Why don’t numbers with a difference of 2 have any common factors beside 1 and possibly 2?
Prove divisibility law $\,b\mid a,c\,\Rightarrow\, b\mid ka + lc$ for all $k,l\in\Bbb Z$
How many copies of the DeLorean time machine co-existed in the Back to the Future series? so i have a 30A disconnect available for use at a customer site for running 2 5HP motors ( at 480V)  the only question i have though is if the line is sized for a 10HP motor would both the 5HP motors need additional fuse protection?   Or could i get away without having addtional fuses protecting the motors ? If $x+y+z=xyz$, prove $\frac{2x}{1-x^2}+\frac{2y}{1-y^2}+\frac{2z}{1-z^2}=\frac{2x}{1-x^2}\times\frac{2y}{1-y^2}\times\frac{2z}{1-z^2}$ given that $x^2~,~y^2~,~z^2\ne1$  I came across this question in an ancient ($19$th century) Trigonometry book, and this is the method they use to prove the result (please note: I understand this method fully): Let $x=\tan A$ ,$~y=\tan B$ and $z=\tan C$ which is acceptable without any loss of generality. This means that we are saying that $$\tan A+\tan B+\tan C=\tan A\tan B\tan C$$ Consider $\tan (A+B+C)$: $$\tan (A+B+C)=\frac{\tan A+\tan B+\tan C-\tan A\tan B\tan C}{1- \tan A \tan B- \tan C \tan A - \tan B \tan C}$$ So if $~\tan A+\tan B+\tan C=\tan A\tan B\tan C~~$ then $~~\tan (A+B+C)=0$. Hence, let $~A+B+C=\pi$. Now consider $\tan (2A+2B+2C)$: $$\tan (2A+2B+2C)=\frac{\tan 2A+\tan 2B+\tan 2C-\tan 2A\tan 2B\tan 2C}{1- \tan 2A \tan 2B- \tan 2C \tan 2A - \tan 2B \tan 2C}=0$$ $$\implies \tan 2A+\tan 2B+\tan 2C=\tan 2A\tan 2B\tan 2C$$ $$\implies \frac{2\tan A}{1-\tan^2 A}+\frac{2\tan B}{1-\tan^2 B}+\frac{2\tan C}{1-\tan^2 C}=\frac{2\tan A}{1-\tan^2 A}\times\frac{2\tan B}{1-\tan^2 B}\times\frac{2\tan C}{1-\tan^2 C}$$ $$\therefore\frac{2x}{1-x^2}+\frac{2y}{1-y^2}+\frac{2z}{1-z^2}=\frac{2x}{1-x^2}\times\frac{2y}{1-y^2}\times\frac{2z}{1-z^2}$$ as required.  My question is, is there any other way of proving this without this rather heavy use of trigonometry? I also would prefer not to work through heaps of algabraic manipulation and expansion to obtain the required result, although if that's necessary I will put up with it ;) Thank you for your help. In my flagging summary under the statistics, there is 1 flag listed as "disputed".  This is the first time I've ever seen a disputed flag in my summary, so I'm just curious, what is a disputed flag? How would you even dispute a flag? Is this the same as voting a flag as invalid? I will quote a question from my textbook, to prevent misinterpretation:     Let $G$ be a finite abelian group and let $m$ be the least common multiple of the orders of its elements. Prove that $G$ contains an element of order $m$.   I figured that, if $|G|=n$, then I should interpret the part with the least common multiple as $lcm(|x_1|,\dots,|x_n|)=m$, where $x_i\in G$ for $0\leq i\leq n$, thus, for all such $x_i$, $\exists a_i\in\mathbb{N}$ such that $m=|x_i|a_i$. I guess I should use the fact that $|x_i|$ divides $|G|$, so $\exists k\in \mathbb{N}$ such that $|G|=k|x_i|$ for all $x_i\in G$. I'm not really sure how to go from here, in particular how I should use the fact that $G$ is abelian. Classification of automorphisms of projective space Problem using atan in pgfplots Canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism biblatex ieee style same authors Where does gravitational waves' energy go? Following the measurement of gravitational waves, many sources described them and explained they carry energy away. What I don't get is how this energy will get transfered back to anything else.  If the fabric of space-time itself is vibrating, it would seem to be in impossible for any physical object to gain this energy.    What am I missing?  How would one hypothetically get energy out of gravitational waves?  If impossible, does the universe end up with nothing but GW? Universe being flat and why we can't see or access the space "behind" our universe plane? I'm a layman, but i watched some intereting videos about big bang on youtube[michio kaku, hawking this kind of things, not some crackpots :)]   I described everything on my picture:     So is there any possibility to see that inner side, for example, by creating wormhole, or entering new dimension? Or i get something wrong, on the way, and my model is totally wrong?  And another question. I heard that even if you would fly a ship across the universe you would eventually end in the place where you started, because we live in a 3d space that is defined on a surface of big sphere. But why a big sphere? Universe might be a big mobius strip, and the same principle of ending at start will apply. Is it possible?  Thanks in advance, and please don't bash me for silly questions. Im just curious, but i don't have any degree in physics. Im also not an english native speaker. How to get text under mouse in any open window If $S_n$ has cyclic subgroups of these orders or not? This question was asked in my quiz and I am having difficulties in solving it .  Question : For a positive integer $n$ , let $S_n$ denote permutation group on $n$ symbols. Choose correct statements from below. (a) For a positive integer $n$ and for every $m$ with $1\leq m \leq n , S_n $ has a cyclic subgroup of order $m$; (b) For every positive integer $n$ and for every $m$ with $n&lt;m &lt;n!$, $S_n$ has a cyclic subgroup of order $m$; (c) There exists positive integers $n$ and $m$ with $n&lt;m&lt;n!$ such that $S_n$ has a cyclic subgroup of order $m$; (d) For every positive integer $n$ and for every group $G$ of order $n$, $G$ is isomorphic to a subgroup of order $S_n$ .  Attempt: (a) is true as I can choose an element of length $m$ which will generate cyclic subgroup of order $m$. (b) Choose $n=4$ and $m=19$ to contradict (b). (c) I dont know if there can be found any such case and if yes how to do so . (d) I dont know about it . kindly tell. Answer:   A,C,D are true How do I detect orientation on app launch for splash screen animation on iPad! Comparing $\pi^e$ and $e^\pi$ without calculating them How can I  compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ? How to redirect and append both stdout and stderr to a file with Bash? If I have the following code:  MyClass pClass = new MyClass(); pClass.MyEvent += MyFunction; pClass = null;   Will pClass be garbage collected? Or will it hang around still firing its events whenever they occur? Will I need to do the following in order to allow garbage collection?  MyClass pClass = new MyClass(); pClass.MyEvent += MyFunction; pClass.MyEvent -= MyFunction; pClass = null; Proof of properties of injective and surjective functions. I'd like to see if these proofs are correct/have them critiqued.     Let $g: A \to B$ and $f: B \to C$ be functions.  Then:   (a) If $g$ and $f$ are one-to-one, then $f \circ g$ is one-to-one.   (b) If $g$ and $f$ are onto, then $f \circ g$ is onto.   (c) If $f \circ g$ is one-to-one, then $f$ is one-to-one?   (d) If $f \circ g$ is one-to-one, then $g$ is one-to-one?   (e) If $f \circ g$ is onto, then $f$ is onto?   (f) If $f \circ g$ is onto, then $g$ is onto?     (a)  Let $a,b \in A$.  If $(f \circ g)(a)=(f \circ g)(b)$, then $f(g(a))=f(g(b))$.  Since $f$ is one-to-one, we know that $g(a)=g(b)$.  And, since $g$ is one-to-one is must be that $a=b$.  Hence $f \circ g$ is one-to-one.    (b) Since $f$ is surjective we know that for all $c \in C$ there is a $b\in B$ such that $f(b)=c$.  Since $g$ is surjective, there is a $a \in A$ such that $g(a)=b$.  Hence, for all $c \in C$ there is an $a \in A$ such that $(f \circ g)(a)=f(g(a))=f(b)=c$.    (c)  This is false.  Let $A=\{1\}$, $B=\{1,2\}$, and $C=\{1\}$. Define $g(1)=1$ and $f(1)=f(2)=1$. Then $f \circ g$ is injective, but $f$ is not injective.    (d)  This is true.  Let $a,b \in A$ where $a \neq b$.  If $(f\circ g)(a)\neq(f\circ g)(b)$, then $f(g(a))\neq f(g(b))$.  Hence $g(a) \neq g(b)$.  (e)  Since $f \circ g$ is surjective, then for every $ c \in C$ there is an $a \in A$ such that $f(g(a))=c$.   Let $b = g(a)$.  Then $b \in B$ and $f(b)=g(f(a))=c$.  Thus $f(b)=c$.  Hence $f$ is surjective.  (f)  Using the same set up as part (c), we have that $g$ is not surjective since there is nothing in $A$ such that $g$ will go to 1. "Think of" versus "think about" Could someone help me to understand when I should use think of and when think about in sentences?  What is the difference between using one or the other? How can I upload files asynchronously with jQuery? I would like to upload a file asynchronously with jQuery.     $(document).ready(function () {     $("#uploadbutton").click(function () {         var filename = $("#file").val();          $.ajax({             type: "POST",             url: "addFile.do",             enctype: 'multipart/form-data',             data: {                 file: filename             },             success: function () {                 alert("Data Uploaded: ");             }         });     }); }); &lt;script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.2.0/jquery.min.js"&gt;&lt;/script&gt; &lt;span&gt;File&lt;/span&gt; &lt;input type="file" id="file" name="file" size="10"/&gt; &lt;input id="uploadbutton" type="button" value="Upload"/&gt;     Instead of the file being uploaded, I am only getting the filename. What can I do to fix this problem?