I am studying for an exam and came upon the problem of solving the linear congruence $10 x = 15 \pmod{45}$.  I am stuck as the $10$ and $45$ are not coprime but have the $5$ as a common factor.
I know how to compute modular multiplicative inverses for co-prime variables $a$ and $b$, but is there an efficient method for computing variable $x$ where $x &lt; b$ and $a$ and $b$ are not co-prime, given variables $a$, $b$ and $c$, as described by the equation below?   $ a x \equiv c \mod b $   For example, given  $ 154x \equiv 14 \mod 182 $, is there an efficient method for computing all the possibilities of $x$, without pure bruteforce?  Please note that I'm not necessarily asking for a direct solution, just a more optimized one.    I do not believe that the Extended Euclidean Algorithm will work here, because $a$ and $b$ are not co-prime.   Edit:  Follow up question, since the first one had a shortcut:  Could the be computed efficiently as well?  $12260x \equiv 24560 \mod 24755$.  $107$ needs to be one of the computed answers.
How exactly do I install Minecraft Mods and what is Forge? Is ArcSDE no longer needed at ArcGIS 10.1? Story where radically different cultures live together and are defined by rulesets, some choose not to see each other, a house is used as a spaceship I think this novel came out in the 2000s.  There are many cultures that live close together but are very different, for example some pretend to be pre-industrial.  One of the main characters is a youth who grew up in a pretend pre-industrial society and is now exploring the rest of the solar system? ring-world?   Each culture experiences their own reality that is defined by a ruleset.  One of the rulesets becomes aggressively self-propagating and this is the problem that drives the plot.  At some point a tumbling house, just a normal house, is used to travel across space. If I take the $k \in \mathbb{N}$ power of 10 and mod it by a large prime, I notice that the remainders repeat at some point.  For instance $10^k mod~7$ seems to repeat every $8$th value of $k$.  Given $k$ (the power of ten), and a large prime number $p$, how can I compute the size of the cycle?  I tried writing a python script that computes the first instance of a repeat. However, I don't know if this works since a single instance of a repeat may not necessarily mean a full cycle. Word for two collections that do not have any elements in common Why is file transfer speed decreased when copying multiple smaller files? Let's say I'm copying a portable apps directory from PC to an external HDD. An application.exe file is around 4 megabytes in size, but most other files are only tens of kilobytes in size, and most of those files take longer to copy/move than that bigger file having 4 megabytes.  Why is this? I recently got the book "selected problems in real analysis", and I'm stuck solving the very first problem  $(u_n)$ is a binary sequence iff it only contains $0$ and $1$ in the sequence  Let $A$ be the set of all binary sequences  I have to prove that $A$ and ${2}^{\mathbb{N}}$ have the same cardinality, that is to say there exists a 1-1 function from one set to another  I've thought about maybe considering integers as base-2 numbers  Thanks for your help How to write an RSS feed with Java? I'm using Java, and need to generate a simple, standards-compliant RSS feed. How can I go about this? Why didn’t anyone dial Abydos before Daniel solved the mystery of the seventh symbol? From what I recall from the movie, Daniel only finds the seventh symbol... however when he explain they give way to show him the star gate and they put their sequence in the dialing computer. This proves the dialing computer was prepared to dial 7 chevrons.  It’s also stated they’ve also tested the 6 previous symbols in the order they appear on before, one has to assume only to fail when they insert the 7th, making them aware of the fact you can, at least dial 7th chevrons.  With 39 chevrons on the gate and 6 already locked in, that leaves 33 to try to get a lock. Those are not that many not to go and simply try one by one so... how come nobody brute forced the thing? Solving a Second Order Constant Matrix Differential Equation Given that $A$ is a constant coefficient matrix, how would one find a non-zero solution to the following equation: $$ x''+2Ax'+A^2x=0 $$ I am thinking this could incorporate systems of linear equations, but I am not sure how to use $x'=Ax$ or $x'=P(t)x +g(t)$ in this situation.   Any hints or suggestions are appreciated. How do "and" and "or" act with non-boolean values? Why does an object with a glossy shader render in black? New questions, Show Recommended, Sort Newest not working i was stopped by my sister from upgrading Ubuntu, for the reason she explained was that i would have to reinstall all the softwares that I've installed in last 6 months. but i think if that's the case, the upgrade itself includes all softwares that i have installed. so please tell me what's the truth?     please note that i just can't afford mentally to reinstall all apps due to slow INTERNET speed.because it may take 3 to 4 days of constant downloading.   if you are an Indian living in an suburban city,you can understand my situation.   i think the question itself explains everything.still I'll explain everything in detail.i want to upgrade my current Ubuntu 12.10 OS to 13.10(now please don't ask what is 13.10). this option is available from "about this computer" and many other places.the upgrade is of about 700mb to 800 MB.    BTW, i don't know why i think it's a rule here to put some question on questions asked here,however detailed or/and simple it may be.at least that's what i feel after looking at many questions. I am travelling form Delhi (DEL) to San Francisco (SFO), I have to make a connection flight in Vienna that would take me to Frankfurt (FRA). My flight to SFO is from FRA, I have one ticket for entire journey and the flights are operated by Lufthansa or its subsidiaries.  Do I need a transit visa? My flight arrives at Vienna international airport and leaves from the same, and for second leg it arrives at Frankfurt international airport.  I asked this question as an addendum to my  but was advised to make a new post.  I hold Indian passport and valid US visa. Can creatures without hands (and as a consequence fingers) use rings? Pretty much what it says on a label. I have a monster for my next adventure that has a pretty interesting ring in his treasure. Since it is intelligent, and would be able to use the ring, I don't see why the ring should be lying there in a chest waiting to be looted.  Monster would use the ring, if it only had hands or even fingers. Hold a second, are fingers even necessary in 5e?  For reference, assume it only has grasping / manipulating tentacles like a  (although it is not one); Said tentacles are very flexible, capable of fine manipulation, and have varying thickness. Indian passport threading coming off I'm a physics teacher and I would like to find a simulation software that could be used to give exact numerical answers to high school level problems in Newtonian mechanics.  I want the students to first calcultate a numerical answer to a physics problem and then use the simulator to visualize the problem in order to understand it better and verify that the simlator gives the same answer that they got when doing the calculation themselves.  Here are some examples of what kind of problems I want the students to be able to solve with the simulator:   A slope with a specific angle of inclination and a ball starts rolling at a certain point. Determine the speed, angular momentum and total mechanical energy of the ball after a specific time t. A ladder of known mass and length is leaning against the wall at a specific angle. The contact between the ladder and the wall is frictionless. Calcultate the x and y components of the force between the floor and the ladder.   I would like the software to have the following features:  - Simulation in 2D. No 3D simulations are necessary and the software should at least have the option of doing simulations only in 2D.  - Position all objects exactly by writing numbers (just positioning the objects using the mouse is not precise enough).  - Show animations of the time evolution of the system.  - Show forces, velocities and accelerations as arrows.  - Give exact numerical results so that the students can use the software to verify their calculation.  Ideally the software would have a graphical user interface, but if no suitable software with graphical interface exixts, I could also consider a software with text-based inputs. I would consider both free and commercial software.  My question is: Is there any software that would meet these requirements and if there is, where could I get it? At what speed does our universe expand? Mac-like alt-tab replacement for Windows 7? Is there an alt-tab replacement for Windows 7 that works in the same manner as command-tab and command-tick (tilde) on OS X?  Requirements:  1. Completely keyboard-shortcut-based, no mouse action should be necessary.  2. 2 keyboard shortcuts with the following behavior:  a.) alt+tab:       cycles through open applications (NOT windows... applications, meaning even if i have 10 firefox windows open, I only want to see firefox once in the list that comes up when i hit this shortcut)  b.) alt+`:     cycles through WINDOWS that are instances of the currently active application (meaning if firefox is my active window and i hit shortcut2, it will cycle through those 10 firefox windows, while ignoring any other windows from ANY other application).    Update 08:40 5/21: I tried out eidylon's suggestion, TopDesk.  It doesn't provide any of the functionality described here.  Other apps I've already tried:   VistaSwitcher Switcher Dexpot (awesome app, still using, but doesn't do what I want in this case... also looked for plugins but couldn't find one that does it) UltraMon Nvidia nView   update: Unfortunately I still haven't found a solution for this.  update: made this a community wiki in case anyone is ever able to find a solution for this.