If $a \mid r$ and $b \mid r$ show that $\operatorname{lcm}(a,b) \mid r$ by using only elementary properties of numbers.
Prove that If $m'$ is a common multiple of $s$ and $t$, then $m | m'$. Here $m$ is the LCM of $s$ and $t$.    Although the statement is intuitively clear to me I don't know how to prove.
Teams of users rails app What is the maximum possible length of a .NET string? What is the longest string that can be created in .NET? The docs for the String class are silent on this question as far as I can see, so an authoritative answer might require some knowledge of internals. Would the maximum change on a 64-bit system?  [This is asked more for curiosity than for practical use - I don't intend to create any code that uses gigantic strings!] What are the benefits/detriments to graduate students joining unions? Difference between "per" and "a" Old TV mini-series with a living brain in a lab bst file not found Why is a Hyperplane called a "Hyper"plane? I just had this curious question. In other fields, the word "hyper" is actually used to refer to something which is "over; beyond; above" as defined by Google. An example of such terms would be Hyperglycaemia, defined by Wikipedia as:     a condition in which an excessive amount of glucose circulates in the blood plasma.   Or Hypersonic, defined by Google as:     relating to speeds of more than five times the speed of sound.   So, my question is, what would be the motive to give a Hyperplane such a name when it is defined as :     a subspace of one dimension less than its ambient space Flattening a shallow list in Python Is there a simple way to flatten a list of iterables with a list comprehension, or failing that, what would you all consider to be the best way to flatten a shallow list like this, balancing performance and readability?  I tried to flatten such a list with a nested list comprehension, like this:  [image for image in menuitem for menuitem in list_of_menuitems]   But I get in trouble of the NameError variety there, because the name 'menuitem' is not defined.  After googling and looking around on Stack Overflow, I got the desired results with a reduce statement:  reduce(list.__add__, map(lambda x: list(x), list_of_menuitems))   But this method is fairly unreadable because I need that list(x) call there because x is a Django QuerySet object.  Conclusion:   Thanks to everyone who contributed to this question.  Here is a summary of what I learned.  I'm also making this a community wiki in case others want to add to or correct these observations.  My original reduce statement is redundant and is better written this way:  &gt;&gt;&gt; reduce(list.__add__, (list(mi) for mi in list_of_menuitems))   This is the correct syntax for a nested list comprehension (Brilliant summary !):  &gt;&gt;&gt; [image for mi in list_of_menuitems for image in mi]   But neither of these methods are as efficient as using itertools.chain:  &gt;&gt;&gt; from itertools import chain &gt;&gt;&gt; list(chain(*list_of_menuitems))   And as @cdleary notes, it's probably better style to avoid * operator magic by using chain.from_iterable like so:  &gt;&gt;&gt; chain = itertools.chain.from_iterable([[1,2],[3],[5,89],[],[6]]) &gt;&gt;&gt; print(list(chain)) &gt;&gt;&gt; [1, 2, 3, 5, 89, 6] Install Ubuntu on Intel Mac I've recently taken up a position in an all Mac workshop - We have Intel based Apple machines. Can I dual boot Ubuntu on my machine, if so how? In MK (Morse-Kelley) set theory life is easy: $\forall X\forall y\left(y\in\bigcap X\leftrightarrow\forall x\left(x\in X\rightarrow y\in x\right)\right)$. If $X=\left\{\right\}$ then $\bigcap X=U$, where $U$ is the universal class. So the (unary) intersection of the empty set is the class that contains all sets as elements. In ZF (Zermelo-Fraenkel) set theory, instead, proper classes are not allowed. So, how can I define $\bigcap X$ in ZF? I tried with the following definitions:   $\forall X\left(X\not=\left\{\right\}\rightarrow\forall y\left(y\in\bigcap X\leftrightarrow\forall x\left(x\in X\rightarrow y\in x\right)\right)\right)$. This means that $\bigcap\left\{\right\}$ is undefined, which is not that good. $\forall X\forall y\left(y\in\bigcap X\leftrightarrow X\not=\left\{\right\}\land\forall x\left(x\in X\rightarrow y\in x\right)\right)$. This means that $\bigcap\left\{\right\}=\left\{\right\}$, which is the opposite of MK.   I couldn't find any other valuable definition. Any ideas? Thank you. create mesh then add vertices to it in python Limit the post for differents custom post type in the same wp_query I have three custom post type:    books magazine videos   Now i want to limit the number of post for custom post type, for example:   books => 3 post magazine => 2 post videos => 1 post   I used a query post like this:  query_posts( array(     'post_type' =&gt; array(     'books',     'magazine',     'videos' ), 'paged' =&gt; $paged, 'posts_per_page' =&gt; 6,  )); $counter = 1; if (have_posts()) :     while (have_posts()) : the_post();   if (($counter == 2) || ($counter == 3) || ($counter == 4)) {         $div = 'alpha';     } elseif(($counter == 6)) {         $div = 'omega';     } else {         $div = '';     }   The problems is i cant limit the numbers by custom post type.  Any idea?  Thank you :) Busy Device on Umount I often experience a problem to umount a directory:  umount /mnt/dir umount: /mnt/dir: device is busy   There are many reasons why the device is busy. Sometimes there are processes running which have open locks on it, sometimes there are other directories mounted on top of /mnt/dir.  My question:  What are the steps to check why a directory couldn't be unmounted.  I know there are many reasons, but it's ok if you explain a specific solution.  [EDIT]  [X]  running processes on mounted volumes.   [X] another volume is mounted on top of a volume we want to unmount   [_] NFS locks the volume we want to unmount Get Response header when 401 occurred $\hbar \rightarrow 0$ in quantum mechanics We often see a limit $\hbar \rightarrow 0$ in quantum mechanics and sometimes it's related with symmetry breaking. Can someone briefly write the story behind this limit? Basic Web Development IDE/Editor like Dreamweaver? I've used Windows for many years, using Dreamweaver as my editor as it had a file tree which is based on a project (via Local, FTP or SFTP).  I need a web development IDE with the following functionality:   syntax highlighting for PHP, HTML, CSS, JavaScript  file tree - Local, FTP, SFTP, Project-Based (e.g. via the methods of Local/FTP/SFTP) in other words, like Dreamweaver can open multiple files (tabbed)   Additionally, I wouldn't mind features like debugging, class listings, etc, as long as I can hide them away. I've always wondered this - why can't you declare variables after a case label in a switch statement?  In C++ you can declare variables pretty much anywhere (and declaring them close to first use is obviously a good thing) but the following still won't work:  switch (val)   {   case VAL:     // This won't work   int newVal = 42;     break; case ANOTHER_VAL:     ...   break; }     The above gives me the following error (MSC):     initialization of 'newVal' is skipped by 'case' label   This seems to be a limitation in other languages too.  Why is this such a problem? There are a lot of legendaries available in ORAS, however a lot of them will only show up/be available if certain prerequisites are met.  I know Groudon/Kyogre are available through the Story, and      Rayquaza &amp; Deoxys   through the postgame Delta episode, but what about the other available legendaries? What do I need to do to catch them? Can I be admitted into a PhD program with non-thesis MS degree? If I do a non-thesis MS program in statistics, will I be eligible for an admission in to a PhD program in Statistics later in my life? The MS program I am looking into is a non-thesis program but I still have to complete a research project requirement (which involves oral presentation as well as some form of a written report) towards the end of the program...I am still not sure if it can lead to a PhD program though.  Thanks! For my own understanding, I am interested in manually replicating the calculation of the standard errors of estimated coefficients as, for example, come with the output of the lm() function in R, but haven't been able to pin it down. What is the formula / implementation used?