I'd like to call this modular multiplication but all the proofs I can find prove the following :   If $a_1 \equiv b1 \pmod{n}$ and $a_2 \equiv b_2 \pmod{n}$, then $a_1a_2 \equiv b_1b_2 \pmod{n}$  How do you go about proving this?  I want to try to show the following, but I am not sure how to do so: $$ab \equiv (a \bmod{c})(b \bmod{C}) \pmod{C}$$
If we multiply 2 number and take the mod with prime this is equivalent to first taking mod with the prime of individual number and then multiplying the result and again taking mod.  $$ ab\bmod p = ((a\bmod p)(b\bmod p))\bmod p$$  does there exist any proof for this?  Does it work for composite moduli too? Then I can use the Chinese remainder Theorem to calculate the result does there any other way apart from Chinese remainder Theorem to solve the problem?
How to toggle or turn off text selection being sent to the clipboard I would like to disable the default behavior that seems to happen with every Linux distribution that I've ever tried that any selected text is immediately sent to the clipboard (Mint, Ubuntu, Fedora, SuSE, etc.) and every window manager (Gnome, KDE, Cinnamon), and instead to behave more like the Windows implementation.    I know that this is a beloved behavior by many in Linux, and I'm sure many will think I'm an idiot.  The reason I want to do this, is that I'm a keyboard junky when navigating a GUI.  (e.g.  when I'm in Linux and I copy a URL and then switch to my browser and type Ctrl+L, it selects the address bar and moves my intended paste down one notch and replaces it with what I'm trying to overwrite.)  I know there are MANY workarounds, but I don't really care about that, what I'd prefer is to be able to toggle the behavior for the clipboard. Proving $v(s,p)=2^{p-1}(2s-1)$ is a bijection of natural numbers and $f(s)=2s-1$ is a bijection between natural numbers and odd numbers. How do I prove this function is bijective? $$ v(s,p)=2^{p-1}(2s-1). $$ The domain is natural numbers and the codomain is also the natural numbers.  And this one: $$ f(s)=2s-1. $$ The domain is the natural numbers, and the codomain is the odd numbers in the natural numbers.  With this one I would do this to show it's injective: $$ \begin{align} v(s)&amp;=v(s_1)\\ \implies 2s-1&amp;=2s_1-1\\ \implies (2s_1)/2&amp;=(2s_2)/2\\ \implies s=s_1 \end{align} $$ So it's injective since if $v(s)=v(s_1)$ then $s=s_1$.  And to show it's surjective $f(s)=y$: $$ \begin{align} y&amp;=2s-1\\ \implies s&amp;=(y+1)/2 \end{align} $$ Then the function must be surjective since every $y$ is the same as the codomain for $f$.  Am I correct? What is the difference between the | and || or operators? I have always used || (two pipes) in OR expressions, both in C# and PHP. Occasionally I see a single pipe used: |. What is the difference between those two usages? Are there any caveats when using one over the other or are they interchangeable? Why won't Blender run on Windows XP? I downloaded Blender 64 bit and the OS indicated that the program was not compatible with my computer chip. I then downloaded the 32 bit version which seemed to allow me to install. However the blender Icon in windows when pressed informs me that the program is not a window application. I have an Intel(R) Core (TM)2 Duo CPU E7300 @2.66Ghz if that is important and I am running XP. I am fairly sure I am doing something wrong here- can anyone help please? There are five boxes consisting of several identical coins. There are five boxes consisting of several identical coins. The first box contains 4 coins, the second box contains 9 coins, the third box contains 7 coins, the fourth box contains 10 coins, and the fifth box contains 3 coins. Some coins are taken from the boxes such that at least a coin taken from each box. In how many ways in taking the coins, such that the numbers of the coins taken are 12?  My attempt was solving it using repetition combination of which, there are only seven coins remaining considering that at least a coin taken from each box. By using the repetition combination, i.e. taking 7 from 9 (five boxes and four borders), resulting in 36 ways. However, there is no 36 ways in the choices. Probability that an integer number having Poisson distribution is even The probability $P(X=n)$ that an event X takes place $n$ times in a fixed period of time follows the Poisson distribution with parameter $\lambda$ i.e.  $$ P(X = n) = e^{-\lambda} \frac{\lambda ^ n}{n!}$$  I have to evaluate the probability that the event $X$ takes place an even number of times. I know that:   $$ P(X \text{ is even} ) = e^{-\lambda} \cdot \sum_{k=0}^{+\infty} \frac{\lambda^{2k}}{(2k)!}$$  but I can't solve the series.  I guess that I have to use the fact that $e^\lambda = \sum_{n = 0}^{+\infty} \lambda^n/n!$, but I got stuck.  How can I evaluate $P(X \text{ is even})$ (alternative solutions appreciated). Possible Duplicate:    When we tell a joke or narrate a story in simple present can we use 'would' in some cases? For example:  He goes up to this man and punches him in the face. The man looks straight into his eyes and tells him that he will take revenge He goes up to this man and punches him in the face. The man looks straight into his eyes and tells him that he would take revenge For wall-sized graphics and large banners (e.g. 3m × 5m), what is an acceptable PPI for print? As I understand it 300 ppi is the typical for 'small' artworks (esp. for clean text resolution). However, for small artworks, the audience usually engages at a much closer distance. Therefore, in the interests of keeping a bitmap file down to a reasonable size, can you get away with a much-reduced PPI for a large format/banner print? How to get directory size in PHP Cannot install a version of GDAL via Anaconda that permits reading BigTIFFs How to calculate number of days between two given dates? If I have two dates (ex. '8/18/2008' and '9/26/2008'), what is the best way to get the number of days between these two dates? Converting date to specified format and two date comparisions I stumbled upon a  with the following paragraph:     MBIE has launched an investigation into claims Chinese galvanised   steel coil imports have been dumped on the local market, causing   "material injury" to New Zealand industry, and the case will end up on   the table of newly appointed Commerce Minister Jacqui Dean.   I thought that sentence was grammatically wrong because the author should put the word "that" (as a conjunction) right after the word "claim". I don't see why the author could drop the conjunction. Is this a prevailing phenomenon in American or Australian English? I would like to see the icons in the 'Hot Network Questions' right panel clickable, leading to the homepage of the site where the question comes from.  It would make it more comfortable to switch to the site when you want rather to visit it than to see that specific "hot" question. Why is the gravitational force always attractive? Why is the gravitational force always attractive? Is there another way to explain this without the curvature of space time?   PS: If the simple answer to this question is that mass makes space-time curve in a concave fashion, I can rephrase the question as why does mass make space-time always curve concavely? Find $\sum_{n=1}^\infty \frac{n^3}{n!}$. I have to find the sum of the following series:  $$\sum\limits_{n=1}^\infty \frac{n^3}{n!}$$  I know how to prove the convergence of this series, but how do i find the sum. I can't use the properties of derivation or integration of the series, i have to do it using the definition that a series converges iff it's sequence of partial sum converges. In terminal, how can I write 'sagi' instead of 'sudo apt-get install' What does the double slash mean in `${f// /_}`? I'm learning Bash, and I want to replace space characters with other "non blank" characters. I'm using a for loop:  for f in *\ *; do mv "$f" "${f// /_}"; done   My question is, why are the double slash and the space in ${f// /_}? What does ${f// /_} do? Removing all 'stable' theme's CSS from a Drupal 8 custom theme I would like to remove all CSSs automatically included from stable theme:  /core/themes/stable/css/system/components/*   I can remove them one by one inside the .info.yml file:  stylesheets-remove:   - 'core/themes/stable/css/system/components/ajax-progress.module.css'   - 'core/themes/stable/css/system/components/autocomplete-loading.module.css'   - ...   Is there a way to remove all of them at once?   Something like:  stylesheets-remove:   - 'core/themes/stable/css/system/components/*' How to answer a negative question in English? Q: "Don't you know?"   If I really don't know, how should I respond to this question? Yes or No?  There's a similar question on EL&amp;U, but the answer isn't clear how it works:      Also here on ELU too:      These answers are confusing. I want to know what one word answer I can use. In other words I want to know:   What will native speakers understand if I reply No to this question? What will native speakers understand if I reply Yes to this question?