How can I prove/disprove if ma ≡ mb (mod n), then a ≡ b (mod n) for all positive integers a, b, m, n. I know the fact that  if (m,n) = d, then ma ≡ mb (mod n) if and only if a≡b (mod n/d)   I have no idea what should I do?
I was working on a problem that resulted in the calculation: $20 \equiv 10x \pmod{11}$.  I got the answer $x \equiv 2 \pmod{11}$ with the thought process: Since $10$ is a factor of $20$ I can rewrite $20 \pmod{11}$ as $10x \pmod{11}$ with $x=2$. But isn't this basically the same as using division, which we aren't supposed to do in modular spaces?  If I solve by multiplying $20$ by the multiplicative inverse of $10 \pmod{11}$ which is $10$, I get $200 \pmod{11}$ which simplifies to $2 \pmod{11}$ anyway. Did these numbers just happen to be the same or is my original method valid?
Case insensitive regular expression without re.compile? In Python, I can compile a regular expression to be case-insensitive using re.compile:  &gt;&gt;&gt; s = 'TeSt' &gt;&gt;&gt; casesensitive = re.compile('test') &gt;&gt;&gt; ignorecase = re.compile('test', re.IGNORECASE) &gt;&gt;&gt;  &gt;&gt;&gt; print casesensitive.match(s) None &gt;&gt;&gt; print ignorecase.match(s) &lt;_sre.SRE_Match object at 0x02F0B608&gt;   Is there a way to do the same, but without using re.compile. I can't find anything like Perl's i suffix (e.g. m/test/i) in the documentation. Replace dynamic content in XML file Does anyone know of any interesting subsets of the Cantor set? When I first started thinking about this, I thought that since the Cantor set $C$ is the intersection of disjoint unions of closed sets, those particular closed sets would be subsets of $C$. Those sets, however, happen to be intervals! The Cantor set contains no isolated points nor (open) intervals, so then what subsets does it contain besides $\varnothing$ and itself? Which Windows 7 log file contains device connection/disconnection information? Windows 7 frequently plays the usb/new device connection/disconnection sound. It happens both when the computer is idle and under load. All usb device connections have been checked (though I can't rule out a bad cable yet). I thought a good way to track down the issue would be to wait for the sound and then check the log file for the latest connected/disconnected device. I just need to know which log file has this information.  Edit: I'm going to leave the answer accepted, however the issue persists. I get that sound seemingly randomly throughout the day. Anyone else have any ideas? String concatenation: concat() vs "+" operator Assuming String a and b:  a += b a = a.concat(b)   Under the hood, are they the same thing?  Here is concat decompiled as reference. I'd like to be able to decompile the + operator as well to see what that does.  public String concat(String s) {      int i = s.length();     if (i == 0) {         return this;     }     else {         char ac[] = new char[count + i];         getChars(0, count, ac, 0);         s.getChars(0, i, ac, count);         return new String(0, count + i, ac);     } } How might I schedule a C# Windows Service to perform a task daily? How should you interview for QA *Automation* positions? In $\mathbb{Z}\left [ x \right ]$, let $I = \left \{ f\left ( x \right ) \in \mathbb{Z}\left [ x \right ]: f\left ( 0 \right ) \text{ is an even integer} \right \}$. Is $I$ a Prime ideal of $\mathbb{Z}\left [ x \right ]$?  Let $f_{1}\left ( x \right ),f_{2}\left ( x \right ) \in \mathbb{Z}\left [ x \right ].$ Let $f_{1}\left ( x \right )f_{2}\left ( x \right ) \in I$. Clearly, I is a proper ideal of $\mathbb{Z}\left [ x \right ]$ since the elements in $\mathbb{Z}\left [ x \right ]$ evaluated at 0 can produce an odd integer but this cannot be true in I. Now, $f_{1}\left ( 0 \right )f_{2}\left ( 0 \right )$ is an even integer in I. Hence, either $f_{1}\left ( 0 \right )$ has an even constant integer or $f_{2}\left ( 0 \right )$ has an even constant integer. Thus, I is a prime ideal by definition. Is this correct? I would now like to show if I is a maximal ideal. Again, clearly, I is a proper ideal by the above argument. Let B be an ideal of $\mathbb{Z}\left [ x \right ]$ and I $\subseteq B\subseteq \mathbb{Z}\left [ x \right ]$. How should B be defined so that I would conclude either $B=I$ or $B=\mathbb{Z}[x]$? I did a dumb thing... I forgot that Ubuntu 10.04 (Lucid Lynx) switched to &nbsp;2 which puts a ton of *.mod files (kernel modules) in /boot/grub. I thought they were soundtrack files put there erroneously, and I moved them. Needless to say, the next reboot was traumatic. I was presented with something I had no memory of ever seeing... a 'grub rescue>' prompt.  With the help of  however, I was able to recover...   I discovered that GRUB rescue does not have 'cd', 'cp' or any other filesystem commands except its own variation of 'ls'. So first I had to find the partition with the /boot directory containing vmlinuz file and other boot image files... (approximation from memory of failed attempts, as well as blank lines for clarity, added 2014-07-10 by docsalvage)    grub rescue&gt;  ls   (hd0,4) (hd0,3) (hd0,2) (hd0,1)    grub rescue&gt;  ls (hd0,4)/boot ... some kind of 'not found' message  grub rescue&gt;  ls (hd0,3)/boot ... some kind of 'not found' message  grub rescue&gt;  ls (hd0,2)/boot ... grub ... initrd.img-2.6.32-33-generic ... vmlinuz-2.6.32-33-generic   I found a /boot directory containing the vmlinuz file vmlinuz-2.6.32-33-generic on partition (hd0,2). Then I manually booted from the 'grub rescue>' prompt. The following commands will...   Set the root to use the /boot directory on partition (hd0,2). Load kernel module linux. Set that module to use the kernel image vmlinuz-2.6.32-33-generic. Set initrd(initialize RAM disk) to use the image initrd.img-2.6.32-33-generic. Boot Linux.  grub rescue&gt;  set root=(hd0,2)/boot   grub rescue&gt;  insmod linux   grub rescue&gt;  linux (hd0,2)/boot/vmlinuz-2.6.32-33-generic root=/dev/sda2 grub rescue&gt;  initrd (hd0,2)/boot/initrd.img-2.6.32-33-generic   grub rescue&gt;  boot    This boots and crashes to the BusyBox prompt which DOES have some rudimentary filesystem commands. Then I moved the *.mod files back to the /boot/grub directory...  busybox&gt;  cd /boot   busybox&gt;  mv mod/* grub busybox&gt;  reboot  The reboot was successful, but that was a lot of work.   Is there an easier way? Solve differential equation, $x'=x^2-2t^{-2}$ Solve differential equation:  $x'=x^2- \frac{2}{t^2}$  Maybe is it sth connected with homogeneous equation? I have no idea how to solve it. Is there any way in unix to find out who accessed certain file in last 1 week? It may be user or some script ftp it to some other place. Can I get a list of user name who accessed certain file? How can I find out who is accessing particular file?? JSON+Node.js - Unexpected token o How prove this $\lim_{n\to \infty}\sin{n^m}$ divergent. show that $$\lim_{n\to \infty}\sin{(n^m)}$$  divergent,where $m\in N^{+}$  I have kown $$\lim_{n\to\infty}\sin{n^2}$$  to be divergent and dense in $[-1,1]$. This is very  famous problem,the problem is first is post $AMM$( The American Mathematical Monthly,1970-1975) problem,and after some years  $AMM$ post $\sin{(n^{14})}$ in $[-1,1]$ was dense? and this is open problem.so I Guess  $$\sin{(n^m)}$$ is dense in $[-1,1]$. Why is the Association Bonus ignored when trying to answer a protected question? I registered on  only to post an answer to .  However, since it's a protected question, I am not allowed to answer it. (I found this info from ).  Worth to mention, I can comment on the above question, just not post an answer.  I was under the impression that the whole point of association bonus was to bypass such limits, so why isn't that the case for answering protected questions? Why isn't the eigenclass equivalent to self.class, when it looks so similar? While looking at online code samples, I have sometimes come across an assignment of a String constant to a String object via the use of the new operator.  For example:  String s; ... s = new String("Hello World");   This, of course, compared to  s = "Hello World";   I'm not familiar with this syntax and have no idea what the purpose or effect would be.  Since String constants typically get stored in the constant pool and then in whatever representation the JVM has for dealing with String constants, would anything even be allocated on the heap? How to run GUI applications remotely on a headless server? I'm new to Ubuntu - sorry for novice questions. What I have - is Ubuntu server, to which I have access only via ftp and putty. What I would like to do - is to run firefox (or any other browser) from that Ubuntu machine.  I've managed to install firefox , but when I'm trying to run something like this:  x-www-browser http://google.com   It is saying:  Error: no display specified   I don't understand - there is no display at all at Ubuntu sever - it is just computer case on rack, the only display I have - is mine.  Don't know, how to redirect this display output to my machine (if this is possible at all). Proof Check: Every Cauchy Sequence is Bounded Sorry if I keep asking for proof checks. I'll try to keep it to a minimum after this. I know this has a well-known proof. I understand that proof as well but I thought I'd do a proof that made sense to me and seemed, in some ways, simpler. Trouble is I'm not sure if it's totally correct. It's quite short though. I was just hoping someone could look it over and see if it is a valid proof. Thank you!  Lemma: Every Cauchy sequence is bounded.  Proof: Let $(a_{n})$ be Cauchy. We choose $ 0&lt;\epsilon_{0}$. So $ \forall \; n&gt;m\geq N_{0}$ we have that $\vert a_{n}-a_{m} \vert &lt; \epsilon_{0}$. Therefore $(a_{n})$ is bounded for all $ m \geq N_{0} $ by $ \epsilon_{0} $. Since $ \mathbb{N}_{N_{0}}$ is finite, it is bounded. So, for all $ m&lt;N_{0} $, $ (a_{n})$ is bounded. Therefore $(a_{n})$ is bounded.  I realize I haven't said what the bounds are but I think that's sort of irrelevant. So long as we know it's bounded. Any help is much appreciated! Changing URLs in Google's Index I recently submitted a sitemap.xml in Google's Webmaster Tools and some of the pages already got indexed. The problem now is, that I cleaned my URLs with MOD_REWRITE in the .htaccess file afterwards and I would like the new ones to be indexed. The old ones should be removed from the Google Index.  Indexed Page:  www.example.com/article.php?=article=this%is%an%example   New URL:  www.example.com/article-this-is-an-example   Do I even have to remove them manually from the index in any way or do they refresh if Google crawls my new sitemap? I really wan't to prevent duplicate content.  Thanks in advance. Removing non numeric characters from a string