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Fastest Gun in the West Problem
I'm having some troubles understanding the solution (going from step $2$ to step $3$) of this linear congruence equation. Here's the problem with the solution:   Solve the congruence $2x + 11 \equiv 7 \pmod 3$.   First, we reduce all the coefficients $\pmod 3$:  $$2x + 2 \equiv 1 \pmod 3. \tag{1}$$  Next, we add $1$ to both sides, using the fact that $2 + 1 \equiv 0 \pmod 3$:  $$2x \equiv 2 \pmod 3. \tag{2}$$  Finally, we multiply both sides by $2$, using the fact that $2 · 2 = 4 \equiv 1 \pmod 3$:  $$x \equiv 1 \pmod 3. \tag{3}$$  That is, any number in the set $\{\ldots, −5, −2, 1, 4, \ldots\}$ will solve the original congruence.    Like I said my problem is the step between $(2)$ and $(3)$. If I multiply both sides of $(2)$ by $2$ I get:  $$4x \equiv 4 \pmod 3.$$  Now, I understand that $4$ is congruent to $1 \pmod 3$ so the RHS "simplifies" to $1 \pmod 3$ but I don't understand why the LHS "simplifies" to $x$. Can someone help me with this? Google Chrome: Have different settings on different computers? Alternatives to 'respectively' Send file using POST from a Python script Is there a way to send a file using POST from a Python script? How to remove a vector QgsVectorLayer from QGIS python plugin I managed to add QgsVectorLayer in QGIS through a python plugin, for example:  vl = self.iface.addVectorLayer(uri.uri(), layerName, self.dbConn.getProviderName())   But how can I remove it? When quoting a quotation, how do you handle the double quotes? Assume there is original source text:     This restaurant is amazing with "delicious lasagna" and great service.   Make sure you go 6-7pm for the quietest times.   I want to quote "delicious lasagna" but not sure how to handle the outer double quotes.  If I was quoting more than just that quotation (e.g. is amazing with "delicious lasagna" and great service) I'd just do:     "...is amazing with 'delicious lasagna' and great service."   However, when quoting just that quotation, it looks like:     "'delicious lasagna'"   Is this the correct way or is there something better? An idiom to describe someone who thinks he/she is wiser than others Master database is corrupt, instance won't start - what are my options? Help! My master database is corrupt, I can't even bring the SQL instance online! What are my options to get my server back up?   I do have a backup of master, but the  asks me to start the instance in single-user mode, which I can't do!  (Note: I'm leaving this question unspecified as to SQL version, so as to be a more widely-applicable reference. There are some similar questions on DBA.SE, but none that involve the server unable to start.) Definite article before name of languages Should I use the definite article before name of languages?   (the?) Persian language (the?) German language (the?) Arabic language (the?) Russian language (the?) English language   Some language name may also refer to other things, so you need language word after their name. How to set systemwide "XP style DPI scaling" in Windows 8.1? I run my ThinkPad with a system DPI setting of 140%. In Windows 7 and 8, I enabled the "Use Windows XP style DPI scaling" checkbox in the Custom DPI Setting dialog. When this setting is off, non-DPI-aware applications get DPI virtualization via bitmap stretching, which results in large but fuzzy text. Turning on the checkbox disables this bitmap stretching, so non-DPI-aware apps get smaller text but without the fuzzy stretching. I find most apps to be more readable with the smaller crisper text, so I enable XP style scaling (i.e. disable DPI virtualization).  Windows 8.1 RTM has removed this option completely. After upgrading, all of my non-DPI-aware apps are running with ugly bitmap stretching. I'm able to fix this by going into the Properties for each app and setting the "Disable diplay scaling on high DPI settings" box in the Compatibility tab, but that is a bit of a pain.  Is there any way to restore the systemwide XP-style DPI scaling in Windows 8.1? Another Question in Hatcher Area of an ellipse. (Calculus) Find the sum of $\sum \limits_{n=1}^{\infty} \frac{2^{-n}(n+3)}{(n+1)(n+2)}$ Entries I can safely exclude doing backups What is Reflection in Java Can anyone explain? How do I get an XBOX 360 controller working? I have a Microsoft XBOX 360 gamepad with the wireless adapter and I was wondering what I would have to install to use them with Ubuntu I do know that when using this on Windows I have to manually install the drivers for the wireless receiver (if that is relevant.) It is well established that:   'In quantum mechanics, the behavior of an electron in an atom is described by an orbital, which is a probability distribution rather than an orbit.   There are also many graphs describing this fact: : (hydrogen atomic orbital - one electron) In the figure, the shading indicates the relative probability to "find" the electron, having the energy corresponding to the given quantum numbers, at that point.  My question is: How do these symmetries shown in the above article occur?                 What about the 'preferable' axis of symmetries? Why these? In my game, XP needed to reach the next level is Current Level × Level Threshold.  Given that, how can I get my current level from the total XP ever earned?    For example:  Level Threshold = 50 Current Level = 1   Starting at level 1, I would need (1 × 50) = 50 XP to get to level 2, and so on.  Level 1: 50 XP needed to reach level 2 Level 2: 100 more XP needed to reach level 3 Level 3: 150 more XP needed to reach level 4   In other words, the progression table is as follows:  Level 1: 0 XP to 49 XP Level 2: 50 XP to 149 XP  Level 3: 150 XP to 299 XP Level 4: 300 XP to 499 XP   If I have 300 XP, I'll just reached level 4. How can I calculate that generally? What is the best way to include R code in the Latex paragraph I am a newbie in Latex. I have two simple questions regarding including R code in Latex.  Q1: For a block of R code, I use 'listings' package, as follows.  \documentclass[12pt]{report} \usepackage{listings} \begin{document} \begin{lstlisting}[language=R] &gt; dpareto(4, 2, 3) [1] 0.09375 &gt; dpareto(4, -2, 3) [1] NaN &gt; dpareto(seq(-1, 4, 1), 2, 3) [1] 0.0000000 0.0000000 0.0000000 0.0000000 0.2962963 0.0937500 \end{lstlisting} \end{document}   I feel it's not professional. What is the best way to present the block of R code?  Q2: For code pieces in the paragraph, for example I wish to include a R function, or variable, or package name in my Latex context, what is the best way to do that? Does Earth's Rotation Affect Its Shape? The question I am working on is, &quot;Consider the following. (a) Find the angular speed of Earth's rotation about its axis. rad/s (b) How does this rotation affect the shape of Earth?&quot;  I am fully capable of solving part (a); however, I am not sure how to describe the effect earth's rotation on its shape. I tried to search my textbook for the answer, but could not find anything. Is there an actual effect on the shape?