What is different $(a_0, a_1, ..., a_d) = (1)$ and $\gcd(a_0, a_1, ..., a_d) = 1$ in arbitrary UFD?
Does existence of $\gcd$ implies PID?
Given 4 integers, $a, b, c, d > 0$, does $\frac{a}{b} < \frac{c}{d}$ imply $\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$? How can I downgrade a package to an older version via apt-get?  Other tools are also acceptable but apt-get is preferred. Why is Euler's Totient function always even? I want to prove why $\phi(n)$ is even for $n&gt;3$.  So far I am attempting to split this into 2 cases.  Case 1: $n$ is a power of $2$. Hence $n=2^k$. So $\phi(n)=2^k-2^{k-1}$. Clearly that will always be even.  Case 2: $n$ is not a power of $2$. This is where I am unsure where to go. I figure I will end up using the fact that $\phi(n)$ is multiplicative, and I think I'll get a $(p-1)$ somewhere in the resulting product which will make the whole thing positive, as $p$ is prime implies $(p-1)$ is even. As a native speaker of English, the gerund version of this sentence sounds better:  infinitive:     When used together in chains,   extension methods are an unprecedented   tool to produce extremely concise   code.   gerund:     When used together in chains,   extension methods are an unprecedented   tool for producing extremely concise   code.   But how can I explain to someone learning English how to decide in situations like these whether to use the infinitive or the gerund? I know base 2 and base 10 , MB and MiB, but i'm not understanding the byte counts shown for a directory, by du and by windows explorer The du command shows 43G in use. I'm not sure whether that's base 2 in the sense of 2^30 or if it's base 10, in the sense of 10^9.  But anyhow -   Running this du command from cygwin, to see the size of the windows directory (done from an administrative command prompt - since it seemed to have some issues reading some directories otherwise)  user@comp ~ $ du -sh /cygdrive/c/windows 43G     /cygdrive/c/windows   Windows Explorer  which I understand shows it in 'base 2', (a larger unit for KB,MB,GB than base 10 KiB, MiB e.t.c.), and hence a lower quantity of that larger unit is required to give the same result.    Windows explorer shows      46.6GB  , over 50 billion bytes.    (according to windows explorer)  or  46.8GB  , over 50 billion bytes. (according to windows explorer)  So, du aside.  Even if I multiply the larger figure, 46.8GB  by 1.048576 then I get 49.0733568 that's not over 50 billion. So that's part A of the question,  but then also, part B of the question, why is the du figure so low. Relationship between Binomial and Beta distributions I'm more of a programmer than a statistician, so I hope this question isn't too naive.  It happens in sampling program executions at random times. If I take N=10 random-time samples of the program's state, I could see function Foo being executed on, for example, I=3 of those samples. I'm interested in what that tells me about the actual fraction of time F that Foo is in execution.  I understand that I is binomially distributed with mean F*N. I also know that, given I and N, F follows a beta distribution. In fact I've verified by program the relationship between those two distributions, which is  cdfBeta(I, N-I+1, F) + cdfBinomial(N, F, I-1) = 1   The problem is I don't have an intuitive feel for the relationship. I can't "picture" why it works.  EDIT: All the answers were challenging, especially @whuber's, which I still need to grok, but bringing in order statistics was very helpful. Nevertheless I've realized I should have asked a more basic question: Given I and N, what is the distribution for F? Everyone has pointed out that it's Beta, which I knew. I finally figured out from Wikipedia () that it appears to be Beta(I+1, N-I+1). After exploring it with a program, it appears to be the right answer. So, I would like to know if I'm wrong. And, I'm still confused about the relationship between the two cdfs shown above, why they sum to 1, and if they even have anything to do with what I really wanted to know. No Drupal Site Found when clearing cache I installed drush using composer following the instructions here:   When trying to run drush cc all I get this error:     No Drupal site found, only 'drush' cache was cleared.[warning] 'all'   cache was cleared. [success]   I added this line:  export PATH="/Applications/MAMP/bin/php/php5.6.2/bin:$PATH"   to my .bash_profile and restarted bash and MAMP. Still no luck. Any ideas?   This is my drush status:   Drupal version         :  7.36                                                               Site URI               :  http://default                                                     Database driver        :  mysql                                                              Database hostname      :  localhost                                                          Database port          :                                                                     Database username      :  root                                                               Database name          :  srod                                                     PHP executable         :  /usr/bin/php                                                       PHP configuration      :                                                                     PHP OS                 :  Darwin                                                             Drush script           :  /Users/srod/.composer/vendor/drush/drush/drush.php   Drush version          :  7.0-dev                                                            Drush temp directory   :  /tmp                                                               Drush configuration    :                                                                     Drush alias files      :                                                                     Drupal root            :  /Applications/MAMP/htdocs/srodsite                             Site path              :  sites/default      I'm running it from Drupal project folder. Modern textbook on statistical field theory with an emphasis on applications to non-equilibrium phenomena? Seeking podcasts related to geospatial technologies? I would like to make a list of podcasts related to geospatial technologies.  Do you listen to any that you find interesting and informative? How can I speed up D&D 3.5e combat? I'm looking for ways to speed up the combat in my D&amp;D 3.5 game. It seems that turns - both player and non-player - simply take too long. I have thought about giving everyone a 30 second timer, but I'm afraid that might inhibit their ability to come up with strategy and tactics. Flagging summary improvement: filters Quieting down a Dell PowerEdge r710 I run a Dell PowerEdge r710 server out of my house and I'm moving in with roommates in a couple weeks. Unfortunately I don't think they'll appreciate the ~70dBA it generates and I'm looking for ways to make it quieter.  I looked into replacing the fans with something less noisy, but unfortunately none of the quieter options move nearly enough air. The best one I found has been the , but at less than half the CFM (30.3 vs 67.85 on the stock  fans) it doesn't seem powerful enough.  I've also thought about putting  around the server to quiet it down, but acoustic panels also have the downside of acting like insulation, so heat could turn into an issue. Changing subject and verb positions in statements and questions Stone-Weierstrass theorem with $p(1/x)$ Using awk to sum the values of a column, based on the values of another column I am trying to sum certain numbers in a column using awk. I would like to sum just column 3 of the "smiths" to get a total of 212. I can sum the whole column using awk but not just the "smiths". I have:  awk 'BEGIN {FS = "|"} ; {sum+=$3} END {print sum}' filename.txt   Also I am using putty. Thank you for any help.  smiths|Login|2 olivert|Login|10 denniss|Payroll|100 smiths|Time|200 smiths|Logout|10 No field properly between $\mathbb Q$ and $E$ iff $G(K/\mathbb Q) \cong A_4$ or $S_4$ Need clarification on the Draw 4 rule. - Player 1 lays down a Draw 4 Card and says the color is Red. - Player 2 does not have a Red card but has a Draw 4 Card. Can Player 2 lay down his Draw 4 Card and have Player 3 draw 8 cards? Pass in an array of Deferreds to $.when() 100 apples are/is considered as a large number of apples How to convert numbers between hexadecimal and decimal