I'm trying to prove\show the associative property for the $\gcd(a,b)$ function.   $$\gcd( a ,\space \gcd(b,c) ) = \gcd(\space \gcd(a,b) , c )$$  Let:  $$e = \gcd( a ,\space \gcd(b,c) )\text{ and }f = \gcd(\space \gcd(a,b) , c )$$  From here you can see that  $$ e\mid a,\gcd(b,c) \Longleftrightarrow e\mid a,b,c $$  And similarly,  \begin{align*} f|\gcd(a,b),c &amp;\Longleftrightarrow f\mid a,b,c\\ &amp;\Longleftrightarrow f\mid e\\ &amp;\Longleftrightarrow e\mid f \end{align*}  If $ e\mid f$ and $f\mid e \Longrightarrow |e| = |f| $ and since $e$ and $f$ are both non-negative we have the result, $e = f$, as required.  Is this proof sound?
Prove $\gcd(a,b,c)=\gcd(\gcd(a,b),c)$ for $0\ne a,b,c\in \Bbb{Z}$. I tried solving it with sets but I sense there are some details I am missing. I would truly appreciate your reference.
Prove that:$\int_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) dx$ Very thin white line in colortbl Whenever I use the colortbl package to format tables, a thin white space between the cells sometimes shows up on screen (not in print for as far as I can tell). For example in this table:    Is there an easy way to enlarge each rectangle by a tiny amount so that the white space is not visible anymore?  (My guess is that the white space appears due to rounding in my pdf viewer, evince. I don't know if other pdf viewers have the same issue. Regardless, I would like to have my pdf file look good in common pdf readers) What do two question marks together mean in C#? Ran across this line of code:  FormsAuth = formsAuth ?? new FormsAuthenticationWrapper();   What do the two question marks mean, is it some kind of ternary operator? It's hard to look up in Google. Many researchers list out their submitted manuscripts on their website. 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Is there a convenient way to have equations print the label along with the tag/number, so that I can easily check the label in the compiled document?  Something like this would be perfect:  x = y               (14.1) eq:my_label How to configure logback to skip logging messages from org.package.* with all levels below WARN? Rubik's Cube Combination Could anyone explain why the number of legal or reachable combinations of a $3\times 3\times 3$ Rubik's Cube is $1/12\mbox{th}$ of the total. I understood the logic behind the total number of combinations: $8! \cdot 2^{12} + 3^{12} \cdot 12!$. What I am not able to understand is why do we divide this quantity by $2 \cdot 3\cdot 2$. I don't have any idea about parity or group/set theory. Is there another possible explanation for it? How do berry mutations work? What do potential postdoc advisers seek in research statement letters? I am putting together my application for a postdoctoral position in theoretical condensed matter physics. 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So when I come to the next step of running database updates, no hook_update_N functions are run because the module thinks it's already the correct version.  The only method I can see is to manually set the module version in hook_enable, however I'll then need to update the module code to remove the hook_enable to make sure it doesn't run if the module is ever re-enabled in the future, so obviously this approach is a bit dirty.  I'm hoping there's a better solution? Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ Possible Duplicate:          On Wikipedia, it says that:     Matrix representation of complex numbers   Complex numbers $z=a+ib$ can also be represented by $2\times2$ matrices that have the following form: $$\pmatrix{a&amp;-b\\b&amp;a}$$   I don't understand why they can be represented by these matrices or where these matrices come from. Plural or singular verb for collective term Is there a program equivalent to lint for LaTeX? Is there a program equivalent to lint for LaTeX? (lint checks C code for syntax errors and possible mistakes.) I'm writing a simple bash script, but I need it to check whether it's being run as root or not. I know there's probably a very simple way to do that, but I have no idea how.  Just to be clear: What's a simple way to write a script foo.sh, so that the command ./foo.sh outputs 0, and the command sudo ./foo.sh outputs 1? My machine has a SSD (fairly small) and a Hard disk (large). I want to have windows and a couple of programs on the SSD but I want to have the documents folders (such as my documents and my downloads) on the HD. I also want the default install directory pointing at the larger HD. Can anyone advise or point me to a relevent guide? Thanks. Why is my query suddenly slower than it was yesterday? [Salutations]  (check one)  [ ] Well trained professional, [ ] Casual reader, [ ] Hapless wanderer,   I have a (check all that apply)  [ ] query [ ] stored procedure [ ] database thing maybe     that was running fine (if applicable)   [ ] yesterday [ ] in recent memory [ ] at some point    but is suddenly slower now.  I've already checked to make sure it's not being blocked, and that it's not the victim of some long running maintenance task, report, or other out of band process.  What is the problem, what should I do, and what information can I provide to get some help?  [*Insert appropriate closing remarks*] How do I transfer iPhone contacts to an Android phone?