I need to solve for $x$ in this congruence:  $$17x+7 \equiv 3 \pmod 6$$
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?
Knife tool doesn't cut obscured edges I'm using Blender 2.66,  I find that the knife tool (K) doesn't work the way it used to.     Dragging across an obscured edge, even if selected, doesn't cut it. There seems to be no way to activate the option to do a midpoint cut or multiple cuts. Normed vector spaces over finite fields Normed vector spaces are typically defined over the reals or complex numbers.  Is there any "standard," well-behaved construction that generalizes this to a vector space over a finite field, such as $\Bbb F_2$?  I'm looking for something kind of like the class of $\ell_p$ norms, except designed with finite fields in mind.  Ideally, something that has deep fundamental properties making it well-behaved in the same way that the Euclidean norm is. How can I copy location info from geotagged phone images to images taken with my other camera around the same time? So i did quite a big trip recently, and took lots of pictures with my DSLR and occasionally Android phone. Only after coming back did i read about the whole GPX file, tracklog options in Lightroom. meaning i don't have such a tracklog file.  The android phone did geotag most photos it took. (at least, when it got a gps fix i guess)  So my question is, is there some software or way that will automatically copy the geotag from the android's photos to photos taken on the same time with my DSLR?  Or alternatively, generate a tracklog from the geotagged data in the android pictures? I'm trying to wrap my head around the math/physics of how illuminance on the film/sensor plane remains equal for a given f-number regardless of the focal length of the lens.  Since a longer lens will have a larger entrance pupil at a given f-stop than a shorter lens at the same f-stop, how can the light falling on the film/sensor be equivalent?  Empirically, I know this to be true.  Say I take an incident meter reading (not reflected/TTL) off my subject, and get f/5.6 at 1/100s, I'll get proper exposure at the point at which i took the meter reading with that setting whether I use a 50mm lens or a 200mm lens (obviously composition will be different).    Wikipedia's  reads:     A 100 mm focal length f/4 lens has an entrance pupil diameter of 25   mm. A 200 mm focal length f/4 lens has an entrance pupil diameter of   50 mm. The 200 mm lens's entrance pupil has four times the area of the   100 mm lens's entrance pupil, and thus collects four times as much   light from each object in the lens's field of view. But compared to   the 100 mm lens, the 200 mm lens projects an image of each object   twice as high and twice as wide, covering four times the area, and so   both lenses produce the same illuminance at the focal plane when   imaging a scene of a given luminance.   I'm not sure I agree with the explanation on wikipedia.  The film/sensor doesn't care how big the image is that falls outside the bounds of the sensor; the film/sensor "sees" what it sees, and that's it. Select list of node titles for a view exposed filter What is a good book to study linear algebra? I'm looking for a book to learn Algebra. The programme is the following. The units marked with a $\star$ are the ones I'm most interested in (in the sense I know nothing about) and those with a $\circ$ are those which I'm mildly comfortable with. The ones that aren't marked shouldn't be of importance. Any important topic inside a unite will be boldfaced.  U1: Vector Algebra. Points in the $n$-dimensional space. Vectors. Scalar product. Norm. Lines and planes. Vectorial product.  $\circ$ U2: Vector Spaces. Definition. Subspaces. Linear independence. Linear combination. Generating systems. Basis. Dimesion. Sum and intersection of subspaces. Direct sum. Spaces with inner products.  $\circ$  U3: Matrices and determinants. Matrix Spaces. Sum and product of matrices. Linear ecuations. Gauss-Jordan elimination. Range. Roché Frobenius Theorem. Determinants. Properties. Determinant of a product. Determinants and inverses.  $\star$ U4: Linear transformations. Definition. Nucleus and image. Monomorphisms, epimorphisms and isomorphisms. Composition of linear transformations. Inverse linear tranforms.  U5: Complex numbers and polynomials. Complex numbers. Operations. Binomial and trigonometric form. De Möivre's Theorem.  Solving equations. Polynomials. Degree. Operations. Roots. Remainder theorem. Factorial decomposition. FTA. Lagrange interpolation.  $\star$ U6: Linear transformations and matrices. Matrix of a linear transformation. Matrix of the composition. Matrix of the inverse. Base changes.   $\star$ U7: Eigen values and eigen vectors Eigen values and eigen vectors. Characteristc polynomial. Aplications. Invariant subspaces. Diagonalization.   To let you know, I own a copy of Apostol's Calculus $\mathrm I $ which has some of those topics, precisely:   Linear Spaces Linear Transformations and Matrices.   I also have a copy of Apostol's second book of Calc $\mathrm II$which continues with   Determinants Eigenvalues and eigenvectors Eigenvalues of operators in Euclidean spaces.   I was reccommended Linear Algebra by Armando Rojo and have Linear Algebra by , which seems quite a good text.   What do you reccomend? I have some questions with string theory:   Why is it that there is exactly 4 large spacetime dimensions while the rest remain small? It is a nonlocal QFT. How could that fit in GR? What does "at 19.45 hours" mean? I was reading a book and this sentence seemed a little odd to me:     At approximately 19.45 hours the two men rose, stretched and yawned. They picked up their gear and stood at the door, ...   What does the first part mean? If it's the exact time, shouldn't it be 19:45? And if it's duration, shouldn't it be "After 19.45 hours"?  (The story here happens at an airbase, if it's of any relevance)    Edit: The book I've quoted is this:   by John Creed.   The author is Irish and the text is set in northern England/Ireland. How long are the life phases of a dragon? "Back" and "Install Now" buttons grayed out in 14.04 installer The "Back" and "Install Now" buttons are grayed out in the 14.04 installer. How do I restore them? Does anyone know what the Inverse Cumulative Distribution Function of Normal Distribution is? Does it have a closed form expression? I did not find any good answer using Google. I am starting to learn using blender, and at some point I managed to do something to the 3D view camera.  Usually the camera orbits around an object, but now the rotation seems to happen (atleast very close to) around the "camera's origin". Neither the input setting in "User Preferences", the pivot point selection nor Ctrl-Alt-Shift-C (as suggested on some forum) do anything to the issue. Does the selection of sRGB or Adobe RGB in camera when shooting RAW ever matter? I shoot RAW essentially 100% of the time, and have the in-camera color space setting at AdobeRGB (sRGB is also offered). The raw converter in turn is set to produce sRGB output files. As I understand the process, the general flow is for raw shooting: raw sensor data readout (including A/D conversion), compression, write to a raw file, then in post-processing rather than in-camera apply color space/color temperature/etc; for JPEG shooting: raw sensor data readout, apply color space/color temperature/etc, JPEG compression, write to memory card (in camera).  Hence, when a colleague said that I should leave the camera set to sRGB for best/consistent results despite using the camera's raw format, it didn't really make much sense to me. I have mulled it over for a while now and it still doesn't really make any sense.   seems to support my view of the differences in the data recorded in RAW and JPEG modes, respectively.  I did find , but the answers there seem to focus more on the difference between AdobeRGB and sRGB. My question is, when shooting RAW, does it ever matter (except for the image preview) whether the camera is set to sRGB or AdobeRGB? If it does, then why, and in what situations? Is my understanding of the "raw" format processing chain flawed?  In case it matters, I have a Canon EOS 50D. Best methods to fix broken tracks I've made a couple of PCBs which need the odd few 10 mil tracks, but are mostly 20/30 mil. Unfortunately the etching process isn't fantastic and there are numerous breaks in the traces, most of them small.  I've managed to convince solder lumps to go over the top of the smaller gaps, and have used components legs to bridge some of the larger ones. The solder lumps don't feel particularly reliable though, and there are some gaps which are too small and awkward to get a piece of wire across but too big to be bridged with just the iron.  Are there any standard ways to reliably fix up broken traces?  Update: Thanks for the suggestions everyone!  I found and tried out a silver conductive pen. As a first time user I found it difficult to dispense, and obviously I didn't use enough of it or mix it properly because it didn't even end up being conductive.  It seems like the main problem was that I was trying to use pieces of wire which are too short. I found some small gauge tinned wire and used longer lengths, running it over significant lengths of good track on either side of the fault. That makes it much easier to position, and you can tack them down one end at a time. How can I search for special characters? Situation: It started out as me  for the literal string C-c in  but it ended up omitting the - sign and showing results which contained the strings C/C++ and C &amp; C++.  is an example of a post which contains that string of characters (yet fails to show up in the results).  Things I've tried:   post suggests that I can use the undocumented code: operator, but when I do that, it just wraps the whole term in double quotes like .  As @terdon in Unix &amp; Linux chat has , using this  does yield some relevant results, but it doesn't show me  that is supposed to be there. edit: It appears that the Symbolhound query no longer returns that result.  Finally,  says that you can search for literal characters, but as you can see:    Quoted phrases are exact matches except for case-sensitivity, for example, you can search  or .  The second example (symbols) leads to 0 results.  So by now I assume that the search engine has changed (but the old documentation still lingers on this site). Is there a method for me to search for strings that include special (or is it literal?) characters? Many times when going through the Close queue I see that people are using the too localized reason.  Sometimes the reason for this is in the comments (like the syntax error being pointed out), but a lot of times it's in the answer, so then I have to click on the question and look at the answers to see if it is really too localized.  Could we please get the answers shown underneath the question in the Close review queue? Revert new tab page behavior in Google Chrome 29+ How exactly does Knuth's Up-Arrow notation work? Solutions of bivariate cubic Diophantine equation The problem is what are the integer solutions to  $$7x^2 - 40xy + 7y^2 = (|(x - y)| + 2)^3$$  I only got $x - y = 4 (\mod 13)$. Also, how to solve it? At the very end of Deathly Hallows, after Harry repaired his own wand with the Elder Wand, he tells Dumbledore's Portrait in the office:     “I’m putting the Elder Wand,” he told Dumbledore, who was watching him with    enormous affection and admiration, “back where it came from. It can stay there. If I die a    natural death like Ignotus, its power will be broken, won’t it? The previous master will    never have been defeated. That’ll be the end of it.   Now, this seems incredibly foolish to me.  "The previous master will never have been defeated" would only have been assuredely true if Harry had died right after depositing the Elder Wand back into Dumbledore's Tomb.   If not, Harry (the Wand owner) would still be alive, and at the risk of being Disarmed (as was shown with Harry and Draco Malfoy, it doesn't matter if the Elder Wand's owner is defeated by death or simple disarming, AND it's not even required that the Disarming would involve the Elder Wand!).  So, if Harry merely put the Elder Wand back into the Tomb, and some years later, someone would disarm him (or, even worse, kill him) - then that someone would become the Elder Wand owner, either by accident or by design:    By accident - Harry is likely to get into plenty of magical duels in the future. Even discounting friendly one (teaching people the Expelliramus), he plans on becoming an Auror!  By design - plenty of people have been in the Great Hall hearing Harry explaining to Voldemort that he, Harry, was now the owner of the Elder Wand, so the fact of his ownership would most assuredly NOT remain a secret, inviting countless people tempted by owning the Deathstick to hunt Harry down. Oh, and the location is now also known!   So "its power will be broken" is quite unlikely to come true!  Is Harry hopelessly wrong? Or am I missing something in the above logic? Was this addressed in canon/JKR info?