I know how to get them (extended Euclidean algorithm using the mod and the number you want the inverse of if their gcd is 1, the inverse is the number you get * the one you have in the linear combination), but I don't understand what this means.  I know inverses are important for decryption for example. I just don't quite understand how it works. And why exactly is it that you can only have an inverse when gcd(a,m)=1 ? (a being any number, m being the mod your are using)
Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.  Can someone help me with this?
Editing older revision (not most recent) no longer includes older title and tags, and automatic rollback note Who is the eponymous Jedi in Return of the Jedi? Calculation of $\lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$ Calculation of  $\displaystyle \lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$  $\bf{My\; Solution::}$  Let $y=\displaystyle \lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$, Now taking $\log_{e}$ on both side  $\displaystyle \log_{e}(y) = \lim_{n\to \infty}\log_{e}\left(\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}\right)=\lim_{n\to \infty}\frac{1}{n}\cdot \sum_{r=1}^{n}\log\left(1+\frac{r}{n}\right)$  Now Using Reinman Sum of Integral ...  $\displaystyle \log_{e}(y) = \lim_{n\to\infty}\frac{1}{n}\cdot \sum_{r=1}^{n}\log\left(1+\frac{r}{n}\right) = \int_{0}^{1}\ln(1+x)dx = \log_{e}(4)-\log_{e}(e) = \log_{e}\left(\frac{4}{e}\right)$  So $\displaystyle y =\lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}} =\frac{4}{e}$  But how can i solve above question using Stirling Approximation,  May I ask for help?  Thank you in advance! Turning fullscreen on made Minecraft crash, and now it crashes on start every time I switched to full screen on minecraft, the screen turned black as if it were loading it took a while so i closed minecraft and reopened it. When I logged on it gave me a program on my taskbar saying "minecraft minecraft" then when i closed the program it gave me this error message:   java(tm) platform SE binary has stopped working.    Is there a way to fix this? Is the converse of Cayley-Hamilton Theorem true? The question is motivated from the following problem:     Let $I\neq A\neq -I$, where $I$ is the identity matrix and $A$ is a real $2\times 2$ matrix. If $A=A^{-1}$, then the trace of $A$ is   $$ (A) 2 \quad(B)1 \quad(C)0 \quad (D)-1 \quad (E)-2$$   Since $A=A^{-1}$, $A^2=I$. If the converse of  is true, then $\lambda^2=1$ and thus $\lambda=\pm1$. And then $\rm{trace}(A)=1+(-1)=0$.   Here are my questions:   Is  $C$ the answer to the quoted problem?  Is the converse of Cayley-Hamilton Theorem, i.e.,"for the square real matrix $A$, if $p(A)=0$, then $p(\lambda)$ is the characteristic polynomial of the matrix $A$" true? If it is not, then what's the right method to solve the problem above? Why are loops always compiled into "do...while" style (tail jump)? Background notebook template with spiral Let $X$ be a random variable and let $g,f$ be real-valued, nondecreasing, and bounded.   Show that $\mathbb{E}[f(X)g(X)]\geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$  Having a hard time seeing where to start for some reasons, any hints?  I managed to write:  $\mathbb{E}[f(X)g(X)] = \frac{1}{4}\mathbb{E}[(f(X)+g(X))^2 - (f(X)-g(X))^2]$  But I'm not sure if this is in the right direction or not. How can I get admin access to a Mac without knowing the current password? My Father died last year and I inherited his Mac Mini. My wife has retired and we are trying to set up the mini for her use but the changes we want to make require admin access and we do not have the information my father used to set up the machine.  Is it possible to reset the admin information without having the old information? If so, how would I go about it?   I am not a Mac user usually so any guidance will have to be pretty simple! The most efficient method to obtain the total entries into a mysql table Why use sucrose rather than glucose in an injectable vaccine? Simplest Example of a Poset that is not a Lattice A partially ordered set $(X, \leq)$ is called a lattice if for every pair of elements $x,y \in X$ both the infimum and suprememum of the set $\{x,y\}$ exists. I'm trying to get an intuition for how a partially ordered set can fail to be a lattice. In $\mathbb{R}$, for example, once two elements are selected the completeness of the real numbers guarantees the existence of both the infimum and supremum. Now, if we restrict our attention to a nondegenerate interval $(a,b)$ it is clear that no two points in $(a,b)$ have either a suprememum or infimum in $(a,b)$.   Is this the right way to think of a poset that is not a lattice? Is there perhaps a more fundamental example that would yield further clarity? Difference between generalized linear models & generalized linear mixed models Finding $\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2} dx$ Calculate  $$\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2}dx$$   I have tried to put $\displaystyle x=\frac{1}{t}$ and $\displaystyle dx=-\frac{1}{t^2}dt$  $$ \int^{\infty}_{0}\frac{t^2\ln^2(t)}{(t^2-1)^2}dt$$  $$\frac{1}{2}\int^{\infty}_{0}t\ln^2(t)\frac{2t}{(t^2-1)^2}dt$$  $$ \frac{1}{2}\bigg[-t\ln^2(t)\frac{1}{t^2-1}+\int^{\infty}_{0}\frac{\ln^2(t)}{t^2-1}+2\int^{\infty}_{0}\frac{\ln(t)}{t^2-1}dt\bigg]$$  How can I solve it? I have an email account that has passed 60GB of emails, and currently I'm having a lot of trouble using an email client to archive emails of last year (2011).  Via terminal, I am trying to use  to locate the files between 2011-01-01 and 2011-12-31, but no avail.  How can I find files between two dates?  If relevant, the end goal will be a batch that will move each file found, matching the date interval, to a folder. Origin —  Elementary Number Theory — Jones — p35 — Exercise 2.17 — For which primes $p$ is $p^2 + 2$ also prime? Only for $p = 3$. If $p \neq 3$ then $p = 3q ± 1$ for some integer $q$, so $p^2 + 2 = 9q^2 ± 6q + 3$ is divisible by $3$, and is therefore composite. (1) The key here looks like writing $p = 3q ± 1$. Where does this hail from? I know $3q - 1, 3q, 3q + 1$ are consecutive. $p$ is prime therefore $p \neq 3q$? (2) How can you prefigure $p = 3$ is the only solution? Does $p^2 + 2$ expose this? On an exam, I can't calculate $p^2 + 2$ for many primes $p$ with a computer — or make random conjectures. Skyrim crashes as soon as I boot up into the menu. How do I fix this? Whenever I boot up Skyrim, it crashes as soon as it reaches the main menu with no dialog or anything. I thought this was related to my using dual monitors somehow, but I disconnected the other monitor and the game still crashed. I tried running the game in different resolutions, both full-screen and windowed, at various quality settings; no dice. My computer does meet the minimum requirements to play Skyrim.  How do I fix this problem? Programmatically Create X509 Certificate using OpenSSL Several badge descriptions contain raw HTML I was reading through some badge descriptions and noticed that Synonymizer's description needs to be updated. As it stands, there is raw HTML (an href) in the description:    EDIT: after reading comments by @ColeJohnson, I have come to the understanding that this post comes off as whiny. I want to make it very clear that I wasn't ever trying to complain. My first reaction was to fix the perceived error myself. When I realized I was unable to do so, I figured the best course of action would be to find someone who would be able to make the necessary edits. Hence the post here. Particles rotation not working correct