Prove that if $\gcd(a, b) = 1$ then $\gcd(2a + b, a + 2b) \in \{1, 3\}$   I can get up to: $\gcd(2a + b, a + 2b) | 3(a + b)$  But I am unable to move forwards. I would like a hint?  BY Bezout's, there exists integers $x, y$ such that $ax + by = 1$.   $d = \gcd(2a + b, a + 2b) \implies (2a + b)u + (a + 2b)t = d$ for integers existing $u, t$.  But this isn't helping much. What can I do?
If $a$ and $b$ are relatively prime, prove that $a + 2b$ and $2a+b$ are also relatively prime or have a $\gcd$ of $3$.  I'm very new to number theory so please don't assume I am familiar with some of the terminology.
I know the so-called textbook definition of unit tests and integration tests.  What I am curious about is when it is time to write unit tests... I will write them to cover as many sets of classes as possible.  For example, if I have a Word class, I will write some unit tests for the Word class.  Then, I begin writing my Sentence class, and when it needs to interact with the Word class, I will often write my unit tests such that they test both Sentence and Word... at least in the places where they interact.  Have these tests essentially become integration tests because they now test the integration of these 2 classes, or is it just a unit test that spans 2 classes?  In general, because of this uncertain line, I will rarely actually write integration tests... or is my using the finished product to see if all the pieces work properly the actual integration tests, even though they are manual and rarely repeated beyond the scope of each individual feature?  Am I misunderstanding integration tests, or is there really just very little difference between integration and unit tests? How does Hisoka see Gon exactly? As a rival or as a toy to entertain himself? Hisoka hinted that Gon is his treasure and also his prey, but it can't be both, right? Because in the end, he didn't want Gon to die no matter what.  So it's weird that he didn't want Gon to die but also made him his prey to fight till death. Is it sexual for Hisoka or what? I still remember the 'Schwing' thing. Colored borders around windows after update When two phrases are pronounced alike but have different spelling and meaning, can we call them homophones?  e.g. "ice-cream" and "I scream", "nitrate" and "night rate", "that's tough" and "that stuff".  Or is there another term for them?  What linguistic phenomenon distiguishes these near homophones?  I've checked the putative duplicate at  but it isn't exactly what I'm asking here. Equivalent of "both" when referring to three or more items? I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$.  If $x\in V$, then since $V$ is a manifold, there is a local parametrization $\phi: U\to V$ where $U$ is open in $\mathbb{R}^k$. Without loss of generality, we can require $\phi(0)=x$. Then $T_x(V)$ is defined to be the image of $d\phi_0$ on $\mathbb{R}^k$.   An arbitrary element of $T_x(V)$ looks like $$ d\phi_0(v)=\lim_{t\to 0}\frac{\phi(0+tv)-\phi(0)}{t}=\lim_{t\to 0}\frac{\phi(tv)-x}{t} $$  but this doesn't seem very useful to show $T_x(V)=V$. What is the right approach? Is it possible to produce a context free grammar for Tikz path? node? Even further, I would be happy to see a definition of the tikz language in the usual manner programming languages such as Java and Pascal are defined.   The reason I ask is that I keep getting mysterious error messages for the tiniest typos. Essentially, incorrect use of the underlying languages leads to a "crash" of the compiler. This crash sometimes gives useful error messages, but sometimes they are hard to decipher. Preseed Ubuntu 16.04 not working (preseed file not found) I've been trying to preseed my installation for 2 days now. I'm quite new in this topic, so execuse me if I'm asking noob questions. How do I use boot parameters, how do I know that my preseed file works. When is my preseed file loaded Wayne Dalton Garage So I recent had a Garage door "professional" come out to my house and look at my garage. I originally thought I was having a motor problem because it would only lift door sporadically. After performing a 25 point inspection he believed he located the problem, which was the spring system from my Wayne Dalton Garage (torquemaster). He told me how horrible of a system they are and they don't fix them because of that they only replace the entire system and would cost me $700 or more. So now I'm left trying to figure out what I need, torquemaster springs, bracket assembly, both or if I need something completely else. I need some experience Wayne Dalton experience. Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$ Suppose that $S_0$ and A are positive numbers, let $$S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right)$$ with $n \geq 0 $.      (a)Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$       (b)Show that $S_{n+1} \leq S_n $ , if $n \geq 1$      (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists      (d) find s   (a) Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$  Given $$P_n:  S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \geq \sqrt{A}$$  $$P_0: S_{1} = \frac{1}{2}\left(S_0 +\frac{A}{S_0}\right) \geq \sqrt{A}  $$  We assume that $P_n$ is true  $$P_{n+1}: S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right)$$ by assumption $$S_{n+2}= \frac{1}{2}\left(S_{n+1}\left(1 +\frac{A}{(S_{n+1})^2}\right)\right) \geq \frac{1}{2}\left(\sqrt{A}\left(1 +\frac{A}{(\sqrt{A})^2}\right)\right)$$ $$ S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right) \geq \sqrt{A}  $$  It follows that $S_{n+1} \geq \sqrt{A} $    (b) Show that $S_{n+1} \leq S_n $ , if $n \geq 1$  $$S_{n+1} \leq S_n$$ $$\frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \leq S_n $$ Dividing by $S_n$ $$\frac{1}{2}\left(1 +\frac{A}{S_n^2}\right) \leq 1 $$ $$\frac{A}{2S_n^2} \leq \frac{1}{2}$$ $$A \leq S_n^2$$ $$S_n \geq \sqrt{A}$$ As  $S_{n+1} \leq S_n$ yields a true statement, it follows $S_{n+1} \leq S_n$ is true.    (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists  Since $S_{n+1} \leq S_n$, the sequence is non-increasing,   using the non-increasing theorem stating that      if $\{S_n\}$ is non-increasing then $$\lim\limits_{n \rightarrow &gt; \infty} S_n = \inf\{S_n\} $$     (d) find s    Is the argumentation in (a) and (b) appropriate? Also, I have to admit I m getting less confident in my argumentation (c) and (d). How to proceed in (c) and (d)?  Much appreciated for your input or help. Wouldn't the thermodynamic cost of creating alternate universes make the Many Worlds interpretation implausible? What voltage is a 3-LED light from a LED-TV? I have some "NDV rev1.1 4248 back lights" from a television.   There is a print that sais "94V" on the strip, and on the wire supplying the strips it said "96V".   Every strip has 9x buttons which have 3 LED's in them. So i devided 94 by 9 and figured that every button should have 10.4V roughly, which makes sense, because a white LED is normally 3.5 volts.  I have a 12v Lithium battery which measured 11.1V so i tested one of the LED's with it and it lasted less than one second.   Are the LED's are rated for 10.4V roughly and not more? Why does a translation matrix need extra row? PHP: Static and non Static functions and Objects I recently installed the apache2 package using the Software Centre, I want to create an alias apacheres which will replace the more cumbersome command /etc/init.d/apache2 restart.  I opened the .bashrc file in my home directory, and it contains the following lines:  # Alias definitions. # You may want to put all your additions into a separate file like # ~/.bash_aliases, instead of adding them here directly. # See /usr/share/doc/bash-doc/examples in the bash-doc package.  if [ -f ~/.bash_aliases ]; then     . ~/.bash_aliases fi   So I created a .bash_aliases file in my home directory, and added only this line:  alias apacheres ='/etc/init.d/apache2 restart'  When I started my terminal, I get this error message:  bash: alias: apacheres: not found bash: alias: =/etc/init.d/apache2 restart: not found   The shell has obviously read the file, otherwise it wouldn't know of its existence. I have checked the directory path and I can find the file apache2 there. What do I need to do to make this alias work? I have noticed that some questions get a lot of responses quickly, and the cool fact is that most of the time these initial responses are correct.  I see users are motivated to accept answers. Because many questions take less than 24h to get an acceptable response, I'm unsure what to do when I have a question that takes more than 24h to get a response that I feel is complete enough to warrant being marked as accepted.  And even for acceptable responses, I often hesitate to accept them out of concern that posters will pass my question by if it already has an accepted answer (unless it has lots of upvotes). Does this really deter the higher-quality posters, though? How can I make a camera the active one? In Oracle, I can re-create a view with a single statement, as shown here:  CREATE OR REPLACE VIEW MY_VIEW AS SELECT SOME_FIELD FROM SOME_TABLE WHERE SOME_CONDITIONS   As the syntax implies, this will drop the old view and re-create it with whatever definition I've given.  Is there an equivalent in MSSQL (SQL Server 2005 or later) that will do the same thing? I think I accidentally pressed something to make it like this:    See how the XYZ arrows are at the 3D cursor instead of those vertexes I've selected?  How do I fix that? Whenever I select on a vertex or something, those blue, red, green arrows would always be where I selected, now it just keeps going to the 3D cursor. Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact.  This has been asked before, but all the answers I have seen prove the contrapositive. Realistically, this may be the way to go, but is there way to exhibit an unbounded continuous function (under the assumption $X$ is not compact) without appealing to results beyond introductory real analysis (e.g., the solution I've seen involves the Tietze Extension theorem)?  Because we're working with metric spaces, it's clear that the assumption that $X$ is noncompact (towards proving contrapositive) will lead us to extract a sequence of points in the space with no convergent subsequences. But how can we infer the existence of an unbounded continuous on the space knowing only about some sequence of points in this space (without using anything too advanced)?