Let $a$ be a postive integer and let $A=\{a,a+1,a+2,\ldots,a+9\}.$ Show that there exists some $i$ such that for any $j\neq i$ we have $(a+i,a+j)=1$
Given any $10$ consecutive positive  integers , does there exist one integer which is relatively prime to the product of the rest ?
Ubuntu can't be installed on my system because of what the title says. This is because support Intel HD Graphics 530 was only just added in Linux kernel 4.3 and all versions of Ubuntu only have kernel versions up to 4.2 as of 15.10. My computer's processor is the new Intel Skylake 6600K if you need to know, everything else is generic. It has a UEFI BIOS motherboard. Help? How did Harry, Ron and Hermione buy presents for each other? I have been re-reading the Harry Potter books again and I noticed that Harry, Ron and Hermione have been giving each other Christmas presents each year at Hogwarts. How have they been able to buy each other presents if they are not allowed to go out of Hogwarts?   From their third year on they could visit the village and buy presents there, but what about the first two years? Did they give their personal belongings? Or did they just happen to have boxes of chocolates or books to give each other? `du -sh` reports different used size than `df -h` Possible Duplicates:             df -h says that I have 494G used on /var:  Filesystem            Size  Used Avail Use% Mounted on /dev/sda3             564G  494G   41G  93% /var   But du -sh /var says it's 350G  What makes them different? What is the difference between an "API," a "framework," an "IDE," and an "engine?" How do you attach a debugger to a Web service? I am using a Web service. Can anybody tell me how to debug through it? Adding 2 favorite tags at one time combines tag after page refresh I was adding some tags to my favorite tags today and noticed tags being joined together when the page is refreshed.  Steps to reproduce:   Click edit on Favorite Tags Create the first tag and click add [I added django in this case] Add a second tag and click add [I added python in this case] Refresh page  Favorite tags now show previous tags + djangopython   In the edit section, it appears as if the tags are added separately, but when I refresh the page they get joined  OS: MacOSX-10.8  Browser: Chrome, Firefox Extracting tree crown areas from remote sensing data (visual images and LiDAR) Why is this sentence wrong? "He went to home." I have a PHP script that can encode a PNG image to a Base64 string.  I'd like to do the same thing using JavaScript. I know how to open files, but I'm not sure how to do the encoding. I'm not used to working with binary data. Intuitive understanding of the derivatives of $\sin x$ and $\cos x$ One of the first things ever taught in a differential calculus class:   The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$.   This leads to a rather neat (and convenient?) chain of derivatives:   sin(x) cos(x) -sin(x) -cos(x) sin(x) ...   An analysis of the shape of their graphs confirms some points; for example, when $\sin x$ is at a maximum, $\cos x$ is zero and moving downwards; when $\cos x$  is at a maximum, $\sin x$ is zero and moving upwards.  But these "matching points" only work for multiples of $\pi/4$.  Let us move back towards the original definition(s) of sine and cosine:  At the most basic level, $\sin x$ is defined as -- for a right triangle with internal angle $x$ -- the length of the side opposite of the angle divided by the hypotenuse of the triangle.  To generalize this to the domain of all real numbers, $\sin x$ was then defined as the Y-coordinate of a point on the unit circle that is an angle $x$ from the positive X-axis.  The definition of $\cos x$ was then made the same way, but with adj/hyp and the X-coordinate, as we all know.  Is there anything about this basic definition that allows someone to look at these definitions, alone, and think, "Hey, the derivative of the sine function with respect to angle is the cosine function!"  That is, from the unit circle definition alone.  Or, even more amazingly, the right triangle definition alone.  Ignoring graphical analysis of their plot.  In essence, I am asking, essentially, "Intuitively why is the derivative of the sine the cosine?" Inverse function of $x^x$ How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried using logs, differentiating, etc, etc, but to no avail. Insert a "Let us continue this discussion in chat" comment before thread gets messy I've seen , but there seem to be no proper solution.  I suggest an ability to insert a "Let us continue this discussion in chat" comment before the thread gets messy enough to fit the requirement. The point is to prevent the problem before it occurs.  Perhaps this should become a privilege for users of X reputation or more?  Your opinions on the subject? Recently I have started playing with jQuery, and have been following a couple of tutorials. Now I feel slightly competent with using it (it's pretty easy), and I thought it would be cool if I were able to make a 'console' on my webpage (as in, you press the ` key like you do in  games, etc.), and then have it Ajax itself back to the server in-order to do stuff.  I originally thought the best way would be to just get the text inside the textarea, and then split it, or should I use the keyup event, convert the keycode returned to an ASCII character, append the character to a string and send the string to the server (then empty the string).  I couldn't find any information on getting text from a textarea, all I got was keyup information. Also, how can I convert the keycode returned to an ASCII character? Infinite Series: Fibonacci/ $2^n$ I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner)  In the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, 34, 55,\ldots$ each term after the first two is the sum of the two previous terms. What is the sum to infinity of the series: $$\frac{1}{2} + \frac{1}{4}+ \frac{2}{8} + \frac{3}{16} + \frac{5}{32} +\frac{8}{64} + \frac{13}{128} +\frac{21}{256} +\frac{34}{512}+ \frac{55}{1024} + \cdots$$  Now, I solved this using an infinite geometric matrix series (incorporating the matrix version of the relation $a_n= \frac{a_{n-1}}{2}+ \frac{a_{n-2}}{4}$), and my students, after much hinting on my part, googled the necessary string to stumble across Binet's formula (which allows one to split the series into two simple, if rather messy, geometrics). Both of these are good methods, but neither really seems plausible for a challenge set for 15-18 year olds under exam conditions. So how is one supposed to do it? Here is the dictionary I have  propertyList = {     "id":           "int",     "name":         "char(40)",      "team":         "int",     "realOwner":    "int",      "x":            "int",     "y":            "int",      "description":  "char(255)",      "port":         "bool",     "secret":       "bool",     "dead":         "bool",     "nomadic":      "bool",      "population":   "int",     "slaves":       "int", }   But when I print it out with "\n".join(myDict) I get this  name nomadic dead port realOwner secret slaves team y x population id description   I know that a dictionary is unordered but it comes out the same every time and I've no idea why. Ackermann function - how to calculate the number of times it calls itself Purely for my own amusement I've been playing around with the .  The Ackermann function is a non primitive recursive function defined on non-negative integers by:  $A(m,n) = n+1$, if $m=0$  $A(m,n) = A(m-1,1)$, if $m&gt;0$ and $n=0$  $A(m,n) = A(m-1,A(m,n-1))$, if $m&gt;0$ and $n&gt;0$  I'm not interested in values of the function itself, rather I'm interested in the number of recursions the function takes to arrive at a value. For example:  $A(1,2) = A(0,A(1,1)) = A(0,A(0,A(1,0))) = A(0,A(0,A(0,1))) = A(0,A(0,2)) = A(0,3) = 4$  That's 6 steps, so if we define $RA(m,n)$ as the number of recursions to calculate $A(m,n)$, then $RA(1,2) = 6$.  I made a short Excel macro to calculate $RA(m,n)$ for any given $m$ and $n$. The problem I have is that $RA(m,n)$ grows even faster than $A(m,n)$ and I quickly run out of stackspace. The table below shows $RA(m,n)$ for very small values of $m$ and $n$. The numbers shown for $RA(3,10)$ and $RA(4,1)$ are the number of steps before crashing!    A formula for $RA(1,n)$ is trivial, and it is relatively easy to see that $RA(2,n) = 5(n+1) + 4n(n+1)/2$.  So my questions: can anyone see an explicit formula for $RA(3,n)$? More generally, does a formula exist for $RA(m,n)$, in the same way that $A(m,n)$ can be expressed using Knuth arrows?   For the low values shown of $m$ and $n$, $RA(m,n)$ looks about order of magnitude larger than $A(m,n)$. Can we prove that $RA(m,n) &gt; A(m,n)$ for all $m,n$. Intuitively, it should be.  For bonus points, can $RA(m,n)$ be expressed in terms of $A(m,n)$? In Unity3D, how do I create an efficient scratch card effect? I want to create a scratch-card-like effect in new Unity UI's new Image tag.  I know it's possible with setPixels, but that's prohibitively expensive on mobile devices.  Something like this:    I'd like the user to be able to drag on any part of the image to "scratch off" the top layer and reveal the content underneath it.  How can I do this? Does Kylo not know the fate of Darth Vader? It seems Kylo has an obsession of Darth Vader and has an interest in the dark side because of it.   Does he not realize that Darth Vader turned good and helped destroy the Empire (Spoilers?)? I feel like Kylo would've seen how much better the Light Side was as it was able to turn Vader from the Dark Side if he had known this. How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$ 4 fast cores or 6 slower ones?