Let $a,m,n\in\mathbf{N}$. Show that if $\gcd(m,n)=1$, then $\gcd(a,mn)=\gcd(a,m)\cdot\gcd(a,n)$.  Proof:  Let $u,v\in\mathbf{Z}$ such that $\gcd(m,n)=um+vn=1$. Let $b,c\in\mathbf{Z}$ such that $\gcd(m,a)=ab+cm$. Let $d,e\in\mathbf{Z}$ such that $\gcd(a,n)=ad+en$.  So $\gcd(a,m)\cdot\gcd(a,n)=a^2bd+cmen+aben+emad$.  Where do I go from here?
How can I prove that if $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \times \gcd(b, c)$?  By eea there exists $ax+by=1$ from $\gcd(a,b)=1$ so a and be are co-primes there also exists $dk=a$ and $dj= b$ where $d=\gcd(a,b)=1$ this is all the information I have gathered from the question but I dont know how to approach and solve it. Can anyone help explain to me how to arrive at the answer? Thanks!
How can I use GDAL to batch define a projection? What is the use of percent signs (%) at the end of lines? (Why is my macro creating extra space?) Space after LaTeX commands How to select kernel for SVM? How can I find $\gcd(a^m+1,a^n+1)$ with $a,m,n$ positive integers? How can I find $\gcd(a^m+1,a^n+1)$ with $a,m,n$ positive integers?    I have this idea:  Let $d=\gcd(m,n)$. Then there exist positive integers $x,y$ such that $mx-ny=d$ (WLOG). We shall find $G=\gcd(a^m+1,a^n+1)$.  If $m,n$ are odd, then $d$ is odd, therefore one and only one of $x,y$ is even. We have:  $$a^{ny}(a^d+1)=a^{mx}+a^{ny}=(a^{mx}-1)+(a^{ny}+1)=(a^{mx}+1)+(a^{ny}-1).$$  If $x$ is even and $y$ is odd, then $a^{m}+1\mid a^{mx}-1$ and $a^{n}+1\mid a^{ny}+1$, therefore $G\mid a^{ny}(a^d+1)$. If $x$ is odd and $y$ is even, then $a^{m}+1\mid a^{mx}+1$ and $a^{n}+1\mid a^{ny}-1$, thus $G\mid a^{ny}(a^d+1)$. However, since $\gcd(a^m+1,a^{ny})=\gcd(a^n+1,a^{ny})=1$, so $\gcd(G,a^{ny})=1$, hence $G\mid a^d+1$. We also have $a^d+1\mid a^{m}+1$ and $a^d+1\mid a^{n}+1$, so $a^d+1\mid G$. Thus $G=a^d+1$.  If $v_2(m)=v_2(n)=v_2(d)=k&gt;1$, then there exist some odd numbers $m_1,n_1,d_1$ such that $m=2^km_1,n=2^kn_1,d=2^kd_1$. We shall have $m_1x-n_1y=d_1$, so one and only one of $x,y$ is even, and we can use the same argument when $m,n$ are odd, so $G=a^d+1$.  However, if $v_2(m) \neq v_2(n)$, I cannot find any solutions for this. I think that $G \in \{1,2\}$, but I cannot prove or disprove it. How can I find $G=\gcd(a^m+1,a^n+1)$ if $v_2(m) \neq v_2(n)$ ? Moreover, is there anything from my arguments that can be improved ? What is this rhetorical scheme or literary device called? I can't recall the name of the rhetorical scheme or literary device involved with using the same word more than once within the same sentence but with different meanings:  "The only thing we have to fear is fear itself."    fear - verb fear - noun   "Turn right right here."   right - adv. directional right - adv. in this place   "My feet are but feet away."   feet - noun lower appendage feet - noun distance After migration to MacOSX 10.11 "El Capitan", how to run simple utilities such as "texdoc" and "kpsewhich"? Last night, I migrated my MacBook's operating system from 10.10 "Yosemite" to 10.11 "El Capitan". Everything seemed to go smoothly. Happily, regarding my TeX-related work, I am able to use the GUI front ends "TeXworks" and "TeXshop" just fine since completing the OS upgrade. I use MacTeX2015, with all the latest updates installed.  Oddly, though, since completing the upgrade I can no longer run any MacTeX command-line utilities such as texdoc and kpsewhich. (Actually, as I just discovered, I can no longer run pdflatex either from a command line. However, I never do run pdflatex from a command line prompt, so that's not an urgent problem for me -- at least not for now.)  The suggestions given in the posting  do not seem to be applicable to this case, as they all seem to revolve around setting up /usr/texbin correctly -- /usr/texbin is no longer allowed under MacOSX 10.11. The document  by Herb Schulz deals in depth with adjusting settings in the GUI front ends, but it appears not to touch on what to do about /usr/texbin, which appears to be the now-missing link to TeX's command line utilities.   Pointers and solutions most welcome! Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$ Suppose $f$ is continuous and periodic on the reals with period 1. Prove that if $x\in[0,1]$ is an irrational number, then  $$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$$   Suggestion: First consider $f(t) = e^{2\pi(ikt)}$ where k is an integer.   I can see that this is a limit of a weighted average, but the suggestion throws me off. I've seen the suggestion in fourier transforms but it's not clicking at the moment. Any help would be welcome. LINQ's Distinct() on a particular property I am playing with LINQ to learn about it, but I can't figure out how to use Distinct when I do not have a simple list (a simple list of integers is pretty easy to do, this is not the question). What I if want to use  on a list of an Object on one or more properties of the object?  Example: If an object is Person, with Property Id. How can I get all Person and use Distinct on them with the property Id of the object?  Person1: Id=1, Name="Test1" Person2: Id=1, Name="Test1" Person3: Id=2, Name="Test2"   How can I get just Person1 and Person3? Is that possible?  If it's not possible with LINQ, what would be the best way to have a list of Person depending on some of its properties in .NET 3.5? Unable to post - login problems with Area 51 ... again I recently signed up for Area 51 with my stackexchange network account/ID, verified the account and logged in.  I see the main page  where I confirm that I am logged in.  I click on the discuss link on the black menubar at the top, which takes me to the discussion zone. From here I click on ask question, which asks me to add  a login even though I have already logged in with my stackexchange ID.  Here if I choose the stackexchange login option then it takes me back to the main page, from where the cycle begins again. Why should the minimal class be avoided? Is the Formula Logically Valid? Relation between real roots of a polynomial and real roots of its derivative I have this question which popped in my mind while solving questions of maxima and minima. First Case:Let $f(x)$ be an $n$ degree polynomial which has $r$ real roots. Using this can we say anything about the number of real roots of $f'(x)$?  Second Case:Suppose, $f(x)$ has all  $n$ real roots. Then will all of its derivatives also have all real roots?  Also, if any of its derivatives do not have all real roots, then will $f(x)$ also have not all real roots? If the above is true then what about its converse?  Comment Case:For the third case:Suppose f'(x) is a 5 degree polynomial with 3 real roots.Then f(x) will be a 6 degree polynomial.(correct me if I am wrong).What are the possible no. of roots that f(x) can have(3,4,5 etc.?).Basically I am asking for an example.Also it would be great if you follow all cases with an example like in the 4th case. Attempting official deploy process with single sandbox may reset/remove all existing work? Prove that $\frac{x_1}{1+x_2+x_3+\ldots+x_n}+\frac{x_2}{1+x_1+x_3+\ldots+x_n}+\ldots+\frac{x_n}{1+x_1+x_2+\ldots+x_{n-1}}\ge\frac{n}{2n-1}$. If $x_1,x_2,\ldots,x_n&gt;0$ and $x_1+x_2+\ldots+x_n=1$, prove that $$\frac{x_1}{1+x_2+x_3+\ldots+x_n} + \frac{x_2}{1+x_1+x_3+\ldots+x_n} +\ldots + \frac{x_n}{1+x_1+x_2+\ldots +x_{n-1}} \ge \frac {n}{2n-1}$$  This can easily be simplified: $$\frac{x_1}{2-x_1} + \frac{x_2}{2-x_2} +\ldots + \frac{x_n}{2-x_n} \ge \frac {n}{2n-1}$$  I could try using the Cauchy-Schwarz inequality: $$\frac{x_1}{2-x_1} + \frac{x_2}{2-x_2} +\ldots + \frac{x_n}{2-x_n} = \frac{x_1^2}{2x_1-x_1^2} + \frac{x_2^2}{2x_2-x_2^2} +\ldots + \frac{x_n^2}{2x_n-x_n^2} \ge \frac{(x_1+x_2+\ldots+x_n)^2}{2(x_1+x_2+\ldots+x_n)-(x_1^2+x_2^2+\ldots+x_n^2)}=\frac{1}{2-(x_1^2+x_2^2+\ldots+x_n^2)}=\frac{n}{2n-(x_1^2+x_2^2+\ldots+x_n^2)n}$$  It's left to prove that $$(x_1^2+x_2^2+\ldots+x_n^2)n \ge 1$$ or $$(x_1^2+x_2^2+\ldots+x_n^2)n=1$$I can continue using the Cauchy-Schwarz inequality again: $$(x_1^2+x_2^2+\ldots+x_n^2)n=(x_1^2+x_2^2+\ldots+x_n^2)(1+1+\ldots+1)\ge(x_1+x_2+\ldots+x_n)^2=1$$ So I've proved it myself. (I've shown this proof after editing, I didn't post the question with the solution in the details. I found the proof a while after putting the question here). What does The Doctor's Speech in The Rings of Akhaten refer to? I recently watched the brilliant speech from The Rings of Akhaten again. For those who haven't seen it or don't remember it:     I walked away from the last Great Time War. I marked the passing of   the Time Lords. I saw the birth of the universe and I watched as time   ran out, moment by moment, until nothing remained. No time. No space.   Just me. I walked in universes where the laws of physics were devised   by the mind of a mad man. I've watched universes freeze and creations   burn. I've seen things you wouldn't believe. I have lost things you   will never understand. And I know things. Secrets that must never be   told. Knowledge that must never be spoken. Knowledge that will make   parasite gods blaze. So come on, then. Take it! Take it all, baby!   Have it! You have it all!   Some of these I understand:     I walked away from the last Great Time War.    Clear enough     I marked the passing of the Time Lords.   Again makes sense, given he "killed" them.     I saw the birth of the universe and I watched as time ran out, moment by moment, until nothing remained. No time. No space. Just me. I walked in universes where the laws of physics were devised by the mind of a mad man. I've watched universes freeze and creations burn.   I'm very vague on this put is some of it related to The Big Bang?     I've seen things you wouldn't believe. I have lost things you will never understand. And I know things. Secrets that must never be told. Knowledge that must never be spoken. Knowledge that will make parasite gods blaze.   Possibly relating to the question which must never be answered, but at this point we've not been to Trenzalore so does he even know what knowledge is and why?  What do the unknown sections of these speech refer to? I remember (vaguely) reading somewhere that it is wrong to use the present perfect when the subject of a sentence is someone who is long dead:     *Einstein has visited the Philippines.   My question is, What if the dead person is not the subject but an object:     Leo is the best president our country has/had ever had. (The ones who   succeeded him were inferior, even our current president.)   Which tense is proper here How much money do I need to make in order to generate 1k/month for the rest of my life? Is that even possible? If yes, in what way? Edit: I am 30 years old. Let $\mathbb F$ be a field and $\mathbb  K $ be an extension field of $\mathbb F$  such that $\mathbb K$ is algebraically closed.   Let $\mathbb L$ be the field of all elements of $\mathbb K$ which are algebraic over $\mathbb F$. Then $\mathbb L_{|\mathbb F}$ is an algebraic extension.   My question is : Is $\mathbb L$ algebraically closed ?  I am trying to prove the existence of algebraic closure, so please don't assume that every field has an algebraic closure. If in a UFD every maximal ideal is principal then it is a PID