Suppose x is a positive integer with $n$ digits, say $x = d_1d_2d_3\ldots d_n.$ If $9$ is a divisor of $d_1 + d_2 + \ldots d_n$, prove then $9$ is a divisor of $x$.  My attempt: suppose $x = 4518.$ Therefore $d_1 = 4, d_2 = 5, d_3 = 1, d_4 = 8$ those added together equals $18$, where $9$ is a divisor.   With that in mind $4518$ can be written as $4000 + 500 + 10 + 8.$ How do you show that $9$ is a divisor of this entire number from the information given?
I don't know if this is a well-known fact, but I have observed that every number, no matter how large, that is equally divided by $9$, will equal $9$ if you add all the numbers it is made from until there is $1$ digit. A quick example of what I mean:  $9*99 = 891$ $8+9+1 = 18$ $1+8 = 9$  This works even with really long numbers like $4376331$ Why is that? This doesn't work with any other number. Similarly for $11$ and alternating digits sums.
Steam doesn't start after upgrade to Ubuntu 16.04 After upgrade to Ubuntu 16.04, Steam doesn't start. When I execute it from the command line, I get the following error:  $ steam Running Steam on ubuntu 16.04 64-bit STEAM_RUNTIME is enabled automatically grep: symbol lookup error: grep: undefined symbol: pcre_jit_stack_alloc grep: symbol lookup error: grep: undefined symbol: pcre_jit_stack_alloc grep: symbol lookup error: grep: undefined symbol: pcre_jit_stack_alloc Installing breakpad exception handler for appid(steam)/version(1459463254) Reading the target of a .lnk file in Python? I'm trying to read the target file/directory of a shortcut (.lnk) file from Python. Is there a headache-free way to do it? The  is way over my head. I don't mind using Windows-only APIs.  My ultimate goal is to find the "(My) Videos" folder on Windows XP and Vista. On XP, by default, it's at %HOMEPATH%\My Documents\My Videos, and on Vista it's %HOMEPATH%\Videos. However, the user can relocate this folder. In the case, the %HOMEPATH%\Videos folder ceases to exists and is replaced by %HOMEPATH%\Videos.lnk which points to the new "My Videos" folder. And I want its absolute location. How many ways can you divide $9$ students into three unlabeled teams where one team contains $4$ people, one contains $3$ people and the last contains $2$ people? Unlabeled, meaning that groups with abc = bca = cba, etc  I understand how to do this if the teams are labeled:  $$\frac{9!}{4!3!2!}$$  But there is a term missing in the denominator when the teams are unlabeled and I'm having difficulty understanding how to calculate how many ways the teams can be organized.  There are $3!$ ways to organize the same first group of $3$, $4!$ ways to organize the same second group of $4$ and $2!$ ways to organize the last group of $2$. Why wouldn't you multiply $3!4!2!$ in the denominator?   For instance:  (ABC, DEFG) = (ABC, DEGF)  = (ABC, DFEG) = (ACB, DFGE), etc How to run Visual Studio post-build events for debug build only How can I limit my post-build events to running only for one type of build?  I'm using the events to copy DLL files to a local IIS virtual directory, but I don't want this happening on the build server in release mode. Force applied off center on an object Assume there is a rigid body in deep space with mass $m$ and moment of inertia $I$. A force that varies with time, $F(t)$, is applied to the body off-center at a distance $r$ from its center of mass. How do I calculate the instantaneous acceleration, rotational acceleration, and trajectory of this object, assuming it starts from rest? Set default folder view in Windows In Windows Vista, whenever I open a folder, the selection of columns is nonsensical: album, date taken, etc., for folders that contain no music or pictures whatsoever. I can select the correct columns for a particular folder, but all the other folders are still wrong, even when I go in to Folder Options and click the button to set the current options to all folders of this type.  How can I tell Windows to use the current column selection for all folders, unless otherwise specified? Finding $\int_0^\infty \frac{\cos(ax)-\cos(bx)}{x}dx$ I am practicing calculus section of GRE Math Subject test, and I can't figure out a way to do the following integral: $$\int_0^\infty \frac{\cos (ax) - \cos (bx)}{x}dx.$$  I absolutely have no clue how to do this. Can someone show me the solution explicitly? Thank you. Let $X,Y$ be topological and $f:X \to Y$ be open and continuos.  I am studying Baire space and I would like to try the following facts :      $(i)$ The inverse image of dense set is dense.       $(ii)$ The inverse image of comeager set is comeager.   I am studying Baire space and I would like to try the following facts  Can we collaborate to verify the above claims?    Thanks. How to parseInt a string with leading 0 I'd like to be able to search for questions by multiple tags; for example "questions tagged 'nhibernate' and 'web-services'".  I can get close by searching for "nhibernate web-services" but the results tend to be a few questions that relate to both terms--but much more largely just questions that relate to one or the other. Can unconscious characters be willing? This situation came up last week during our session. The party was facing a pretty difficult mob of enemies and one of the party members was knocked unconscious. Seeing the situation worsening, one of the players decided to use the  (DMG p. 197). In the text, it says:     The rod then instantly transports you and up to 199 other willing creatures you can see to a paradise that exists in an extraplanar space.   At first glance I thought this was actually a pretty good time to use the rod. The question came up though, what about the party member who is unconscious? In the end I made the call that the character went with them to keep things moving and more fun.  However, I'm feeling like back pedaling a bit and notifying my players that for future uses, the line "... willing creatures..." will be more strictly enforced as I don't feel like an unconscious creature could be willing.  Did I handle this the right way? Posts flagged for attention but there are none The comment with the Flagged Posts that seems to be new as of today states there are posts with flags awaiting my attention but the counter shows zero and when I go into flagged posts there is nothing waiting for me. Possible Duplicate:          Well, the question is in the title. I always had the impression that "talk to someone" refers to situations when some information must be conveyed to someone else, while "talk with someone" refers to cases that involve a more complex two-sided interaction. E.g., I "talk to my colleague" to tell him that I'm going for a lunch break (no response is required apart from maybe an acknowledgment), while I go and "talk with my colleague" if I have to discuss something with him in detail. Is this correct, or is there any difference between the two phrases, or is there no difference at all? Add new alias to Gmail without SMTP (forwarding-only address) I used to be able to go to Accounts and Import → Add another email address you own and add an email account that's only a forwarding address. That is, it's not a mailbox and I don't have SMTP access to it.  Gmail would send me a confirmation email to that address, I'd confirm it's mine, and done. Now it only gives me the option to add SMTP credentials.  Did they stop supporting the previous method? Is there any other way I can get that option back? I am using Ubuntu 16.04 with GTX 1080. On my first boot, the screen turn black after I selected Ubuntu in grub. After I added nomodeset, the black screen gone however it only shows the Ubuntu default background image. It can't reach the login screen. Studying for an exam, a review question...  Given $G=\langle x,y|x^4=y^4=e,xyxy^{-1}=e\rangle$.    Show $|G|\leq16$.   For this, I want to consider that $x^3=x^{-1}$ and $y^3=y^{-1}$ based on our assumptions. I am a little lost as to how to put the second part, $xyxy^{-1}=e$ to show there are no more than 16 elements.   If $|G|=16$, find the center of the group and find a group that is isomorphic to $G/\langle y^2\rangle$.   I'm pretty sure the group is centerless. I have noticed that such terms as model hyperparameter and model parameter have been used interchangeably on the web without prior clarification. I think this is incorrect and needs explanation. Consider a machine learning model, an SVM/NN/NB based classificator or image recognizer, just anything that first springs to mind.   What are the hyperparameters and parameters of the model? Give your examples please. Why won't any faces show up in the UV Image Editor? Full Wi-fi/ethernet signal but no internet I'm really hoping someone here can help me, as I'm getting pretty frustrated with my macbook now. I have a late 2011 model 15" macbook pro running OSX 10.9.5 that refuses to connect to Internet on my home network. I have the same problem whether it's WiFi or through ethernet cable, I have full signal but not getting online. Every other device connects fine; android phones and tablets,xbox one, iPhone and other macbooks, all online and browsing the net. I have had no problems connecting to other networks with my macbook pro before and I just tried at a friend's house to check, with no problems. A little something, that may be a clue to someone: when I open network diagnostic tool, my macbook pro connects for something that seems like a microsecond and it tells me I'm connected to Internet and everything seems fine. My IP is 192.168.0.5 subnet mask 255.255.255.0 and IPv6 is set to automatic. None of the options are checked under the proxies tab. Now I've tried the following, in different order and a number of times, to no avail:   Renewing/turning off&amp;on/inputting manually DHCP.  Restarting/resetting the router and the macbook.  Turning WiFi on/off. Remove &amp; add services (WiFi &amp; ethernet). Checked and copied all network settings from another macbook pro that's connected and online on my network. Ran network diagnostic and assistant tools Duplicated/removed/added location Removed the /Library/Preferences/SystemConfiguration folder. Removed network password from keychain.      I've tried other things too, but can't remember what else right now.  Response from ifconfig en1:  en1: flags=8863&lt;UP,BROADCAST,SMART,RUNNING,SIMPLEX,MULTICAST&gt; mtu 1500              ether 28:cf:da:e6:95:0a              inet6 fe80::2acf:daff:fee6:950a%en1 prefixlen 64 scopeid 0x5               inet 192.168.0.5 netmask 0xffffff00 broadcast 192.168.0.255              nd6 options=1&lt;PERFORMNUD&gt;              media: autoselect               status: active    I'm praying that someone out there has the magic words or the 2-click solution for me, but I would be grateful for a long thorough guidelines to solve it too! &lt;3        Regards Trond What are "schlieren" in English? The German word for these patterns is "Schliere, pl. Schlieren", and apparently this word has entered the English language as a loanword (cf. article  in Wikipedia or entry  in OED). But the German loanword is probably not something most native speakers of English understand without the help of a dictionary, so what would you call, or how would you describe, the following patterns?  Concrete:    Air:    Oil on water:    Coffee and milk:    Paint:    Smoke:    Rock:    Marbled paper:    And so on.    "Waves" is not applicable, because waves are vertical undulations of the surface, while these are a pattern on, or inside, the surface level, with no amplitude.  "Striæ" does not really fit either, because those refer more to the parallel layers, than to the curly forms.