If L.C.M of three natural numbers $a, b, c$ is $p^2q^2r^2$(where $p, q, r$ are different prime numbers) such that sum of all possible triplets (a,b,c) are given by $m^n$ (where $m$ and $n$ are prime), then find the value of ($m-4n$)  This is how solved. As the L.C.M of the given numbers is $p^2q^2r^2$, this means that none of the numbers a,b,c can have $p^0 orp^1,q^0orq^1,r^0 or r^1$ as their factors. But they can have $p,q,r$ as their factors in powers of two or higher. For example if $a=p^2q^5r^7$,$b=p^7q^2r^3$ and $c=p^9q^8r^2$ then the L.C.M of a,b,c is $p^2q^2r^2$. Hence there should be infinite solution i.e. infinite ordered triplets. But the author has provided with the following solution: L.C.M. $p^2$ can be obtained from the numbers a, b, c in the following combinations. $$p^2 p^2 p^2    ⇒    1 arrangement$$ $$p^2 p^1 p^2    ⇒    3 arrangement$$ $$p^2 p^1 p^1    ⇒    1 arrangement$$ $$p^2 p^2 p^0    ⇒    3 arrangement$$ $$p^2 p^1 p^0    ⇒    6 arrangement$$ $$p^2 p^0 p^0    ⇒    3 arrangement$$ $$———————————––—                       19 arrangement$$ So if L.C.M. is $p^2q^2r^2$ total combinations are = $19^3$ = $m^n$ ⇒  m – 4n = 7. This is where I am confused and unable to comprehend as why the solution is provided  in such a way, because in the above arrangements L.C.M will not be $p^2q^2r^2$.
Let $a=p_1^{x_1}p_2^{x_2}\cdots\cdot p_q^{x_q}$ and $b=p_1^{y_1}p_2^{y_2}\cdots p_r^{x_r}$ ,   say — Since gcd(a,b) is the largest common divisor of a and b and is divisible by any other common divisor of the two,   GCD$(a,b)=p_1^{\min(x_1,y_1)}p_2^{\min (x_2,y_2)} \cdots p_q^{\min (x_q,y_q)}$  LCM$(a,b)= p_1^{\max(x_1,y_1)}p_2^{\max(x_2,y_2)}\cdots p_r^{\max (x_r,y_r)}$  I understand GCD(a,b) has to divide both $a,b$. Therefore the exponent of any prime factor $p_i$ in GCD has to be in both $a,b$  — therefore $p_i^{\min(x_i, y_i)}$. But I'm muddled and anxious. Why does $\min$ appear in GREATEST common divisor? Why does $\max$ appear in LOWEST common multiple?
Will yearly moderation statistics be published on all sites this year, and future years? Strange book where teen protagonist is sent to a warzone, uses illusory magic to grow male genitalia What recourse do I have if I believe a moderator has abused their privileges? El Capitan froze and rebooted (Macbook Pro Retina Mid 2012) 2.3 Ghz Intel Core i7 8GB DDR3: Was running NetAdmin Pro which is monitoring  my office LAN, processors consumption was at  ~15 - 20%. Simultaneously I ran WiFI Explorer,  3.7% no more than 5% CPU usage. And I was  updating Xcode to v 7.3.1 and was browsing and  watching video from playground.ru Watch the videos until half of it (20mins) the Video froze I moved the mouse and turned out  the whole OSX is frozen. littleSnitch v 3.7 has no reaction I immediately pull the Ethernet adaptor and  MacBook just went to black screen. I then  switch off the wireless router. (My MacBook Pro  is connected to the network via Ethernet  adaptor and via WiFi router bearing different IP, I was thinking of cutting both network  communication, in case it was an intrusion) MacBook just restarted itself to login screen,  after 15-20 sec of black screen and total  silence) I again restarted and booted into Recovery  mode use disk utility. After completing disk  check. Login and previously opened programs resume  back (NetAdmin Pro, WiFi Explorer, Yandex  browser, Activity Monitor and AppStore) AppStore doesn't resume the download  everything else is functioning normally and in Addition I have got my calendar open after login (I did not launch accidental nor was it set to  start with system login)   Question 1: Are there any way to know what was happening and what was OS El Capitan last reaction before it froze and rebooted?  Question 2: Are there any software better than "Console" v10.11 to interpret what was going on before it crashes?  P.S. Thank you in advance, I will try to upload the Console messages later from the MacBook Pro    DDT-2:~ Chaleune$ syslog | grep -i "shutdown cause"  Jun 22 14:15:37 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 22 20:34:46 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 23 15:18:23 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 24 17:58:32 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 24 20:09:52 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 24 22:58:15 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5Jun 25 14:43:36 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 25 21:55:31 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 3  Jun 25 22:20:57 localhost kernel[0] &lt;Notice&gt;: Kext com.apple.driver.AppleOSXWatchdog failed to load (0xdc008012).Previous shutdown cause: 5  Jun 26 10:18:44 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 26 12:09:45 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 26 18:00:31 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 27 13:05:58 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 27 22:56:11 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 28 14:18:27 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  *Jun 29 18:56:21 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: -128*  Jun 29 23:31:22 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: -128      shows the following   dsIllInstErr                  = 3,    /illegal instruction error/ dsChkErr                      = 5,    /check trap error/ userCanceledErr               = -128    gave the following interpretation:   03 Illegal Instruction The computer has a specific vocabulary of machine language instructions it can understand. If a computer tries to execute an instruction that isn't in its vocabulary, you see this error code. It's less likely than error 02, but still very common. 05 Range Check Error Programmers can use an instruction in the Motorola 68000 to check if a number is within a certain range. This error indicates that the number tested isn't in the specified range. -128  userCanceledErr         User canceled an operation Is $\frac{1}{\sqrt{2}}$ a fraction or not?   I just came across a Grade 8 student's class work where the teacher has given the above as an example to differentiate fractions from rational numbers. I think this is wrong fundamentally, as $\sqrt2$ is irrational while all fractions would be rational. Please advise. Thanks! I use a Asus laptop with dual-boot Ubuntu and Windows 8.1  Wifi is on but I am not able to detect Wifi networks.  When I run :  lspci -knn | grep Net -A3; rfkill list   This is what happens:    I get  acer-wireless: Wireless LAN     Soft blocked: yes I'd like to show that the class of all limit ordinals, LIM, is cofinal in the class of all ordinals ON.  I thought that I could do this by contradiction and assume that it wasn't cofinal in ON. Then there is an $\alpha \in$ ON such that for all $\beta \in $ LIM: $\beta &lt; \alpha$, i.e. $\beta \in \alpha$ and hence LIM $\subset \alpha$.  This means that LIM is a set so we can consider $\bigcup$ LIM. Then we have $\beta &lt; \bigcup$ LIM for all $\beta \in$ LIM by definition of $\bigcup$ LIM. So LIM $\subseteq \bigcup$ LIM.   $\bigcup$ LIM is either a limit or a successor ordinal.  If it is a limit ordinal then $\bigcup$ LIM $\in$ LIM and by transitivity of ON, $\bigcup$ LIM $\subseteq$ LIM, so $LIM = \bigcup$ LIM in this case which would mean that $\bigcup$ LIM $\in$ LIM $= \bigcup$ LIM which would be a contradiction because sets cannot contain themselves. Hence $\bigcup$ LIM cannot be a limit ordinal.  If $\bigcup$ LIM is a successor ordinal then $\bigcup$ LIM = $\beta \cup \{&#160;\beta \}$ for some $\beta \in $ ON.  Now I'm not sure how to proceed from here. I'm also not sure about whether I'm on the right track with this proof because I think there should be a shorter way  to prove this.  Thanks for your help. Generating data with a given sample covariance matrix Colorize symbols with lstdefinelanguage in latex I want to put some symbols in green color with lstdefinelanguage in latex, but it seems that symbols are not recognized.  The symbols I need to be colored are =, > and &lt;. Is it possible? How can I do this?  The main file:  \documentclass{article}   \usepackage{graphicx,color} \usepackage{booktabs} \usepackage{listings}   \lstdefinelanguage{NeoIDL}{   sensitive = true,   keywords = [1]{module, resource, enum, annotation, for, import, entity, path,   @get, @post, @put, @delete, require, ensure, otherwise, call},   %ndkeywords={int, \char{}  },   %ndkeywordstyle=\color{blue}\bfseries,   morekeywords=[2]{&gt;, &lt;, ==},   keywordstyle=[2]\color{green},   numbers=left,   numberstyle=\scriptsize,   stepnumber=1,   numbersep=8pt,   showstringspaces=false,   breaklines=true,   frame=top,   comment=[l]{//},   morecomment=[s]{/*}{*/},   commentstyle=\color{purple}\ttfamily,   stringstyle=\color{red}\ttfamily,   morestring=[b]',   morestring=[b]"   }  \begin{document}  \begin{figure}[htb] \begin{small} \lstinputlisting[language=NeoIDL,firstnumber=1]{store_pos_servico.neo} \end{small} \caption{Service 1} \end{figure}   \end{document}   The file to be formatted:  module store {    (...)     resource order  {      path = "/order/{id}";       @post   int postOrder (Order order)     require (call store.getOrder(id) == "NotFound"),     otherwise "InvalidPrecondition"    }; } How to get rights of admin after I disabled all admin accounts in my computer I accidentally disabled my admin account.  After I login to another account I found I can not get admin rights, because all admin accounts on my computer are disabled.  I clicked on 'run as administrator' but only see a smart card choice (all the admin accounts are disabled so no choice provided)    I don't want to re-install my OS, help!    Details:  I have account A on my computer.    I got a new computer so I want to give the old one to my mother.    Creating a new user account B for her and disable account A.    Logout and restart.    Successfully login B, now I found I could not get admin rights because no account choice is provided.    More informations about my machine:  OS is Windows 10 and my admin account is a Microsoft Account.  I have an arch-linux installed on my computer and dual-boot it using GRUB, so maybe I can not use safe mode. How to Backup and Restore Android OS without Rooting Falsely accused witch befriends transforming dragon she was to be sacrificed to Story is in a medievalish setting and starts with a woman who has been falsely accused of being a witch being taken by 2 men to be sacrificed to a dragon.  She ends up befriending the (male) dragon, who is able to transform into various forms, including human, and can speak only the language of whatever form he is in. e.g. can speak English in human form, but can't in dragon form.  He needs to return to his dragon form every day at sunrise or he dies.  Contact with iron or steel causes him pain and prevents him from transforming.  Near the end of the book, he's been put in handcuffs by one of the two men from the start and the woman has to get the assistance of the other one (village blacksmith?) to remove them before sunrise, which they manage to do just in time.  I think I read it maybe 10 years ago.    I also remember it was a very short book, probably only about 100 pages. Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$. Let $a,b,c $ are non-negative real numbers, and $a+b+c=3$.  How to prove inequality $$ ab^2+bc^2+ca^2\le 4.\tag{*} $$    In other words, if $a,b,c$ are non-negative real numbers, then how to prove inequality $$ 27(ab^2+bc^2+ca^2)\le 4(a+b+c)^3.\tag {**} $$ $\color{gray}{\mbox{(Without using "universal" Lagrange multipliers method).}}$  Thanks! "It looks straightforward, but actually it isn't" In , I asked about proof of statements which are simple but incorrect.   Here, I ask about statements which seems, at a first glance, straightforward, but if we try to write a proof, we can see it's much harder than it looked. So I expect the answers to contain:   the statement; why it looks easy to prove; why actually it isn't. Why Was It 'Essential' That Voldemort Kill Harry Potter? ‘So the boy ... the boy must die?’ asked Snape, quite calmly.   ‘And Voldemort himself must do it, Severus. That is essential.’   Deathly Hallows - page 551 - UK Hardcover - chapter 33, The Prince's Tale   Dumbledore tells Snape it is 'essential' that Voldemort be the one to kill Harry, I'm assuming in order for the piece of Voldemort's soul in Harry to be properly killed as well.  But why?   Hermione destroyed the Hufflepuff cup; Ron destroyed the Slytherin locket; Neville killed Nagini; Harry destroyed the diadem and Tom Riddle's diary. So the Horcruxes were not immune to destruction at the hand of someone other than Voldemort. And as it ended up, Harry himself wasn't even a Horcrux, but rather just an unknowing host to a parasitic bit of Voldemort's soul.   Why was it 'essential' that Voldemort be the one to kill Harry in order for the piece of Voldemort's soul to die? Prove that there exists at least one $x_0\in\mathbb{R}$, such that $f(x_0)+f''(x_0)=0$ Let $f:\mathbb{R}\to\mathbb{R}$ be a function, two times   differentiable with $\left|f(x)\right|\leq1,\forall x\in\mathbb{R}$   and $f^2(0)+\left(f'(0)\right)^2=4$. Prove that there exists at least   one $x_0\in\mathbb{R}$, such that $f(x_0)+f''(x_0)=0$.   I have tried the usual Rolle method by $e^x$, but didn't go far... Any help available? A standard proof for press needs crop marks and color bars as below    With the package crop, the crop marks can be easily added.  But how about the color bar ?  Is there a way or a package to make it ?  An example is   \documentclass[b5paper,10pt]{article} \usepackage[utf8]{inputenc} \usepackage[cross,a4,center,info,axes]{crop}  \title{Test} \author{User} \begin{document} \maketitle \section{Fox} The quick brown fox jumps over a lazy dog. \end{document} How do Horcruxes work? A few questions regarding the Horcruxes.   Had Voldemort actually succeeded in using a Horcrux, would he come back the same age (biologically and/or aesthetically) he was when he made the Horcrux, or the age he would biologically be if he had never died (half died)? If Harry was a Horcrux, then in the graveyard, when Wormtail added Harry's blood to the potion to bring Voldemort back, didn't that technically use up the Horcrux inside of Harry? So in Deathly Hallows when Voldemort kills Harry, shouldn't Harry have actually died rather than just have the Horcrux inside him destroyed, seeing as it was technically used in the Goblet of Fire? What are all the common undefined behaviours that a C++ programmer should know about?  Say, like:  a[i] = i++; Calling a method in java and returning beer cost