At least how many numbers are needed to be taken to be sure that there are at least 11 numbers among these numbers where the difference between any two is divisible by 7?
Prove that, for any $n+1$ integers $a_{1},a_{2},....,a_{n+1}$, there exist two of the integers $a_{i}$ and $a_{j}$ with $i \neq j$, such that $a_{i} - a_{j}$ is divisible by $n$.  Please help me about this problem,I know there is lots of answers can find from Google, but I am not really understand them. I hope someone can give me more details explanation. I would really appreciate your help.
Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose that the system of linear equations $AX = B$ has a solution in $\mathbb{R}^n$. Does it necessarily have a solution in $\mathbb{Q}^n$?  Where do I start?    I feel I can use this to :  $$x_i=\frac{1}{a'_{ii}}\left(b'_i - \sum_{j=i+1}^{k} a'_{ij} x_j \right).$$    Here is a thought that just crossed my mind, but I'm not sure if this is legal:  Suppose $Y\in \{\mathbb{R}\setminus \mathbb{Q}\}^n$ and $Y'\in\mathbb{Q}^n$ are solutions to $AX=B$. Then $AY=AY'=B$, or $A(Y-Y')=A(Y+(-Y'))=0$, but  addition between $\{\mathbb{R}\setminus \mathbb{Q}\}^n$ and $\mathbb{Q}^n$ is . . . Typesetting annuity and life-insurance symbols in ConTeXt How do I typeset annuity and life-insurance symbols, actuarial notation in ConTeXt. I see there are packages available but not for ConTeXt.    Thanks Wizard sickness I remember in an interview, or during a Q&amp;A, Rowling stating that Wizards live longer than Muggles. This got me thinking about Wizard deaths in general, and then deaths being attributed to illness.  So, I ask, has there ever been a Wizard, that has died from something like cancer? Is there anything, even briefly mentioning a Wizard becoming ill?  And finally, are there any illnesses that are exclusive to Wizards? How do I survive in the Nether? Whenever I travel to the Nether in Minecraft, I almost instantly get blasted to smithereens and chopped up into little pieces.  How do I survive in the Nether, at least long enough to build some sort of shelter and get situated? How can I remove malicious spyware, malware, adware, viruses, trojans or rootkits from my PC? What should I do if my Windows computer seems to be infected with a virus or malware?  What are the symptoms of an infection? What should I do after noticing an infection? What can I do to get rid of it? how to prevent from infection by malware?   This question comes up frequently, and the suggested solutions are usually the same. This community wiki is an attempt to serve as the definitive, most comprehensive answer possible. Feel free to add your contributions via edits. Q&A blank in Stack Exchange Android app v1.0.95 - S10e/Android 10 Recently upgraded to a Samsung S10e and Android 10, but all Q&amp;A in the Stack Exchange app v1.0.95 are blank.  The title and comments are visible, but question/answer posts are just white space. Accessing the About page in the app is also blank.  I tried clearing data, cache and re-installing the app, but I still get the same behaviour. Switch statement fallthrough in C#? I've been looking all over for a proper explanation of how to delete photos from the device (nexus 5) to reclaim some space, while being sure that the pictures are still in the Google+ backup. Unfortunatley i have not found one, other than "wild claims" it works somehow, while other say it does not. Plus deleting photos via the Pictures app actually warns you (now?) that it will delete the backed up copies too. Do i have to delete them in the Gallery app?  I did delete some photos recently, and i was quite sure they ware still backed up (still showed up on the Google+ website) but looking now i see they are in the trash in the device.  Is there really no way to be able to know for sure if the pictures are backed up savely while i can delete them from the device?  Alternatively i'm also open for alternative solutions to this problem, or i just go back to the "good old copy and paste when plugging the phone". How do I get a Cron like scheduler in Python? I'm looking for a library in Python which will provide at and cron like functionality.  I'd quite like have a pure Python solution, rather than relying on tools installed on the box; this way I run on machines with no cron.  For those unfamiliar with cron: you can schedule tasks based upon an expression like:    0 2 * * 7 /usr/bin/run-backup # run the backups at 0200 on Every Sunday  0 9-17/2 * * 1-5 /usr/bin/purge-temps # run the purge temps command, every 2 hours between 9am and 5pm on Mondays to Fridays.   The cron time expression syntax is less important, but I would like to have something with this sort of flexibility.   If there isn't something that does this for me out-the-box, any suggestions for the building blocks to make something like this would be gratefully received.  Edit I'm not interested in launching processes, just "jobs" also written in Python - python functions. By necessity I think this would be a different thread, but not in a different process.  To this end, I'm looking for the expressivity of the cron time expression, but in Python.   Cron has been around for years, but I'm trying to be as portable as possible. I cannot rely on its presence. In this case what is correct?        Keep Tom and I updated.      or         Keep Tom and me updated.      I understand that me is an object pronoun. And therefore I feel the second option is correct.  Especially, because I would say "Keep him updated" and also "Keep me updated". Instead of "Keep he updated" or "Keep I updated". But something about saying "Keep him and me updated" feels wrong.  However I am not sure, because several people have told me that in this case I is the correct option because it is being used in conjunction with a name (Tom) and not a pronoun. Is that correct? I have an old Facebook account that I created, uploaded a couple of photos and simply ignored (i.e. no friends added). Now, I can't remember the email address associated with the account and I don't remember the password either. How can I find out which email address I had associated with that account?  P.S. I have created a new Facebook account and my picture can be used to identify me (if I can get someone at Facebook to take a look). Other than that, I'm willing to have that old account deleted altogether if recovery isn't possible.  Is there a way to resolve this issue? I posted a comment to a duplicate question on . But I decided to delete it. Afterwards I voted as duplicate and thought a comment would be generated like     possible duplicate of ...   But it didn't. My guess is that the duplicate-comment won't be generated if a comment exists with that link, even if deleted. Given a list ["foo", "bar", "baz"] and an item in the list "bar", how do I get its index (1) in Python? Is there a fast way to get to the bottom of the Death Star? In Tiny Death Star, you can get to the top of your tower by tapping on the top of the screen while avoiding the "Add Level" button. However, the only way I can find to go to the bottom of the tower is to wait for someone to want an elevator ride, and tap on the button to give them that ride. Is there any other hot-spot for getting to the bottom of your Death Star? $15x\equiv 20 \pmod{88}$ Euclid's algorithm Use Euclid's algorithm to find a multiplicative inverse of 15 mod 88, hence solve the linear congruence $15x\equiv 20 \pmod{88}$  So far I have:   $88=5\cdot15+13$  $15=1\cdot13+2$  Backwards substitution gives  $15v+18w=1$   $1=15-1\cdot13$   $=15-1(88-5\cdot15)$   $=15\cdot15-1\cdot88$   now very stuck what to do from here:( Show that if $f(z)$ is an entire function such that it takes real values on the lines $Im(z)=0$ and $Im(z)=a$ (for some $a \in \mathbb{R}_{\neq 0})$, then $f$ is periodic with period $2 i a$.    So here is my thought process: First of all by reflection principle $f(\bar{z})=\overline{f(z)}$ so if our function $f$ is given in the strip $0 \leq Im(z) \leq a$, then its values on $-a \leq Im(z) \leq a$ follow by reflection principle. Then by imposing $f(z)=f(z+2 ia)$ we can intuitively see that this will define an entire function by looking at how the strips $na \leq Im(z) \leq 2na$ seamlessly stack on top of each other. Is this logical? And how would one prove this analytically? Keep processes running after SSH session disconnects Is temperature a Lorentz invariant in relativity? Background Video is shown in Rendered mode but not when szene is rendered i have a 3D Object and want to add a background image( if possible even a movie) to this 3D Object. I use Cycles to render and tried a plane with the  image as texture. When i go into render mode, it looks like this(picture with background) and when rendered, it has no background. The key also looks into  a different direction.  Should i even use Cycles for this or use motion tracking? i tried to use that but had several problems.     Rendered mode   When scene is rendered Why shouldn't I use "Hungarian Notation"?