Find all couples $(n,k)$ , $n \geq 2$ and  $0\leq $ k $ \leq n - 1$, such as : $\binom nk$ and $\binom {n}{k+1}$ are relatively prime. I first tried finding a link between the two using combinatorics, but it wasn't that easy, since they don't have a particular order, and any could be greater than the other. Then I recalled that if $n$ is prime, it's guaranteed that it divides both of them, I tried to apply this in a more general scenario, using a prime divisor of $n$, but it didn't succeed. Any ideas or tips?
Is the following conjecture true?     Given an $n&gt;1$ and a $k$ such that $0&lt;k&lt;n$ then $\gcd\left(\binom{n}{k}, n\right)&gt;1$   This was a question asked in my old Number Theory course that I recently remembered. It was posed after we learned that $p\mid\binom{p}{k}$ for $0&lt;k&lt;p$. We never resolved it either way. I expect it to be true and have an elegant answer, but I haven't been able to prove it.
Is GTA Online cross-platform? Grand Theft Auto Online exists on five different platforms: XBox 360, PS3, XBox One, PS4, and PC. Do any of these play on the same servers, allowing people on differing platforms to play together? My daughter asked me to cosign on a car loan.  Somehow I ended up as the borrower and she a the cosigner.  Is that legal? A pitcher contains $X + 1$ blue balls and $Y + 1$ red balls. It is known that $X, Y$ are independent random variables, and it is given that $X \sim \mathrm{Poisson}(n), Y \sim\mathrm{Poisson}(2n)$. Calculate the limit: $\displaystyle\lim_{n\to \infty}\mathbb P (X + n = Y) $   I tried to condition on the value of $Y$ and to use the Law of total probability but it did not lead to a solution. ( I tried to sum: $ \sum_{i=0}^\infty\mathbb P(X+n=Y|Y=i)\mathbb P(Y=i) $ and get an expression that I can calculate its limit as n goes to infinity) Consider the following two versions of a question:1   Version 1: What is the difference in meaning between "[to be] not invited" and similar negation forms?  Version 2: What is the difference in meaning among "[to be] not invited" and similar negated forms?    Personally, I think Version 1 might subtly suggest a Star Topology (see below) with the target sentence "[to be] not invited"at the center.  And Version 2 might subtly suggest a more free-form comparison like the Mesh or Fully Connected, with the target sentence simply "one among many":   Between approx equal "Star" and among approx equal "Fully Connected"?  Don't get me wrong: I do think this is quite subtle, not always applicable, and I don't think anyone would intuit such a "network topology" meaning without further explanation.   So the main question is, are both grammatically correct?  If yes, is there a semantic difference to choose one over another, perhaps as I suggested?  Or does the plurality aspect of between and among trump other semantic and stylistic issues?    1. In my question , another user provided a very helpful edit. However, the change of the word between to among prompted this question. in my bash loop over a list of some servers, if the ssh connects the bash script exits I've got a capture the flag arena. One issue we keep running into during play, is that of clutter.  We have /gamerule keepInventory false, so the pitch gets rather cluttered with armor, weapons and building materials, such that it becomes difficult to pick the flag up, due to full inventories.  I've discovered the following command to remove all drops from the arena:  /kill @e[type=Item]   However, this removes all the floating items from the ground. I'd like, if possible, to preserve the flag from being destroyed.  I have this command for detecting the flag:  /testfor @e[type=Item,x=40,y=1,z=40,dx=40,dy=27,dz=80] {Item:{id:"minecraft:banner",Damage:4s}}   But I'm not sure how to combine the two in a way that it will destroy everything except for the flag.  Any ideas? Boxplot interpretation: is it correct that a boxplot is missing a whisker? In League of Legends, there are a number of effects which revive a champion upon death.  The item  is the most well-known and widely used of these.  I was wondering, however, whether its effect would come before or after other revive effects?  I have two examples.  First, Zilean's ultimate  will provide a champion with a Guardian Angel -like buff for a limited time.  If a champion has this buff in addition to a Guardian Angel, which effect will happen first?  Second, Anivia's passive  allows her to come back to life if her egg is not destroyed after she is killed.  If she has a Guardian Angel, will her passive be triggered before or after it? How does this compare to  and ?  It seems like in one case it is more beneficial to have Guardian Angel's precedence be greater, and in the other it is more beneficial to have it be less.  In the first scenario, if the Guardian Angel was triggered first, then Zilean would have a good chance of simply wasting his ultimate.  On the other hand, if it is triggered after these revival effects, then Anivia could not get the full benefit of her passive.  Is there a set order of precedence, or does each revival effect have their own unique "priority level" that is explicitly detailed for each ability? I can show this best by a picture:    This cross also shows when I export it as raster or pdf. What is it and how can I get rid of it?  I am using Qgis 2.4 and the map was done with the help of the Grass plugin AniMove on Linux Mint. Intel Corporation Device [8086:0d4f] is not supported on Ubuntu 18.04.3 I am currently unable to install ubuntu server 18.04.03 LTS on the latest intel NUC10i7FNH.  Ubuntu always states both network adapters as unclaimed. The previous version (NUC8i7BEH) has the same network controller I219-V and worked like a charm.  *-network:0 UNCLAIMED      description:  Network controller      product: Intel Corporation      vendor: Intel Corporation      physical id: 14.3      ...   Without network it is not even possible to follow most suggestions how to get it enabled, because it is impossible to fetch new packages from internet.  Here are some more information as requested:  The output from: modprobe -v e1000e &amp;&amp; dmesg | grep e10   [ 3447.889394] e1000e: Intel(R) PRO/1000 Network Driver - 3.2.6-k [ 3447.889395] e1000e: Copyright(c) 1999 - 2015 Intel Corporation.   The output from: sudo lspci -knn  00:1f.6 Ethernet controller [0200]: Intel Corporation Device [8086:0d4f]     Subsystem: Intel Corporation Device [8086:2081] 01:00.0 PCI bridge [0604]: Intel Corporation JHL7540 Thunderbolt 3 Bridge [Titan Ridge 2C 2018] [8086:15e7] (rev 06)    For me this looks somehow, as the kernal driver would not be in use :-/  Here are a few more information i got from playiung around with 19.10 desktop version.  [    0.000000] Linux version 5.3.0-29-generic (buildd@lcy01-amd64-024) (gcc version 9.2.1 20191008 (Ubuntu 9.2.1-9ubuntu2)) #31-Ubuntu SMP Fri Jan 17 17:27:26 UTC 2020 (Ubuntu 5.3.0-29.31-generic 5.3.13)   output from lshw:    *-network:0        Beschreibung: Kabellose Verbindung        Produkt: Intel Corporation        Hersteller: Intel Corporation        Physische ID: 14.3        Bus-Informationen: pci@0000:00:14.3        Logischer Name: wlp0s20f3        Version: 00        Seriennummer: c8:09:a8:9c:cd:e0        Breite: 64 bits        Takt: 33MHz        Fähigkeiten: pm msi pciexpress msix bus_master cap_list ethernet physical wireless        Konfiguration: broadcast=yes driver=iwlwifi driverversion=5.3.0-29-generic firmware=48.4fa0041f.0 ip=192.168.178.109 latency=0 link=yes multicast=yes wireless=IEEE 802.11        Ressourcen: iomemory:600-5ff irq:16 memory:6023114000-6023117fff   The output from: modprobe -v e1000e &amp;&amp; dmesg | grep e10  [    2.357430] e1000e: Intel(R) PRO/1000 Network Driver - 3.2.6-k [    2.357430] e1000e: Copyright(c) 1999 - 2015 Intel Corporation. [    2.357616] e1000e 0000:00:1f.6: Interrupt Throttling Rate (ints/sec) set to dynamic conservative mode [    2.562796] e1000e 0000:00:1f.6 0000:00:1f.6 (uninitialized): registered PHC clock [    2.634139] e1000e 0000:00:1f.6 eth0: (PCI Express:2.5GT/s:Width x1) 1c:69:7a:60:44:f0 [    2.634140] e1000e 0000:00:1f.6 eth0: Intel(R) PRO/1000 Network Connection [    2.634258] e1000e 0000:00:1f.6 eth0: MAC: 13, PHY: 12, PBA No: FFFFFF-0FF [    2.634991] e1000e 0000:00:1f.6 eno1: renamed from eth0 [    5.103285] checking generic (4000000000 e10000) vs hw (4000000000 10000000) [   15.806697] e1000e: eno1 NIC Link is Up 1000 Mbps Full Duplex, Flow Control: None    output from lspci:  00:1f.6 Ethernet controller [0200]: Intel Corporation Ethernet Connection (10) I219-V [8086:0d4f]     DeviceName:  LAN     Subsystem: Intel Corporation Ethernet Connection (10) I219-V [8086:2081]     Kernel driver in use: e1000e     Kernel modules: e1000e What's the best way to deal with hot/stuck pixels in long exposure night photographs? I discovered red/blue/white pixels on my night photographs tonight. Trying to confirm this, I took a black image with the lens on and indeed I saw a whole lot of them (I didn't count them but at least a good fifty on the whole picture).   I'm not exactly sure if they are hot pixels, stuck pixels or anything else.  I quickly gave a check to my last exported JPEG files (I get RAW out of the camera) and couldn't find anything. After a quick search, I've seen that Lightroom is caring about those pixels after RAW conversion.  I've read that, with Canon DSLRs, there is a software process that could take care of that directly on my camera. I don't know if I can use it without getting things even more messy.   My question is: Considering my crazy pixels are not visible on my pictures, should I get too emotional about them? My camera is under warranty but is it really worth it to return it to Canon? If it's only to receive it back 1 month later after Canon just used the very same menu to map out the pixels, it's a bit ridiculous. Could this be a sign that my camera has a bigger problem?  It seems related to shutter speed. When shooting a black image at 1/250s I get nothing. When shooting the same setup but 1s, I get a few of them. The more I increase the shutter speed, the more crazy pixels I get. "To mentor someone during a project" vs. "to mentor someone on a project" ..., whom I mentored during his final semester's project.   ..., whom I mentored on his final semester's project.      Which of these two is grammatically correct? Since I am not talking about the final semester but the final semester's project, I am not very sure about during. Can someone help me with this? Ubuntu gets stuck in a login loop My Ubuntu is stuck in a login loop when trying to enter my desktop. When I login, the screen gets black and soon after that the login screen comes back.   I've read that the problem might be caused by an error depending on the graphics, here's my graphics card: ATI Radeon 7670M How to boot WUBI installed Ubuntu within Windows 8? I have Windows 8 Release Customer Preview on my laptop. I tried to install Ubuntu on it with Ubuntu Windows installer (I have always used this to install, although that was on Windows 7), it installs and asks to reboot, then immediately Windows 8 boots up.  I do not have the option to choose what OS to boot. How can I boot to Ubuntu? VM is not exactly what I would like to use, but it is an option, however a last resort. Any help? Highest power of a prime $p$ dividing $N!$ How does one find the highest power of a prime $p$ that divides $N!$ and other related products?  Related question: How many zeros are there at the end of $N!$?    This is being done to reduce abstract duplicates. See   and  for more details. Why are lamps and mesh lights not illuminating anything? How can I install Gimp 2.8's help documentation on Ubuntu 12.04? Workflow for UV orientation How to I deal with unwrapped UVs orientation?   Delete the default Cube. Add a Cylinder Edit mode and edge select the Top and the Bottom and a single connecting edge. Make them into seams UV Unwrap (standard Unwrap mode)   The unwrapped UV now shows two circles above a row of quads. I can save this and open it in an image editor and paint over those areas and get the texture to align on the cylinder back in blender.  But, the pattern(just happens to be text) that was inserted in place of the row of quads is now upside down on the model. Any attempt to flip or align just makes things worse, from upside down to mirrored or the Top/Bottom not aligning.  So my real question is how could I have known that the quads represented an upside down view of the world?  Do other blender users export UVs and mark them with arrows and colours just to see how they align on a model before actually painting for real?  I tried selecting the edges in step 3 in different orders before making them into seams and it made no difference.  I also looked at the Normals but they appear to be only Inside/Outside not Up/Down.  Blender 2.70 Is Ubuntu an open source software? Is Ubuntu an open source software (OSS)? Many spells have "At higher levels" that provides for more targets or more damage.  For example Magic Missile can be cast as a second level spell and shoot 4 missiles instead of it's 3 as a first level spell.  Can a spell without that verbiage, such as Firestorm be cast in higher level slots for the same effect as the lower level spell?