I believe that the following statement is true, but I couldn't prove it.  Let $u,m,k\in \Bbb Z$. If $\gcd(u,m)=1$ then there exists an integer $k_1\in\Bbb Z$ such that $\gcd(mk_1+u,mk)=1$ Thanks.
I can't crack this one.      Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that  $a,b,c,z \in \mathbb{Z}$ and $c\neq 0)$
irrationality of $\sqrt{2}^{\sqrt{2}}$. The fact that there exists irrational number $a,b$ such that $a^b$ is rational is proved by the law of excluded middle, but I read somewhere that irrationality of $\sqrt{2}^{\sqrt{2}}$ is proved constructively. Do you know the proof? How do I programmatically determine operating system in Java? I would like to determine the operating system of the host that my Java program is running programmatically (for example: I would like to be able to load different properties based on whether I am on a Windows or Unix platform). What is the safest way to do this with 100% reliability? How to prove whether a matrix has rank $1$ If $u$ $∈ \mathbb R^{m \times 1}$ and $v ∈ \mathbb R^{n \times 1}$ how do you show that the $(m \times  n)$ matrix $uv^T$ has rank $1$? Would providing an example be sufficient to prove it? Deciding what answer to accept: the correct one-liner or the later elaborate one with code? Lately I've been getting some good answers to some specific questions. However, I've been hesitating on what answer I should accept.  Do I go with the short quick answer or do I accept a later answer where someone has elaborated on the previous answer with some code?  It's not really important but it kind of bugs me to not accept an answer when people have taken the trouble of replying.  Here's two examples: Given a set of curved geodesics in 2D space, is there a way to determine the geometrical shape of said space? How can we get "Developer" bugs logged with Technical Support? I recently found a bug, which I mentioned on , as well as submitting a case, and after a few days fumbling around with the case, they closed it by saying that they only provide this type of support for Premier Support customers.  Is there a better channel for reporting bugs than by submitting a case, or is there something "special" we have to do? I realize this question is similar to , except that I've already gone the route of , as suggested by the answer there (i.e. submitting a case), to no avail.  How can I make sure that a bug like this can find its way to someone capable of at least confirming that the behavior was expected? This behavior isn't documented anywhere, and so I assume that it's a bug that needs to be addressed. Why is the Apps flash storage so high and how can it be reduced? This is the disk storage info:    This is the Applications folder info:    Why is the storage usage so high and how can it be reduced? This is on a MacBook Pro Retina 13" with OSX 10.9.5.  Edit: I had tried all the suggestions made by @user3439894 in his answer but didn't mention it before posting. Why must a singularity form inside a black hole? Can the Room of Requirement provide anything if needed? For example:   James Potter wanted to hide from Voldemort, it would have served as a perfect hiding place. Harry Potter wanted all the Hocruxes, he should have gone straight to the room of requirement where he would have found all the Hocruxes along with the Sword. All the muggle borns of the wizarding community could have taken refuge in the room of requirement. Can someone meet a dead person in the room of Requirement if needed? Would it have been possible for Harry to go in past in order to see Dumbledore for the explanation of many things? Can the room duplicate something? For instance, Harry needed the Elder wand; with Voldemort having it, could he still get it? Someone wishes to be knowledgeable than Dumbledore, would the seeker find his desired wish by finding books and things?   Who created the room of requirement (surely not Salazar)? Was it beyond Dumbledore's magic to create such a place? Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$ Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$.  How? What's the proof? Or maybe it is self apparent just looking at the above?   PS: This problem is known as "The sum of the first $n$ positive integers". How can I see my progress for different badges? Call C++ library in C# I have a lot of libraries written in C++. I want to call these libraries from C#, however, I have met many problems. I want to know if there is a book or guideline to tell me how to do that. What is the mechanism behind the slowdown of light/photons in a transparent medium? So light travels slower in glass (for example) than in a vacuum. What causes light to slow down? Or: How does it slow down? If light passes through the medium, is it not essentially traveling in the "vacuum" between the atoms? Cyclic Quadrilateral Problems Let ABCD be a cyclic quadrilateral. The diagonals AC and BD meet at P, and DA and CB produced meet at Q. The midpoint of AB is E. Prove that if PQ is perpendicular to AC, then PE is perpendicular to BC. multiplicative group of infinite fields Behavior of $\sum a_n x^n$ given $\sum |a_n - a_{n-1}| < \infty$ Let $(a_n)_{n=0}^{\infty}$  be a sequence of real numbers satisfying $\sum_{1}^{\infty} |a_n - a_{n-1}| &lt; \infty$   Which of the following conclusions must then be true?  The series $\sum a_nx^n$ converges   Nowhere on $ \mathbb{R}$ Everywhere on $  \mathbb{R}$ On some interval containing $(-1,1)$ Only on $(-1,1)$   What I have tried is Make an inequality with $|a -b| \leq |a| +|b|$ .. but i cant proceed further .. Any sort of help will be deeply appreciable . Will the Windows 10 Free Upgrade stay in the Professional channel? Will wolves attack Ocelots? Will wolves attack and kill ocelots in Minecraft? I know they kill sheep. I just don't want my cats being eaten. How do you write something that was discussed in the past, while the issue still exists in the present?        The discussion resulted in the committee members highlighting crucial areas that need to be addressed.   The discussion resulted in the committee members highlighting crucial areas that needed to be addressed.      ......................     1'. The discussion resulted in the committee members highlighting crucial areas that need to be addressed if one is to achieve the desired outcome.      2'. The discussion resulted in the committee members highlighting   crucial areas that needed to be addressed if one were to   achieve the desired outcome. Change the selected value of a drop-down list with jQuery I have a drop-down list with known values. What I'm trying to do is set the drop down list to a particular value that I know exists using jQuery.  Using regular JavaScript, I would do something like:  ddl = document.getElementById("ID of element goes here"); ddl.value = 2; // 2 being the value I want to set it too.   However, I need to do this with jQuery, because I'm using a CSS class for my selector (stupid  client ids...).  Here are a few things I've tried:  $("._statusDDL").val(2); // Doesn't find 2 as a value. $("._statusDDL").children("option").val(2) // Also failed.      How can I do it with jQuery?     Update  So as it turns out, I had it right the first time with:  $("._statusDDL").val(2);   When I put an alert just above it works fine, but when I remove the alert and let it run at full speed, I get the error     Could not set the selected property. Invalid Index   I'm not sure if it's a bug with jQuery or Internet Explorer 6 (I'm guessing Internet Explorer 6), but it's terribly annoying.