I came across the following problem while self studying:  Let  \begin{equation} A =  \begin{bmatrix} 2 &amp; 1 &amp; 3 &amp; 7 &amp; 5\\ 3 &amp; 8 &amp; 7 &amp; 9 &amp; 8\\ 3 &amp; 4 &amp; 1 &amp; 6 &amp; 2\\ 4 &amp; 0 &amp; 2 &amp; 2 &amp; 3\\ 7 &amp; 9 &amp; 1 &amp; 5 &amp; 4\\ \end{bmatrix} \end{equation}  Use the fact that 21375, 38798, 34162, 40223, and 79154 are divisible by 19 to show, without evaluating, that $\det[A]$ is divisible by 19.  I noticed that each of these numbers are the entries in the rows of A, but I don't see how that helps me.
Let $n\in\mathbb N$ and let $a_1,\ldots,a_n$ be natural numbers smaller than $10^n$. Write each $a_k$ in base $10$ and add $0$'s to the left of each decimal expansion, if needed, so that each $a_k$ is written with $n$ digits. Consider the matrix such that the entries of the $k$th row are the digits of $a_k$ (written in the same order). Let $d\in\mathbb N$ be such that $d$ divides each $a_k$. Prove that $d\mid\det A$.   For instance, suppose that $n=4$ and that your numbers are $3876$, $2784$, $684$, and $8388$, each of which is a multiple of $12$. Then$$A=\begin{bmatrix}3 &amp; 8 &amp; 7 &amp; 6 \\  2 &amp; 7 &amp; 8 &amp; 4 \\ 0 &amp; 6 &amp; 8 &amp; 4 \\  8 &amp; 3 &amp; 8 &amp; 8\end{bmatrix}$$and $\det A=-360$, which is, in fact, a multiple of $12$.  I learned about this problem yesterday. I found it quite cute and I decided to share it with all of you.  Note: I know how to prove it.
Android app chokes on per-site meta profiles Steps to reproduce:    Go to a per-site meta. Find any post. Touch the user that posted it. See something like this (click to enlarge):           As you can see, it says "Stack Overflow" as the site, and shows my SO rep and badges, but it shows my Meta.Code Golf posts. When should com­pound words be writ­ten as one word, with hy­phens, or with spaces? Some compound words are written without hyphens (nonaggression, nonbeliever), some with hyphens (well-intentioned), and others with spaces (post office).  Is there a rule or good guide as to which option should be used? For a certain Hibernate entity we have a requirement to store its creation time and the last time it was updated.  How would you design this?   What data types would you use in the database (assuming MySQL, possibly in a different timezone that the JVM)?  Will the data types be timezone-aware? What data types would you use in Java (Date, Calendar, long, ...)? Whom would you make responsible for setting the timestamps&mdash;the database, the ORM framework (Hibernate), or the application programmer? What annotations would you use for the mapping (e.g. @Temporal)?   I'm not only looking for a working solution, but for a safe and well-designed solution. Every differentiable function $f:R \rightarrow R$ with the property that $(2h)f′(x) =f(x+h)−f(x−h)$ for all $x \in R$ and all $h$ has the form $f(x) =ax^2+bx+c$.  I would like to get some help on this one. Taking partial derivatives was an idea, but is that legal? Windows is unable to detect disk partitions I installed Windows 8.1 Preview using the DVD yesterday after wiping off my HDD (I made a backup prior to that). While installing Windows I created only the partitions necessary to install Windows (System Reserved, C: shown in second screenshot). Further I installed Ubuntu 13.04. During Ubuntu installation I created necessary extended partitions (for OS, swap and home directory - all ext4 except swap). Later within Ubuntu I created a NTFS partition at the end (partition 4 in first screenshot). This was a primary partition. The objective behind this was that the partition must be accessible from Windows.  While Ubuntu disk utility reports the partitioning just fine (as below)  Windows disk management is showing all partitions as primary.   Thus within Windows I am able to access only C:! What puzzles me is one can't create so many primary partitions! How can I solve this issue? It has been said that everyone lives by selling something.  Can I substitute the bold part with &quot;it is said&quot;? I think since it is kinda like a saying, using present simple is more appropriate. But I also feel &quot;it is said&quot; can also mean this sentence was said before, and the impact on the present is that now people generally accept that. And present perfect I think carry the implication that people now may or may not say this anymore. I just came over this syntax in some of the questions in this forum, but Google and any other searchengine tends to block out anything but letters and number in the search so it is impossible to search out "=>".  So can anyone tell me what it is and how it is used? This is an exercise from my calculus class. The function is defined as $x\sin (1/y)+y\sin (1/x)$ if $x\neq0 $ and $y\neq0 $, and $0$ if $x=0 $ or $y=0$.  I'm pretty confident the limit exists and should be $0$, because: $$\lim_{(x,y)\to(0,0)}[x\sin (1/y)+y\sin (1/x)]=\lim_{(x,y)\to(0,0)}[x\sin (1/y)]+\lim_{(x,y)\to(0,0)}[y\sin (1/x)]$$  And: $x\leq x\sin(1/y)\leq x$, so $\lim_{(x,y)\to(0,0)}[x\sin (1/y)]=0$ right? (The same can be said for $\lim_{(x,y)\to(0,0)}[y\sin (1/x)]$)  However, I tried checking my answer, and according to Wolfram Alpha the limit doesn't exist. Is this because I'm wrong, or is it just because $x\sin (1/y)+y\sin (1/x)$ is undefined for $y\neq0 $ and $y\neq0 $ A projection operator $P$ is defined as $P^2$=$P$. Use this definition to find the eigenvalues of this operator.   In this question is it necessary to define what the projection operator is? And won't the eigenvalue just be zero? If a spell is cast on a moveable creature/object and lasts longer than a round, and the creature/object moves, does the spell move with it? How to find and replace in linux on a large file? Oh, hi; So I've got this big json file and I'm trying to find and replace a few things but when I've used sed I get an error and I need a sed or awk string to find and replace it, without getting an error.  prajwal  Current: {&quot;telephone_numbers&quot;:[&quot;1-  Future: {&quot;telephone_numbers&quot;:[&quot;+1-  This is the command i tried and the error i got. sed -i 's/{&quot;telephone_numbers&quot;:[&quot;1-/{&quot;telephone_numbers&quot;:[&quot;+1-/g' output.json sed: -e expression #1, char 56: unterminated `s' command I am using the  for GNOME Shell, similar to the old Maximus extension for GNOME 2 / Unity. When I maximise a window, the title and menu bars are hidden.  But I don't want the minimise, maximise or close buttons to appear in the Activity Bar. , so I have been able to remove the minimise and maximise buttons by disabling them in Tweak Tool.  How do I similarly disable/remove the close (X) button?  (I guess this is the opposite of ) Whenever I read about the curvature of spacetime as an explanation for gravity, I see pictures of a sheet (spacetime) with various masses indenting the sheet to form &quot;gravity wells.&quot; Objects which are gravitationally attracted are said to roll down the curved sheet of spacetime into the gravity well. This is troubling to me, because, in order for objects on the locally slanted spacetime sheet to accelerate, gravity must be assumed. Therefore I ask; does the explanation of gravity as the curvature of spacetime assume gravity? If yes, what is the point of the theory? If No, what am I missing? What is port forwarding and what is it used for? There are a lot of questions on , but there doesn't seem to be one that clearly states what it is and what it's used for. So:   What is port forwarding? What is it used for, and why would I need it? Gradient Boosting for Linear Regression - why does it not work? Are these plural or singular? I have a simple spreadsheet, two columns "Amount" (col Q) and "Account" (col O).  I want to get the sum of the "Amount" column where the corresponding "Account" forum equals "Paypal".  This is as far as I can get!  =SUM(Q:Q)*(O:O="Paypal") In what order should one apply max HP reduction and damage? A character at 10/26 HP is hit by a Wraith and fails the Constitution saving throw. They take 21 damage and their maximum HP is reduced to 5.  Are they unconscious because the 11 excess damage is less than 26 or dead because it's greater than 5? Why is "nodev" in /etc/fstab so important? How can character devices be used for hacking? I am learning about linux security and struggling to understand why a USB stick with a character device on it is potentially dangerous.  If I have a USB stick with a bash executable that has setuid root on it, the danger is obvious: Anybody with such a USB stick can gain root privileges on my computer if I have an entry like   /dev/sdb1 /media/usbstick auto defaults 0 0   in my /etc/fstab because defaults includes suid.  But what about character devices? How can I use a character device to gain root privileges or break stuff if a USB stick with a character device on it gets mounted with dev or defaults? Formatting should not count towards the character limit on comments