I want to confirm some of my logic with remainders.  If I know the remainder of $\frac{x}{y}$ is $n$, and the remainder of $ \frac{z}{y}$ is $m$, then can I say that the remainder of $ \frac{xz}{y}$ is $n*m$?  I don't have a formal proof, but I can't find a counter-example.
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?
Why cannot a set be its own element? Should reputation from SO carry over to Meta? I have an upcoming trip where I will be flying to and from Copenhagen, through Toronto.  I notice that my departure and return gate in the Philadelphia airport is not in the international terminal.  Where will I clear customs? Why is saltatory conduction in myelinated axons faster than continuous conduction in unmyelinated axons? Share Wireless connection with Wired Ethernet Port Adventurer enters an other-dimensional realm where he repeatedly dies unexpectedly and wakes up in reality I would like to travel in the USA for a period of 3 - 6 months. But I want to spend some time working for my employer (remote work) and getting paid by the employer. Is this possible under a tourist visa (B1/B2)?  I hope this is OK since I will not be getting paid by a US employer. I am in the IT trade so all that I will be needing is my own laptop and a connection to the internet. The money I earn would actually help me pay for the expenses I get during the holiday. Change the order of integration in:  $$\int_0^{2a}\int_0^\sqrt{2ax-x^2}\frac{\phi'(y)(x^2+y^2)x }{\sqrt{4a^2x^2-(x^2+y^2)^2}}dxdy$$ and hence evaluate it.    I changed the order of integration and got the limits $y=0$ to $a$ and $x=a-\sqrt{a^2-y^2}$ to $a+\sqrt{a^2-y^2}$. I then tried changing to polar coordinates, but it is only making the integrand more complex. This is actually a two part question:   Which methods are there to calculate average slope by unit (say km²) for a given area? Are there any special requirements on the data - like additional information besides height measurements? Which open source GIS implement methods usable to calculate average slope on a grid (e.g. TIF file)? Importance of normal distribution Why did the normal distribution become such a popular (important) distribution? I know one reason is because of CLT. Can you please give more reasons? What software works well for viewing massive TIFF images on Windows 7? Book where wizards learn magic by bonding with crystals that imbue them with powers I read this book a long time ago, but I'm guessing it was written sometime in either the 80's or early 90's. It was a rather long/large book.  Plot summary: Set in a fantasy land of wizards and magic, some kids (maybe) are recruited to learn magic. In the process they find out that wizards are created by being placed in a crashed spaceship that bonds them with crystals that imbue the powers. One of them is able to bond with many/multiple crystals to become very powerful. Do you say IN one of the computers or ON one of the computers? Which is right? How can I create documents in LaTeX using a calligraphic first letter for chapters? Books often have a special letter (often spanning two lines) at the beginning of chapters. How can I create this in LaTeX? Two blue arrows at top right of icons What is the scope of variables in JavaScript? What is the scope of variables in javascript? Do they have the same scope inside as opposed to outside a function? Or does it even matter? Also, where are the variables stored if they are defined globally? Limit of Multivariate Probability Density Function as one or more or all variables approach positive or negative infinity Could someone point out a good source or provide justification as to whether the limit of a multivariate probability density function, as one or more or all variables tends to positive or negative infinity, goes to zero.  $$\lim_{x_{1}\to\pm\infty}f(x_{1},x_{2},...,x_{n})=0$$  $$\lim_{x_{1},x_{2}\to\pm\infty}f(x_{1},x_{2},...,x_{n})=0$$  $$\lim_{x_{1},x_{2},...,x_{n}\to\pm\infty}f(x_{1},x_{2},...,x_{n})=0$$  I have found this related question for the univariate case: Optional argument to \item Slow BIOS bootup This is a new HP DV-7 laptop. When you turn it on SOMETIMES it will wait at 15 minutes on the press escape for menu and sometimes it will boot correctly.  IS this a hardware failure issue or is this a new for a BIOS update? Was voldemort's body going to die? Voldemort has gained "immortality" via his horcruxes, but was his body immortal to age? Looking at all the questions about Voldemort and his horcruxes what was his plan to keep his body alive. We know his first Horcux was created at age 16, yet we see him age still so it seems like eventually his body would die of old age and his soul would be left in the half dead state he was in after his spell rebounded off Harry. What was his plan here, why would he not want something such as the elixir of life to keep his vitality?     While the Elixir of Life does indeed extend life, it must be drunk regularly, for all eternity, if the drinker is to maintain his immortality. Therefore, Voldemort would be entirely dependent on the Elixir, and if it ran out, or was contaminated, or if the Stone was stolen, he would die just like any other man. Thereafter, I am convinced, he intended to continue to rely on his Horcruxes: he would need nothing more, if only he could regain a human form. He was already immortal, you see ... or as close to immortal as any man can be.' (Albus Dumbledore)   Horcruxes seem to fail at something important, keeping you alive, they stop you from dying truly yes, but they don't keep you alive. Harry is shown over and over again to be able to beat better wizards because hes has faster reflexes both because of youth and his genetics, but presumably by 50 harry will not be as fast as his 16 year old self even considering their longer life expectancy. Why operate in a body of a 50 year old?