If $3\mid a$, then $3\mid a^2$
Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
I am attempting to solve this counting problem through combinatorial argument. The following is the equation I am given:   $$\sum_{i=1}^n (i-1)(n-i) = \binom{n}{3}$$  I understand that the right-hand side of this equation represents a set of $n$-elements out of which we choose 3. For example I believe we can say suppose we have a group of $n$ people and we want to choose 3 out of $n$ to be in a committee. However I'm not sure how to express the left-hand side in words. If forming a committee is an appropriate way to tackle this problem then I know the left-hand side must utilize the addition and multiplication principles, but I don't know how to put it into words. Also my intuition tells me that in solving this we should first flip $$(n-i)(i-1)$$  Thanks! How to enable projection transformation in QGIS Use physical harddisk in Virtual Box I'm travelling from Madrid to Brisbane, but I have a 9 hours layover at Dubai Airport. I do not plan to leave the airport, and the connection is with the same airline (Emirates). I'm from El Salvador.    Do I need a transit visa? Check Constraint with Case Statement I want to add a check constraint on a column if entered value for column is greater than 3 than it should be saved as 1 else entered value. For that scenario I am using below query but on insertion of data it shows an error message.  create table tblTestCheckConstraint  ( id int, NewColumn int )   alter table tblTestCheckConstraint add constraint chk_tblTestCheckConstraint_NewColumn1 CHECK   (     CASE         WHEN NewColumn &gt;4 THEN 1         ELSE NewColumn     END = 1 )   Insert Statement:   insert into tblTestCheckConstraint values ( 1,5) insert into tblTestCheckConstraint values ( 1,2)   Error Message:   Msg 547, Level 16, State 0, Line 1 The INSERT statement conflicted with the CHECK constraint "chk_tblTestCheckConstraint_NewColumn1". The conflict occurred in database "DBName" , table "dbo.tblTestCheckConstraint", column 'NewColumn'. The statement has been terminated. Set keyboard caret position in html textbox Does anybody know how to move the keyboard caret in a textbox to a particular position?  For example, if a text-box (e.g. input element, not text-area) has 50 characters in it and I want to position the caret before character 20, how would I go about it?  This is in differentiation from this question:  , which requires jQuery. Find and replace text within a file using commands So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively.  assume c is a constant.  I am having huge trouble to understand the integration of the following expression.  $P(Y&lt;c/u(X))$  $=\int_{t}^{\infty}f_X(x)\int_{0}^{c/u(x)}f_Y(y)dydx +\int_{0}^{t}f_X(x)\int_{c/u(x)}^{\infty}f_Y(y)dydx$  where t is the cross-point that $u(x)$ change sign    here $c/u(x)$ is given by the plot below, t is the point crossing the zero:   Confusion:   I don't understand the second integration of the second term "$\int_{c/u(x)}^{\infty}f_Y(y)dydx$", this isn't right because Y an X only defined for y>0, and x>0. So it's the first quadrant in this plot. Is any aspect of the supernatural testable? What level of proof is possible for the supernatural? Correlation coefficient between a (non-dichotomous) nominal variable and a numeric (interval) or an ordinal variable I've already read all the pages in this site trying to find the answer to my problem but no one seems to be the right one form me...  First I explain you the kind of data I'm working with...  Let's say that I have an array vector with several names of city, one for each of 300 users. I also have another array vector with scores response to a survey of each user or a continuous value for each user.  I would like to know if exist a correlation coefficient that compute the correlation between these two variables so, between a nominal and a numeric/continuous or ordinal variables.  I've searched on the Internet and in some pages they suggest to use the contingency coefficient or Cramer's V or Lambda coefficient or Eta . For each of this measure the just say that they could be applied for such data in which we have a nominal variable and interval or numerical variable. The thing is that searching and searching, trying to understand every one of them, sometime is written or watching the examples that they are reasonable to use them if you have dichotomous nominal variable, except for Cramer's V, other time is not written any requirement for the type of data. A lot of other pages say that is right to apply regression instead, that is right, but I would just simply like to know if there is a coefficient like pearson/spearman for this kind of data.  I also think that is no so properly to use Spearman Correlation coeff since the cities are not sortable.  I have also built the function of Cramer'sV and Eta by myself (I'm working with Matlab) but for Eta they don't talk about any p-value to see if the coefficient is statistically significant...  In the matlabWorks site there is also a nice toolbox that says to compute eta^2 but the kind of input it needs is not understandable.  Is here someone that have done a test like mine? If you need more detail to understand the kind of data I'm using just ask me and I'll try to explain you better. What should I do if two libraries provide a function with the same name generating a conflict? What should I do if I have two libraries that provide functions with equivalent names? Say a creature owns multiple items with wording similar to this:  As an Action, you may speak the command word to [...]  Say the creature has deliberately obtained such items, perhaps by creating them itself, to have the same command word. Can this creature use one action to speak the shared command word and activate multiple items at once? The purpose of this question is not to cheese my DM. I am writing a short story and like to keep my D&amp;D fiction as close to rules-accurate as possible. Using GParted to remove windows partition Frequency count of two column in R Minimize absolute values of errors instead of squares Is there really no English equivalent to German's “Fachidiot”? There was the following paragraph in the article that came under the title, “GOP and the rise of Anti-knowledge” written by  (October 29, 2015):     “English unfortunately doesn’t have a precise word for the German   “Fachidiot,” a narrowly specialized person accomplished in his own   field but a blithering idiot outside it. In any case, a surgeon is   basically a skilled auto mechanic who is not bothered by the sight of   blood and palpitating organs. We need the surgeon’s skills on pain of   agonizing death, and reward him commensurately but that does not make   him a Voltaire.”    In Japanese we have a word, 専門バカ, of which literal translation is “specialist idiot” meaning exactly same as defined the above - a narrowly specialized person accomplished in his own field but a blithering idiot outside it.  I’m really curious about the absence of short descriptions of the specialist who knows nothing more than his specialty area in English like German “Fachidiot,” and Japanese "専門バカ."  Isn’t there really any short word counterpart to “Fachidiot,” specialist’s ignorance, for instance, in English?  P.S.  I found "narrow-minded specialisit" as a German-to-English translation of "fachidiot" in the web site, Interglot translation dictionary. When I wake the laptop from the sleep I need to get the VPN state as I left it (turned on). Can't find the setting of autoconnect in VPN settings... Need some easy way to ask the system to autoconnect the VPN when the internet is available.  Edit based on comments:  Ubuntu 18.04 has no option to Always connect to VPN when using using this connection. When clicking between "Week", "Month", "Quarter", "Year" and "All" on the users page, the name I have typed in to filter on disappears. I feel like it should stay there, so that I can type in my name, and see how I've done in a week/month/etc. See my edit below, this is not totally a duplicate of   My reverse-search function in the CLI is broken and I don't know how to fix it back...  When I hit ctrl+r and search a command I did earlier, find it then press the right arrow key to use it, the output is broken. Here is an example:  # I reverse-searched the word "test" then pressed right arrow yoshiji@x1carbon:~/path/to/git/folder (master) $ch)`test': rake test #                                              ^^^^^^^^^^ weird output #                                                     ^ cursor right there   It also "shifts" my cursor to a different position. If I press enter, the right command is executed (rake test in this case) so it means the text displayed is not the same as what is actually executed.  I know this is most-likely related to my PS1 export in my ~/.bashrc, so here are the last lines of this file:  # Git branch in prompt. git_branch() {   branch=$(__git_ps1 2&gt; /dev/null)   if [ "$branch" = " (master)" ]; then      echo -e "\033[33m${branch}\033[00m" # yellow   elif [ "$branch" = "" ]; then      echo -e "\033[90m (no git)\033[00m" # red   else      echo -e "\033[32m${branch}\033[00m" # green   fi } PS1='\u@\h:\w$(git_branch) \$ '   This is something I wrote to displays the git branch in orange (if master branch) or else in green (or no git in gray).  Do you guys have any idea on what is causing this weird behavior?    Edit: This is a different problem from the  in the way that I am using echo -e "some string" to return a string having color+name of the git branch from the function git_branch(). Using echo -e does not escape the characters \[ and \] (try it yourself in your CLI: echo -e "\[\033[33m\]test").  Ideally, I would like to keep those conditional colors based on the branch's presence &amp; name (gray if no branch, orange if master, else green). Give an example about limits: $\lim\limits_{x\to0}f(x^{2})$ exists but $\lim\limits_{x\to0}f(x)$ does not.