think of a hypernym for clients, partners and members so that the feature remains generic enough for re-use later on other websites. How to display javascript console on Chrome (or a different browser)? I want to quickly try some Javascript code on Android phone.   I cannot find how to open console or something similar on either Chrome or default browser. about:debug doesn't do anything, and I don't want "remote debugging".   I can download another app if necessary (but I don't want to root my device).