Find integers $t,z$ satisfying $(a+b)z+abt=1$
If $(a,b)=1$ then prove $(a+b, ab)=1$.
I've been wondering about this for a while, since getting interested in knife sharpening.  When using a knife, it will get duller. One (the primary) reason being that you "bend" the edge. You fix this using a honing steel.  After a while though, you need to sharpen the knife again.  Another setting: If you have a brittle knife and/or mistreat your knife you will chip it.   As far as I can tell, this should mean that small pieces of metal (dust, shavings, etc.) come lose, and then likely get in the food.  Is this correct? Are there no health considerations when digesting small pieces of very sharp metal? Black and white beads on a circle There are $n$ beads placed on a circle, $n\ge 3$. They are numbered in random order as viewed clockwise. Beads for which the number of the previous bead is less than the number of a next bead are painted in white color,and others - in black.  Two colourations that can be made equal by rotation are considered identical. How many different colourations can occur?  I've write a programm and for $n=3...11$ I've got answers $2 , 1 , 6 , 7 ,  18 , 25 , 58 , 93 , 186$ How to move a window if titlebar is overlapped by Ubuntu menu Somehow the title-bar of chrome was overlapped by Ubuntu menu-bar like attached image.  In this case how to move this chrome window? Switching from MTB to road handlebars How to prove $a_1^m + a_2^m + \cdots + a_n^m \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$ Review-beta: Obviously good answer in low-quality Compactness of $(M, d)$ in terms of continuous functions $f:M\to \mathbb R$ Suppose $(M,d)$ a metric space. I want to show that      If every continuous real-valued function on $M$ attains a maximum, then $(M, d)$ is compact.   I was trying to do this by assuming $M$ non-compact and showing explicitly a function which does not attain a maximum. If $M$ is not bounded it's enough to take the distance from a fixed point as we can always find a sequence which "exits" every compact and pushes the distance at infinity. But what if $M$ is bounded? I'm making a game where you see a spaceship (in 3rd person) travelling at high-speed between planets (it takes 2-3 seconds move between them).  What effect could I use to convey the effect of extreme speed? I already know about the usual effects, change in FOV, white lines around the subjects, etc..   I'm asking about some things that are more space-like, I mean even if it's not how it would be perceived in real physics, I need it to look like it's fast and it's in space.  Don't hesitate to ask for clarification, my English is nowhere near being perfect, maybe I didn't convey my thought perfectly. How to get Total Pupulation From Tables In Sqlserver? Lots of new batteries (for mobile devices, MP3 players, etc) have connectors with 3 pins. I would like to know what is the purpose of this and how should I use these three pins?    They are usually marked as (+) plus, (-) minus, and T. booting windows 10 spends too long In august 2015, I have upgrade my laptop from windows 8.1 to windows 10. But it spends always 10 minutes to boot it. It is not normal that it spends so long. What can I do for boot him faster? Below some information about my laptop.   age: +/- 2 year (on august 2015) edition: Windows 10 Home   File trace are removed.  Update:    I must also say that always there comes a windows update, I have the same problem again form slow booting. Is it good that I turn off the updates? If yes, how can I do that? Also it is in periods that my pc booting to slow. Some time it is like normal (a few seconds) and other times a few minutes. I don't know how it comes that it is fast or slow. What is the symbolic significance of the Night King's message? In Game of Thrones S8E1,        Ned Umber is found pinned to a wall by the Night King with a symbolic message, which looks like tentacles coming out of the body of Ned.   Is there some meaning to that symbolic message? If yes, then what is that? The site description in the hero header is wrong for Meta sites The hero header contains the same description for the Main and Meta sites. Here's the one for MSO:           Meta Stack Overflow is a question and answer site for professional and enthusiast programmers. Join them; it only takes a minute:    This can easily confuse people who are new to the network, so I feel this should be changed to something Meta specific. Perhaps re-use the text from the sidebar, that has a good description:     This site is intended for bugs, features, and discussion of Stack Overflow and the software that powers it. You must have an account on Stack Overflow to participate. Why do I have to say “want you to do” instead of “want you do”? I'm confused about 'want you to do (something)' and 'want you do (something)'.  Example:      "I want you to eat fish" instead of "I want you eat fish" or "I want that you eat fish".    Can anyone explain what is the grammar rule for this? how to fill intersection of 2 shapes in latex tikz package I have to fill a specific region as given in the following figure.  The code I have written is   \begin{tikzpicture}  \draw (0,0) rectangle (10,10);  \draw (3.5,3.5) circle (5) node {A};  \end{tikzpicture}   Now, how to fill the intersecting area (area of circle overlapping the square) Expandably change letter case and use inside \csname, without a package I want to change the letter-case of a single character argument, and use that changed letter inside a \csname.  And I want to do it without an additional package.  For example, I want to pass a macro the letter {S}, and have it execute the macro \csname macros\endcsname, or pass it the letter {T}, and have it execute the macro \csname macrot\endcsname.    In my MWE, I show three approaches:   I didn't expect \lowercase to work, since I recall reading that it is not truly expanded until the last possible moment.  Surprisingly, it goes into an \edef okay, but it must not be truly expanded, because it breaks the \csname. I also formulated an approach where I used \numexpr to add 32 to the ASCII code of the argument and perform a \char upon that result.  Like \lowercase, it gives the appearance of being expandable, but it also fails inside the \csname. I finally achieve success with the stringstrings package.  Having written the package, I know the conversion was not pretty... essentially, it is an exhaustive 26-fold check of the form \if #1Aa\else\if #1Bb\else\if #1Cc\else...\fi\fi\fi, which has the virtue that it can be expandably placed into an \edef, and so it works.   There has to be a better way.  The MWE works only because I use the inefficiently calculated \tmp from METHOD 3 inside the \csname.  If I use \tmp from either of the two earlier methods, it breaks.  \documentclass{article} \usepackage{stringstrings} \def\macros{Expandably converted an S to a s} \def\macrot{Expandably converted a T to a t} \newcommand\convert[1]{%   %MAKE THE ARGUMENT LOWER CASE   1) LOWERCASE\\   \edef\tmp{\lowercase{#1}}% PRETENDS TO EXPAND, BUT FAILS IN \csname   Here is the ``expanded'' conversion: \tmp\\   % OR   2) ASCII CONVERSION\\   \edef\tmp{\expandafter\char\numexpr`#1+32\relax}% PRETENDS TO EXPAND, BUT FAILS IN \csname   Here is the ``expanded'' conversion: \tmp\\   % OR   3) STRINGSTRINGS PACKAGE\\   \caselower[q]{#1}\def\tmp{\thestring}% WORKS BUT USES A PACKAGE   Here is the ``expanded'' conversion: \tmp\\   %   Now use it in a csname: \csname macro\tmp\endcsname% } \begin{document} \convert{S}\par \convert{T} \end{document}       SUMMARY OF ANSWERS:  I have added egreg's code snippets into this MWE to produce a document (and not just terminal output), showing the methods put forth to this point:  \documentclass{article}  %%%%%%%%%%%%%%%%%%%%%%%%%%%% L3 APPROACH  \usepackage{expl3}  \ExplSyntaxOn \cs_set_eq:NN \Xexplower \tl_expandable_lowercase:n \ExplSyntaxOff  %%%%%%%%%%%%%%%%%%%%%%%%%%%% EGREG'S LESS HACKY SOLUTION  \def\explowerchar#1{%   \ifcase\numexpr`#1-`A\relax    a\or b\or c\or d\or e\or f\or g\or h\or i\or j\or k\or l\or m\or    n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or x\or y\or z\else    #1\fi }  \def\explower#1{%   \doexplowerchar#1\relax }  \def\doexplowerchar#1{%   \ifx#1\relax   \else     \explowerchar{#1}\expandafter\doexplowerchar   \fi }  %%%%%%%%%%%%%%%%%%%%%%%%%%%% STEVE'S HACKY PACKAGE APPROACH  \usepackage{stringstrings}  %%%%%%%%%%%%%%%%%%%%%%%%%%%%  \def\macros{Expandably converted an S to a s} \def\macrot{Expandably converted a T to a t}  \begin{document}  \textbf{Steve's hacky approach}  \caselower[q]{S}\csname macro\thestring\endcsname\par \caselower[q]{T}\csname macro\thestring\endcsname  \textbf{egreg's less hacky approach}  \edef\lcstring{\explowerchar{S}}\csname macro\lcstring\endcsname\par \edef\lcstring{\explower{T}}\csname macro\lcstring\endcsname  \textbf{L3 approach}  \edef\lcstring{\Xexplower{S}}\csname macro\lcstring\endcsname\par \edef\lcstring{\Xexplower{T}}\csname macro\lcstring\endcsname  \end{document} Linux GUI development I have a large GUI project that I'd like to port to Linux. What is the most recommended framework to utilize for GUI programming in Linux? Are Frameworks such as KDE / Gnome usable for this objective  Or is better to use something more generic other than X?  I feel like if I chose one of Gnome or KDE, I'm closing the market out for a chunk of the Linux market who have chosen one over the other.  (Yes I know there is overlap)    Is there a better way? Or would I have to create 2 complete GUI apps to have near 100% coverage?  It's not necessary to have a cross-platform solution that will also work on Win32. Determine wether the differential equation is stable or not for the following conditions:  $x'''(0) = x''(0) = x'(0) = x(0) =0$  $\frac{d^3x(t)}{dt^3}+\frac{d^2x(t)}{dt^2}+4\frac{dx(t)}{dt}+4x(t) = e^t(H(t)-H(t-3))$  Now, I got that the solution to this differential equation reduces x(t) by use of Laplace Transforms, however, how do I determine whether or not it is stable?  $x(t) = 0.1e^{-t} + 0.1e^t - 0.1\sin(2t) + 0.1e^3 (\sin(2(t-3)) + e^{-t+3}-e^{t-3})*H(t-3)$  Now I first thought I would look at it graphically and I used Matlab software to get a plot as shown below:    And from the plot above it clearly appears to diverge (though this is not a proof).  Anyone have an idea as to an elegant proof to show the solution is unstable?  Anyone help would be greatly appreciated! A key doubt about the resultant of two polynomials and their common zero locus The problems that appear in difficult math competitions such as the IMO or the Putnam exam are usually very difficult and require some ingenuity to solve. They also usually don't look like they can be solved by simply knowing more advanced theory and the such.  How do people typically come up with these problems? Do they arise naturally from advanced mathematics (the somewhat infamous 'grasshopper problem' from the 2009 IMO comes to mind - to my not exactly knowledgeable mind this problem looks like it popped out of basically nowhere)? What is the perspective that mathematicians take when seemingly "inventing" these problems with no theoretical motivation to them whatsoever?