How would I go about proving $$ a \equiv b\pmod{m}$$ given $$  a \equiv b\pmod{m^2}  $$ I suspected this was false, so I tried to find a counterexample using a , and there does not appear to be a counterexample for $m$ in (2, 16). What I have tried so far: $$  a \equiv b\pmod{m^2}  $$ $$ a = b + m^2k $$ $$ a - b = m^2k $$ $$ \sqrt{a - b} = \sqrt{m^2k} $$ Which would imply that $\sqrt{a-b} \equiv 0\pmod{m}$ assuming that $k$ is a perfect square. It feels like I'm barking up the wrong tree, however, and I'm not sure where to go from here.
I noticed relation between modulo operation and number which is power of two  Example I have to calculate $ 3431242341 \mod 2^5 $, which is $ 5 $ but it is equivalent to  $ ( 3431242341 \mod 2^9 ) \mod 2^5 $   I tried many examples and it seems to be true in general, and I am not sure if it is a coincidence or true in general that I can use first modulo operation ( greater number) and the result will be the same.
Ubuntu 16.04 multitouch gestures I have two dates of the form:  Start Date: 2007-03-24  End Date: 2009-06-26   Now I need to find the difference between these two in the following form:  2 years, 3 months and 2 days   How can I do this in PHP? How to get reflections from 3D to show up on real images Andrew Price released a nice tutorial on how to nicely put 3D objects on top of a real image:   I have seen images, such as this:  which have 3D elements over images and have actual reflections from the object showing up on the wet asphalt.  How can this effect be achieved using render passes (or any other way)?  Here is another example (from ): Given $f_{n} : [a,b]\rightarrow \mathbb R$ is uniformly Lipschitz and $f_{n}\rightarrow f$, how can I show that $f$ is Lipschitz?  $f_{n}$ is uniformly Lipschitz $\leftrightarrow \exists K &gt; 0$ such that for all $n$ and for every $x,y\in [a,b]$ we have $|f_{n}(x)-f_{n}(y)| &lt; K|x-y|$.  $f_{n}\rightarrow f \leftrightarrow \forall~\epsilon&gt;0~~\exists~N$ such that $\forall~n&gt;N$ we have $\mid f_{n}(x) - f(x)\mid&lt;\epsilon~~~(=\mid x-y\mid?)$  Then we have  \begin{align} |f(x)-f(y)| &amp;\leq |f(x)- f_{n}(x)| + |f_{n}(y)-f(y)| + |f_{n}(x)-f_{n}(y)| \\ &amp;\leq |x-y| + |x-y| + K|x-y| \\ &amp;\leq (K + 2) |x - y| \end{align}  which shows that $f$ is Lipschitz. Can anybody tell me if this makes sense? Sorry if layout is poor. I have made a command  \newcommand{\mysec}[2]{%  \textbf{#1} \\ #2 }   I use it with \mysec{A}{B}.  But sometimes I also want a third parameter. I could make a whole new command for those situations where I need 3 parameters. But I thought if I could handle it with an optional parameter in my original command?  So I have to check if the third parameter is set and how to print if it is set. Should I capitalise "or" between examples? On this site, there are often many different options as an answer, for example both with and without the Oxford comma.  When giving multiple options, should I capitalize the word "or" between them?  Option 1:     Eggs, butter and milk are dairy products   Or   Eggs, butter, and milk are dairy products     Option 2:     Eggs, butter and milk are dairy products   or   Eggs, butter, and milk are dairy products     Notice in Option 1 the "or" is capitalised. In Option 2 the "or" isn't. How to extend WIFI from ethernet Here's my current setup.   Bonded "Zone" DSL modem/router on main level serving laptop #1 via Ethernet and laptop #2 via WIFI.  WIFI does not serve basement level well, so I have Ethernet running to basement where it is divided by a switch with one Ethernet going to laptop #3 and another Ethernet going to entertainment system, which houses a Sony Playstation 3. The PS3 has gone on the fritz and is now only good for playing discs, so I recently added a Google Chromecast to the TV. The Chromecast, of course is WIFI only, and is only receiving a weak signal. Plus I need to have a decent WIFI signal to my phone, which is the interface device for the Chromecast. So what I would like to do is to use the Ethernet that was serving the PS3 and broadcast WIFI from it. What device can I plug the Ethernet into that will do this for me? And if you have a recommendation, are there any special setup instructions? Cam I have two GFCIs connected to one double gang box in my laundry room and if so can I do a single line connection so that they both can work independently?  I have a 12/2 going from the panel through conduit (copper wire) to the double gang box. I want to have each GFCI to run independently without using the load. Why, in case one shuts off the other isn't affected. the double gang box is standard. the device is a Leviton 20 amps. I was told that you can run 12/2 from the panel to the first GFCI black and white. Then connect the second GFCI using a jumper and attaching it to he second GFCI. Is this acceptable by code. we use a washer,dryer in the laundry and the garage heavy machinery. so why should this be a problem running two independent? Changing internal URL path for SEO I have a site where Google has already indexed about 1 million pages, and I want to change all the URL paths for SEO purposes, ensuring that we always have our keywords in the path.  Technically I know how to complete it (301 redirects, new canonical URL, update sitemap, etc.) but I'm wondering if Google will penalize the site because of it, or if links will disappear from search results until they are re-crawled? Why are some of my screen-recording/capturing/sharing applications (or app features) not working in Ubuntu 17.10 (and later releases)? After upgrading to Ubuntu 17.10 I noticed some of my screencasting applications are not working, for example screencasting tools like SimpleScreenRecorder and vokoscreen. ()  Also I noticed features like screenshot from GIMP is not working properly, producing blank screenshots (). Shutter too is producing glitchy screenshots ().  Screen-sharing option on Skype is also not working ().  The issue occurs sometimes on later Ubuntu releases too.  Why is this happening and how do I fix this? Should an exactly equal edit be dropped? I just improved an answer which was one second later edited by another user with the same actions, so that the diff is completely empty. I think the later edit should been dropped automatically.  Here is the question I'm talking about: How to change programmatically exposed filters in views I have a page view with exposed filters. One of the filters is term vocabulary, and that's dropdown list, user can select only one category (term).   I want to set programmatically one term from dropdown list when I came from specific URL. I tried with views_query_alter where $category is tid.    $set_exposed_input  = array();  $set_exposed_input['field_category_tid'] = $category;   if ($set_exposed_input) {     $view-&gt;set_exposed_input($set_exposed_input);  } Direct Link to a Comment Along the lines of , what reasons are there (if any) to favor \( ... \) over $ ... $? Prove: If a linear map, $φ: U→V$ is bijective, then $φ(x)=0$ has only the solution $x=0$. Is the ideal $I = \{f\mid f (0) = 0\}$ in the ring $C [0, 1]$ of all continuous real valued functions on $[0, 1]$ a maximal ideal? Is the ideal $I = \{f \mid f (0) = 0\}$ in the ring $C [0, 1]$ of all continuous real valued functions on the interval $[0, 1]$ a maximal ideal? $f$ a real, continuous function, is it measurable? Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function. I need to show that is a measurable function.   I tried working with the definition:  Let $f: X \to \mathbb{R}$ be a function. If $f^{-1}(O)$ is a measurable set for every open subset $O$ of $\mathbb{R}$, then $f$ is called a measurable function.  Since $f^{-1}(O)$ also lies in $\mathbb{R}$, I think it is sufficient to show that every subset of $\mathbb{R}$ is measurable. But is this possible?   So far I concluded that $\mathbb{R}$ itself is measurable, since $$\mu(A) = \mu(A \cap \mathbb{R}) + \mu(A \cap \mathbb{R}^c) = \mu(A) + \mu(\emptyset) = \mu(A).$$ How do I need to approach? Is Linux a Unix? So, there are lots of different versions of Unix out there: HP-UX, AIX, BSD, etc. Linux is considered a Unix clone rather than an implementation of Unix. Are all the "real" Unices actual descendants of the original? If not, what separates Linux from Unix? Get the real width and height of an image with JavaScript? (in Safari/Chrome) Is there some way to get explorer to generate thumbnails without actually having to navigate and scroll through every folder?  All I could get was a command prompt script that would load an explorer window for all subfolders, however only images that were visible would have thumbnails created.