Using the Euclidean algorithm, find $\mathrm{hcf}(86, 100)$, and use this to find integers $s, t$ such that $\mathrm{hcf}(86, 100) = 86 · s + 100 · t$. I have that the HCF is 2 but have forgotten what to do to find $s$ and $t$, any help would be appreciated.
I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea?
Different ways to execute a shell script There are several ways to execute a script. The ones I know are: /path/to/script # using the path (absolute or relative) . script        # using the . (dot) source script   # using the `source` command  Are there any other way? What are the differences between them? Are there situations that I must use one and not another? Why is this simple Excel calculation not netting Zero? It's supposed to be zero but for some reason the result produces a peculiar series of digits at the end.  Also, here's the .xlsx of it: http://83.212.111.36/Book1.xlsx I'm starting out in Visual C++ and I'd like to know how to keep the console window.  For instance this would be a typical "hello world" application:   int _tmain(int argc, _TCHAR* argv[]) {     cout &lt;&lt; "Hello World";     return 0; }   What's the line I'm missing? Choice of $q$ in Baby Rudin's Example 1.1 First, my apologies if this has already been asked/answered.  I wasn't able to find this question via search.  My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch 1, Example 1.1 on p. 2.  In the second version of the proof, showing that sets A and B do not have greatest or lowest elements respectively, he presents a seemingly arbitrary assignment of a number $q$ that satisfies equations (3) and (4), plus other conditions needed to show that $q$ is the right number for the proof.  As an exercise, I tried to derive his choice of $q$ so that I may learn more about the problem.  If we write equations (3) as $q = p - (p^2 - 2)x$, we can write (4) as   $$ q^2 - 2 = (p^2 - 2)[1 - 2px + (p^2 - 2)x^2]. $$  Here, we need a rational $x &gt; 0$, chosen such that the expression in $[...]$ is positive.  Using the quadratic formula and the sign of $(p^2 - 2)$, it can be shown that we need   $$ x \in \left(0, \frac{1}{p + \sqrt{2}}\right) \mbox{ for } p \in A, $$  or, for $p \in B$, $x &lt; 1/\left(p + \sqrt{2}\right)$ or $x &gt; 1/\left(p - \sqrt{2}\right)$.  Notice that there are MANY solutions to these equations! The easiest to see, perhaps, is letting $x = 1/(p + n)$ for $n \geq 2$.  Notice that Rudin chooses $n = 2$ for his answer, but it checks out easily for other $n$.  The Question:  Why does Rudin choose $x = 1/(p + 2)$ specifically?  Is it just to make the expressions work out clearly algebraically?  Why doesn't he comment on his particular choice or the nature of the set of solutions that will work for the proof?  Is there a simpler derivation for the number $q$ that I am missing? How to show that $f'(x)<2f(x)$ This is question B4 in 1999 Putnam Exam. I would appreciate if somebody could help me with this.      Let $f(x),f'(x),f''(x),f'''(x)&gt;0$ , $f'''(x)$ is a continuous function and $f'''(x)&lt;f(x)$ on $\mathbb{R}$. Then show that   $$f'(x)&lt;2f(x),~ \forall x\in \mathbb{R}.$$ Is is true &lt;h1&gt; tag should not have more than 1 on every page for SEO?   Every pages on the header, it includes and &lt;h1&gt; tag. Website Name which show an image from logo class, like this:  &lt;div id="header"&gt;   &lt;h1 class="logo"&gt;      &lt;a href="#"&gt;Website Name&lt;/a&gt;   &lt;/h1&gt; &lt;/div&gt;   and I also include &lt;h1&gt; on the contact page, help, etc and article page for example:  &lt;h1&gt;Contact Us&lt;/h1&gt; &lt;h1&gt;Name of the Article Title here&lt;/h1&gt;   On the homepage, it just has one &lt;h1&gt; for the logo Site Name... other pages have 2 &lt;h1&gt; tags Sometimes after I fry something, I wash it immediately in water because it's a bit easier to wash (before the remains stick to the pan).  Does this damage the pan in any way? We have $q, p$ prime and $tq = (p-1)(p+1)$ for some positive integer $t$. The book I'm reading says that because $q$ is prime, it divides $p-1$ or $p+1$.   Question: Why exactly does it matter that $q$ is prime here? Google Analytics: How can I track between http:// and https:// profiles for a domain? is it possible to combine two fields in one and use that field to group by in reports? for example, i would like to combine ee id and ee name fields in 1 and show as 1 field in the "group by" section on reports. Consecutively Adding Sines One thing, I'm not a mathematician so please be patient. I am still in Algebra II Trig. Leading with that, why does  $$ x_0 = \sin 1, \space  x_1 = x_0 + \sin x_0, \space x_2 = x_1 + \sin x_1 ... $$ and after a while, $$ x = \pi $$  I know this to be true because I have evaluated this on my TI-84 and more deeply evaluated it with this program I made:   To $100000$ Iterations (which I believe is enough)   So what's the math behind it? Why does adding $\sin 1$ in this manner produce $\pi$? Why doesn't doing the same thing using $100$ produce $\pi$? For example,  $$\sin(1) = 0.841470...,\space \sin(1) + \sin(0.8414...), \sin(1) + \sin(0.841470...) + \sin(1.587095126...),\space ... $$ $$ = \pi$$  My question is different from the mentioned because my function is not taking the sin of a sin consecutively, but is this following function: $f(x) = x + \sin x$ and is not $f(x) = \sin x$ and therefore can exhibit different properties I need explained. Robot tank run amok, its trainer trapped in cave on the Moon Another dim memory from my sci-fi reading binges in the 1980s... this one almost definitely in one of the big hardback collections full of late Golden Age short stories.  The story is told from the perspective of a robot (maybe cyborg?) tank that has stopped responding to control signals and is aggressively "defending" some portion of the Moon, to which it was previously assigned. Maybe a war had ended, or maybe it became confused about an exercise that it thought was real?  There was a person trapped in a cave trying to communicate with the tank. The tank mostly ignores him, so the main purpose of what he says is exposition for the reader. I think this person was an expert on this type of robot/cyborg sent to "talk it down", but the tank had attacked, and this guy was the only survivor. The expert was badly wounded -- I think he mentions his suit had to amputate his leg to seal a hole -- or maybe flood it with water which would freeze to seal the hole but mean that his leg had to be amputated later. He keeps trying to convince the tank to stand down, at one point saying that he knew it "back when it was just a two-wheeler an autocyber in boot camp" or something along that line. (Edit: I must have been reading too fast and/or distracted back then, since per the quote below I seem to have misread "motorcycle" for "autocyber"!)  One detail of the attack stands out -- the tank was low on ammunition, I think, so it used a kind of bomb with an electric/magnetic "catapult" (aka "magnapult") to hit the expert's transport. The tank was proud of itself for figuring out a way to hit it far out of the normal range by 1) using its own top speed to add velocity to the projectile, and 2) building up the catapult's energy past its normal operating range, which caused it pain but made the bomb go much further. It catches the transport by surprise, since they thought they were safely out of range of the armaments it had left.  The premise sounds a lot like a Keith Laumer "Bolo" story, but I'm fairly sure that it's not actually one of them. It had a somewhat grimmer feel to it. I could be wrong, though, since it's been 30 years. Merge around a face So, I'm working on a control station that has buttons. It is a one-mesh creation, that means, I don't want to create buttons and things like that separated from the main mesh, but as you can see, I have problems. I would like to make the highlighted parts to be one with the main part around them. I've tried everything, and I have this problem on multiple creations. Any idea?  Update: I've added another example of the problem. I want them to be one face, instead of two. (2nd picture) How to add startup applications in lubuntu? Effect of switching response and explanatory variable in simple linear regression Let's say that there exists some "true" relationship between $y$ and $x$ such that $y = ax + b + \epsilon$, where $a$ and $b$ are constants and $\epsilon$ is i.i.d normal noise. When I randomly generate data from that R code: x &lt;- 1:100; y &lt;- ax + b + rnorm(length(x)) and then fit a model like y ~ x, I obviously get reasonably good estimates for $a$ and $b$.  If I switch the role of the variables as in (x ~ y), however, and then rewrite the result for $y$ to be a function of $x$, the resulting slope is always steeper (either more negative or more positive) than that estimated by the y ~ x regression. I'm trying to understand exactly why that is and would appreciate it if anyone could give me an intuition as to what's going on there. I was given a laptop with a totally dead display. I do not want to pay to replace the broken display, but I would like to use the laptop as a server and connect to it via SSH to control it.  I have pulled out the internal hard drive from the broken laptop, and I want to install Debian GNU/Linux and the SSH server onto this HDD using a working Debian laptop.  However, the hardware in the two laptops is very different, and I am wondering what kind of problems I would run into when I plugged the drive back in to the laptop with the broken display because of this?  How would I make this work? What is the proper way for me to install a bootable copy of Linux onto a HDD that will be used by a computer with different hardware? The SourceForge signon page says I can associate an OpenID with my existing SourceForge account:    I enter my currently-signed-in SourceForge account password I select the custom OpenID URL and enter https://openid.stackexchange.com I'm redirected to https://openid.stackexchange.com/account/login?&lt;session key bits&gt; I enter my openid.stackexchange.com login information there I get the error message:     Cannot Complete Login      Detected an attempt to send an assertion when the identifier () is not owned by the logged in user.    I don't understand what that means. I created this style that shifts the labels and put them on the bottom of the ybars.  However, I have a problem with the bars that have too short length (for example, see the 8.7 and 5.1 in the example below). I want to check if the values are less than some threshold, or the length of the bar is shorter than the label (number) then change its color.  How can I overlay another label on top of the one in white when the bar is shorter? or maybe change the color of the label to a gradient color? or just overlay the part that is outside of the bar?    \documentclass{standalone}  \usepackage{tikz} \usepackage{pgfplots} \usepackage{pgfplotstable} \usetikzlibrary{calc}  \begin{document} \makeatletter \begin{tikzpicture} \begin{axis}[ ybar, every node near coord/.style={   check for zero/.code={ % If meta=0, make the node a coordinate (which doesn't have text)     \pgfmathfloatifflags{\pgfplotspointmeta}{0}{         \pgfkeys{/tikz/coordinate}     }{}   },   check for zero, color=white, text opacity=1, font=\footnotesize, inner ysep=0.5pt, },% % Bottom nodes  calculate full offset/.code={     \pgfkeys{/pgf/fpu=true,/pgf/fpu/output format=fixed}     \pgfmathsetmacro\testmacro{((\pgfplotspointmeta-\pgfkeysvalueof{/pgfplots/ymin})*10^\pgfplots@data@scale@trafo@EXPONENT@y)*\pgfplots@y@veclength}     \pgfkeys{/pgf/fpu=false} },% nodes near coords bottom/.style={     every node near coord/.append style={         /pgfplots/calculate full offset,         yshift=-\testmacro,         rotate=90, anchor=west,%I need the rotate text here and not in the          %general style, as the ybar redefines the style     } },% ymin=0, nodes near coords={\pgfmathprintnumber[fixed zerofill,precision=1]{\pgfplotspointmeta}}, nodes near coords bottom, ] \pgfplotstableread{ % set real imag mag 1 8.7 2 5.1 3 91.8367  }\mydata \addplot table[x index=0,y index=1,header=false] {\mydata}; \end{axis} \end{tikzpicture} \makeatother \end{document}   * EDIT *  I need to create some special gradient on each label that is below the bar height. For example, I need to adapt something like  to each label that is not showing. However, I cannot find a way to put the macro in the style section of the codes. Or is there any equivalent solution? Given the following components  DECLARE @D DATE = '2013-10-13' DECLARE @T TIME(7) = '23:59:59.9999999'   What is the best way of combining them to produce a DATETIME2(7) result with value '2013-10-13 23:59:59.9999999'?  Some things which don't work are listed below.    SELECT @D + @T       Operand data type date is invalid for add operator.     SELECT CAST(@D AS DATETIME2(7)) + @T       Operand data type datetime2 is invalid for add operator.     SELECT DATEADD(NANOSECOND,DATEDIFF(NANOSECOND,CAST('00:00:00.0000000' AS TIME),@T),@D)      The datediff function resulted in an overflow. The number of dateparts   separating two date/time instances is too large. Try to use datediff   with a less precise datepart.   * The overflow can be avoided in Azure SQL Database and SQL Server 2016, using .    SELECT CAST(@D AS DATETIME) + @T       The data types datetime and time are incompatible in the add operator.     SELECT CAST(@D AS DATETIME) + CAST(@T AS DATETIME)      Returns a result but loses precision 2013-10-13 23:59:59.997 In general, when can I omit that from a sentence?   Can I omit that (emphasized) in sentence (2) below?        We say that such algorithms handle concept drift and can learn from time-changing data streams.   When the statistical properties of the target variable change, we say that concept drift has occurred.   She told me that some of her classmates failed.      To me, sentences (1) and (3) sound better without that. If I omit that from (2), it just doesn't sound right when I read it out loud. (I should point out I'm not a native speaker.)  How do you decide when to omit it? Is it just a matter of taste? (I know that is sometimes mandatory.)  Can you think of sentences that "sound better" with that? Can you think of sentences that "sound better" without that? Can you think of a few sentences where it is wrong to omit that?