If $p^2\,$is divisible by 3, why is p also divisible by 3?
Show that $3p^2=q^2$ implies $3|p$ and $3|q$
What should I do to avoid ear pain when flying? I always feel a pain in my ears when flying. Specially when descending. The last time I had extreme pain, feeling it inclusive around my eye balls. It felt like someone was touching the eye and pulling it from the inside. I really had to focus, close my eyes and hold to avoid screaming. I am not sure but I think both things might be related. Such a severe situation only happened once.  I just heard that with a cold the ear pain gets worse. I am bit worried that I might go through that situation again. I have a cold and I am going to fly soon.  What can I do to improve this?  I see that some consider this a duplicate of the question "best way to make my ears pop" but after reading the answers I think my question goes a bit beyond that since my pain is more extreme and I think it goes beyond just popping the ears. I also want to know if other methods, like using earplugs, might be effective. Consider there is a box of mass $m$ at rest on the floor. Most books give an example that we need to do a work of $mgh$ to lift the box $h$ upward.  If we analyze this work done, the external force acting on the box by us should be equal to the weight of the box. Therefore the net force is zero which in turn there is no acceleration. If there is no acceleration and the initial velocity of the box is also zero, how can the box move upward? Prove if f(z) is entire function s.t $|f(z)|\leq M|z|$ then $f(z)$ Is polynomial of degree $1$ Let $f(z)$ be an entire function s.t $|f(z)|\leq M|z|$ ; $M&gt;0$ is a constant then $f(z)$ Is polynomial of degree 1   My try $f(z)$ is entire so it can be written as a $\sum a_n z^n$  $$|\sum a_n z^n|\leq \sum |a_n||z|^n\leq M|z|$$  So $n=1$  Seems that the move $\sum |a_n||z|^n\leq M|z|$ is problematic How to determine if string contains specific substring within the first X characters I want to check whether Value1 below contains "abc" within the first X characters. How would you check this with an if statement?  var Value1 = "ddabcgghh";  if (Value1.Contains("abc")) {     found = true; }   It could be within the first 3, 4 or 5 characters. How to write a function (piecewise) with bracket outside? Find $x,y$ given $\gcd(x,y)$ and ${\rm lcm}(x,y)$ These two exercises I encountered recently seem to develop some type of connection between GCD and LCM I can't quite figure out. Exercise 1:  Find all the numbers $x$ and $y$ such that: $a) \ GCD(x,y)=15, \ LCM(x,y)=150$ $b) \ GCD(x,y)=120 \ LCM(x,y)=1320$ $c) \ GCD(x,y)=100 \ LCM(x,y)=990$  Exercise 2:   Find all the numbers $m,n$ such that $GCD(m,n)=pq , \ LCM(m,n)=p^2qs$  where $p,q,s$ are prime  The first thing that is known to me is that $GCD(x,y) \cdot LCM(x,y)= x \cdot y$ Also $LCM(x,y)$ is at most $x \cdot y$ while $GCD(x,y)$ is at most $\max \{x,y\}$. Last thing is that $GCD(x,y)|LCM(x,y)$. Using all this I tried to solve the first exercise: $a)$ First two obvious pairs are $x=15, y=150$ and $y=15, x=150$. Now neither of the numbers can be bigger than $150$ or smaller than $15$. So we are looking for numbers in the range $15-150$ that satisfy $x \cdot y = 15 \cdot 150$ Another such pair is $(x,y)=(30,75), \ (x,y)=(75,30)$. Similarly for $b)$ we find that the only possible values are permutations of the set {$120,1320$} and in $c)$ since $100$ does not divide $990$ no such numbers exist. Now exercise 2 is what made me think there is actually another connection I'm not quite aware of since now it's about arbitrary prime numbers and the previous method doesn't work anymore. My intuition is that it has something to do with $GCD$ or $LCM$ of the $GCD(x,y), \ LCM(x,y)$ How to merge cells vertically Is it possible to create a table like this one?    The usual structure of tables I use looks like this:  \begin{table}[H]     \begin{adjustbox}{width=\textwidth,center}     % \begin{adjustbox}{center}         \begin{tabular}{lll}             \hline             \multicolumn{1}{l}{\head{XXX}} &amp; \multicolumn{1}{l}{\head{XXX}} &amp; \multicolumn{1}{l}{\head{XXX}} \\             \hline             \ttfamily xxx &amp; \ttfamily xxx &amp; \ttfamily xxx \\             \hline         \end{tabular}     \end{adjustbox}     \vspace{ - 05 mm}     \caption{xxx}     \label{tab:xxx} \end{table} Lost/Deleted dpkg (package manager for Debian). How do I get it back? OS: Ubuntu 12.04  Accidentally deleted dpkg. Now, packages can't be installed. So, needs an dpkg installer which installs packages. A short story about finding an inhabitable planet and meeting aliens So the story goes, humans in a spaceship search for an inhabitable planet, being surprised they found none in the vicinity of Earth. They find one, and it seems uninhabited. At the same time are observed by the aliens, who go down on the planet to meet them. The aliens sit them down and tell them a story about war and conflict that ruled the galaxy, and how all the sentient races in the galaxy got together and got rid of the race that tried to conquer all by blowing up their planets.  In the end you realise man is this terrible conqueror the aliens are talking about. Which is correct: I'll be moving next month or I'll be shifting next month? For changing one's home from one place to another, I've heard people in western part of the world using the sentence:      I'll be moving next month.   In India, even in the English news channels, I've heard many people using the phrase:     I'll be shifting next month.   Which one is the correct usage? Or are both of them correct and the difference in usage is regional? Is `if (condition = value)` the correct syntax for comparison? Restricting T to string and int? I have built myself a generic collection class which is defined like this.   public class StatisticItemHits&lt;T&gt;{...}   This class can be used with int and string values only. However this  public class StatisticItemHits&lt;T&gt; where T : string, int {...}   won't compile. What am I doing wrong? Some prestigious universities like Caltech, Stanford, MIT and Berkeley never disclose their information regarding their cut off major and cumulative GPA for M.Sc and PhD programs in engineering. I also aware of the fact that graduate schools will look at the whole package and make their decision based on sets of factors like letter of recommendation, statement of purpose as well as GPA.   However, since application fee is pretty high it is not worth it to apply to schools that you know you have a slight chance of getting into it so, I would like to know what would be the acceptable GPA (Both major and Cumulative) for applicants to be competitive for top five schools in the world? Find the last two digits of $3^{45}$ I was wondering if there is a simpler way to find the last to digits of a power such as $3^{45}$. I reduced it modulo 100 to get the answer, which is 43. But I was curious if there was a simpler, or more eloquent way to get this result.  Here's my method:  Using the fact that $3^4=81\equiv -19 \pmod{100}$ and $19^5=2,476,099\equiv -1 \pmod{100}$  $$ 3^{45}\equiv 3(3^4)^{11}\equiv 3(-19)^{11}\equiv 3(-19)(19^5)^2\equiv 3(-19)(-1)^2 \equiv -57 \equiv 43 \pmod {100} $$  I just felt like this was sort of a messy way to find out that the last two digits are 43. Is there a way to view &amp; modify the code of standard lightning components that are visible when editing the Napili template in the community builder. I was wondering if I could modify some of the code and use it as lightning out. Chain-calling parent initialisers in python How to force Windows 7 to ask for a "fresh" IP address from DHCP server? I'm troubleshooting a certain issue with my DHCP configuration, and need my Windows machine to ask for a "fresh" IP address, so I can see which address DHCP server gives by default.   When I do ipconfig /release followed by ipconfig /renew, Windows "proposes" its old IP address to the DHCP server (just checked with Wireshark, initial "DHCP Discover" message has Option-50 (requested IP address) with Windows machine's old IP).  Tried disabling/enabling network adapter. Same behavior.  Question: How can I force Windows to just ask for a new IP address, without proposing its old IP address. I'm running CentOS 7.0 build 1503. I'm trying to search for commands I don't know the name of by searching the man descriptions for multiple keywords using AND (e.g. search for a command containing the both words "create" and "group" in the description).  So I came across the command apropos (which is an alias for man -k). The default behaviour when given multiple keywords is to return any command matching any of the keywords, which isn't what I want. According to the man page the option -a specifies using AND, to return matches only when all words match.  But this isn't what I'm seeing in practice, so I need some help here - e.g. using Centos 7, if I do:  apropos create user   I see a huge list of commands, seemingly matching any command with create or user in the description, as expected. Noting the list includes an entry for the command 'useradd (8) - create a new user...'.  If I do:  apropos -a create user   I only get 2 commands (environ and open). This is what I'm confused by - I would expect it to return the command 'useradd (8) - create a new user', because i can see from the previous search that command does contain those words in the description. Also the 2 returned commands don't obviously seem to be anything about creating users. What am I doing wrong? I'm sure it's something embarrassingly obvious, for that I apologise in advance.  Thanks Which is correct: "drive safe" or "drive safely"? When someone is going to drive their car somewhere, I always used to say "drive safely" to them. Recently I was told I should say "drive safe."    (From: )   Which one is correct?  Similarly, is "do good" correct? Do shots fired before the match starts count towards your end-of-game statistics? In Overwatch, there is short countdown for a few minutes before the match begins. During this time, the defending team is allowed to roam the map to set up their defenses, while the attacking team sits in a small lobby until the game starts and the doors open. (Or in KotH-type maps, both teams wait in a lobby.)  While players wait, it's not uncommon for them to fire off random shots, especially since a lot of the environment is destructable (it's entertaining to watch plush onions sail through the air in the arcade).   Do shots fired before the match starts count towards accuracy statistics at the end of the game?