Given that  $$a\cdot b=gcd(a,b)\cdot lcm(a,b)$$  How can we find all the integer solutions $(a,b)$ if $gcd(a,b)=6$ and $lcm(a,b)=540$?  The first thing I did was factorizing using the fundamental theorem of arithmetic.  $$a\cdot b=gcd(a,b)\cdot lcm(a,b)=6\cdot540=2^3\cdot3^4\cdot5$$  I also know that  $$gcd(a,b)=p_1^{min(a_1,b_1)}\cdot p_2^{min(a_2,b_2)}\dots p_n^{min(a_n,b_n)}$$  $$lcm(a,b)=p_1^{max(a_1,b_1)}\cdot p_2^{max(a_2,b_2)}\dots p_n^{max(a_n,b_n)}$$ where $a_1\dots a_n$ and $b_1\dots b_n$ are the exponents in the factorization of $a$ and $b$ respectively.  From there I am not sure where to go to obtain all the possible integers $a$ and $b$.  There is already a similar question on Mathematics SE, however the accepted solution is simply a hint towards the correct procedure to do it, while this question already acknowledged that hint.
These two exercises I encountered recently seem to develop some type of connection between GCD and LCM I can't quite figure out. Exercise 1:  Find all the numbers $x$ and $y$ such that: $a) \ GCD(x,y)=15, \ LCM(x,y)=150$ $b) \ GCD(x,y)=120 \ LCM(x,y)=1320$ $c) \ GCD(x,y)=100 \ LCM(x,y)=990$  Exercise 2:   Find all the numbers $m,n$ such that $GCD(m,n)=pq , \ LCM(m,n)=p^2qs$  where $p,q,s$ are prime  The first thing that is known to me is that $GCD(x,y) \cdot LCM(x,y)= x \cdot y$ Also $LCM(x,y)$ is at most $x \cdot y$ while $GCD(x,y)$ is at most $\max \{x,y\}$. Last thing is that $GCD(x,y)|LCM(x,y)$. Using all this I tried to solve the first exercise: $a)$ First two obvious pairs are $x=15, y=150$ and $y=15, x=150$. Now neither of the numbers can be bigger than $150$ or smaller than $15$. So we are looking for numbers in the range $15-150$ that satisfy $x \cdot y = 15 \cdot 150$ Another such pair is $(x,y)=(30,75), \ (x,y)=(75,30)$. Similarly for $b)$ we find that the only possible values are permutations of the set {$120,1320$} and in $c)$ since $100$ does not divide $990$ no such numbers exist. Now exercise 2 is what made me think there is actually another connection I'm not quite aware of since now it's about arbitrary prime numbers and the previous method doesn't work anymore. My intuition is that it has something to do with $GCD$ or $LCM$ of the $GCD(x,y), \ LCM(x,y)$
Reason behind this self invoking anonymous function variant Find the sum of finite series $S=\sum_{k=1}^{2015}{(-1)^{\frac{k(k+1)}{2}}}k$ I have a project that needs a different voltage (or multiple voltages) than what I have available. Sometimes I need DC instead of AC, or I need a lower or higher voltage. How do I convert from one to another? Should I use resistors or a regulator or something else?  I've heard of converters, transformers, regulators, inverters, step-down, step-up, buck, boost, switching, linear, and other terms. Or acronyms (initialisms) like PMIC and LDO. Can someone shed some light on all this? I cant use lhopital's rule here is my attempt:  Limit of $(\frac{n^{2}-2n+1}{n^{2}-4n+2})^{n}$ =$e^{\lim_{n\to \infty} n\times \ln(\frac{1 -2-n +\frac{1}{n^{2}}}{1 -\frac{4}{n} +\frac{2}{n^{2}}})}$ I got stuck here Poisson Distribution of sum of two random independent variables $X$, $Y$ Need Google Geocoding to return City name in English Find a non-negative function on [0,1] such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable Problem:  Find a non-negative function $f$ on $[0,1]$ such that $$\lim_{t\to\infty} t\cdot m(\{x : f(x) \geq t\}) = 0,$$ but $f$ is not integrable, where $m$ is Lebesgue measure.  My Attempt:  Let $f(x) = \frac{\chi_{(0,1]}}{\sqrt{x}}$. Then,  \begin{align*} \lim_{t\to\infty} t\cdot m(\{x \in [0,1]: f(x) \geq t\}) &amp;= \lim_{t\to\infty} t \cdot m(\{x\in (0,1]: 1/\sqrt{x} \geq t\})\\ &amp;=\lim_{t\to\infty} t \cdot m(\{x\in (0,1]: x \leq (1/t^2)\})\\ &amp;= \lim_{t\to\infty} t \cdot m((0, (1/t^2)))\\ &amp;= \lim_{t\to\infty} \frac{t}{t^2} = 0. \end{align*}  However, $f(x)$ is integrable over this interval. I have tried functions looking like $f(x) = 1/x^p$ but I cannot find any that will work here. There is a hint that says there is a monotonic function that fits this description.    Also, does anyone know of a list of non-Lebesgue integrable functions on $[0,1]$? I feel as though I could use this for many counterexamples if one were to exist. Thanks! I need to label the four corners of a dataframe as shown in the image. Can this be done with dynamic text scripts? How to get a list of the filenames of the figures embedded in a document I have a TeX document that is embedding a huge amount of figures from a fig folder that is full of another humongous amount of other figures.  I would like to move the document along with its figures, but I don't want to take the whole fig folder, just the figures in there that are embedded by the document.  Is there a package (or any other way) that allows me to, for example, print out the filenames of all the figures embedded by the document? Something like:  fig/figurename1.eps fig/figurename2.eps ... etc   I could then just copy-paste that list and add some cp's in a plain text file and automate the copy of those files to my desired location by running the file in my shell:  $ cp fig/figurename1.eps wherever/fig/ $ cp fig/figurename2.eps wherever/fig/ ... etc   Of course this is just an idea, if somebody has an alternative to the "file listing and copy-pasting in a shell file" I would be happy to listen =) Install Ubuntu, you can not continue How do you performance test JavaScript code? Can you launch Remote Desktop via URL? What is the meaning of the "expansion of space"? When we say that "the space between galaxies is expanding," what do we really mean? For instance, if I think of space as being a Cartesian grid, then when space expands should I think of it as adding more grid-points or as making the distance between the grid-points larger? Or is this a flawed picture of space-time even when far from significant mass densities?  It seems to me that it must be the case that we are "adding more grid-points" because otherwise we would not observe the expansion since anything that occupied that part of the grid (e.g. light, my hand, etc.) would also expand by a corresponding amount (otherwise, after a sufficient amount of expansion, we would be able to observe sub-atomic scale processes as being macroscopic in scale). But if it is the case that we are adding more grid-points, then the "expansion" of space seems like a misnomer: shouldn't it be called the "creation" of space?  That being said, either case (adding grid-points or stretching distances) could conceivably be the same process depending on the nature of space-time (i.e. if it were discrete but dense - like the Rationals - rather than either strictly discrete (Integers) or strictly continuous (Reals)... I hope that analogy makes sense).   So what is really going on in intergalactic space? What is the meaning of its expansion? Does it correspond to creating more space, stretching the space that's already there, or something more subtle? Share menu hangs iOS app The Share menu in iOS 13 got a redesign, and the Stack Exchange iOS app is no longer compatible with it. I can still use it to copy the link, but when I try close the menu, it remains stuck. Opening another app and then returning automatically dismisses the menu, but then I can't use the back button anymore to return to the question list. (I'm still able to refresh the page, so it's a navigation issue.)    Ironically, I discovered this bug while trying to mark  as a duplicate; in the app, the quickest way to do that is to use the Share menu to copy the link to the original question to the clipboard...   App Version: 1.6.6.2 Device: iPhone 7 Plus OS Version: Version 13.1 (Build 17A844) I have a finite field $\mathbb{F}_{27}$. I need to find an element of order 13. I know that multiplicative group of this field is cyclic with order 26. So I want to find a generator $g$ of this cyclic group, then $g^2$ will have order 13. Also I know that all elements of this field can be represented by polynomials from $\mathbb{F}_3[t]$, where t is root of irreducible polynom $x^3-x-1$.  Probably we can find generator of $\mathbb{Z}_{26}$ and than build an isomorphism to $\mathbb{F}_{27}\setminus 0$.  Anyway I don't know were to start.  Thanks! How to change file extension at runtime in Java I would like to have multiple bibliographies (reference lists) as described in   ().   However, I'd like to use with a single .tex file, which would contain the bibliography database embedded, as in  ...   Now, I want to use letter documentclass, for which the bibliography needs to be defined ()   So what I end up is with this example (code below, compiled with just pdflatex, no bibtex(which I'd like to avoid in this case)):     ... obviously, something works (multibib recognized primary and secondary cites, assigning them number "1" as start) -- however I don't get two, separate primary and secondary list of references; would that be possible in this context? (doesn't have to be with multibib, but I'd like to avoid a separate .bib file and a bibtex call)  EDIT: I'd also like separate cite keys (i.e. rendered as [a1] and [b1]; not yet sure if multibib can do that; edit: no it cant,  says "You can also edit the .bbl files created by BibTEX and add 'a', 'b',. . . to the labels, but your changes will be overwritten by subsequent BibTEX runs." and recommends \cite[a]{key}) - how possible would that be?  &nbsp;  The code is this:   \documentclass{letter}  % Environment thebibliography undefined when using letter: http://www.latex-community.org/forum/viewtopic.php?f=4&amp;t=3359 \makeatletter \setlength \labelsep {.5em} \newcommand\newblock{\hskip   .11em\@plus.33em\@minus.07em} \let\@openbib@code\@empty \newcommand\refname{References} \newcommand\section{\@startsection   {section}{1}{\z@}%   {-3.5ex \@plus -1ex \@minus -.2ex}%   {2.3ex \@plus.2ex}%   {\normalfont\Large\bfseries}} \newenvironment{thebibliography}[1] {\section*{\refname}%   \@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}%   \list{\@biblabel{\@arabic\c@enumiv}}%   {\settowidth\labelwidth{\@biblabel{#1}}%     \leftmargin\labelwidth \advance\leftmargin\labelsep \@openbib@code     \usecounter{enumiv}%     \let\p@enumiv\@empty \renewcommand\theenumiv{\@arabic\c@enumiv}}%   \sloppy \clubpenalty4000 \@clubpenalty \clubpenalty   \widowpenalty4000%   \sfcode`\.\@m} {\def\@noitemerr {\@latex@warning{Empty       `thebibliography' environment}}%       \endlist} \makeatother  \usepackage{multibib} \newcites{pri}{Primary sources} \newcites{sec}{Secondary sources}  \begin{document}   \begin{letter}{Addressee} Blah, blah - content...  Primary cite: \citepri{lamport94} ...  Secondary cite: \citesec{lamport94} ...   \begin{thebibliography}{9}  \bibitem{lamport94}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1994.  \bibitem{lamport95}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1995.  \end{thebibliography}  \end{letter}   \end{document} Interpretation of log transformed predictor and/or response I'm wondering if it makes a difference in interpretation whether only the dependent, both the dependent and independent, or only the independent variables are log transformed.  Consider the case of  log(DV) = Intercept + B1*IV + Error    I can interpret the IV as the percent increase but how does this change when I have   log(DV) = Intercept + B1*log(IV) + Error   or when I have   DV = Intercept + B1*log(IV) + Error   ? I've seen references to curried functions in several articles and blogs but I can't find a good explanation (or at least one that makes sense!) What do you call someone who pretends to be your friend but is actually your enemy?  A friend suggested spy for me, but that does not nearly describe the word I need for an English project. The character is very good at manipulating how people see him. He's a good liar. And pretends to be your friend but actually is your enemy.