Let $$P=\prod_{k=1}^{2019} \operatorname{lcm}\left(1,2,\dots,\left\lfloor\frac{2019}k\right\rfloor\right).$$ Given that $P=n!$, find $n$.   This is the question provided, I do not really know how to approach this.
I try to prove the following formula:  $$n!=\prod_{k=1}^n \operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)$$  I noticed that  $\upsilon_{p}(\operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)) = s$ iff $\lfloor n/(p^{s+1}) \rfloor &lt; k \leq \lfloor n/(p^{s}) \rfloor $
If an effect would prevent a spell from doing damage, does it continue to prevent that damage after the spell changes zones? Anthony casts , targeting Nathan's .  Nathan casts and resolves , choosing to 1) prevent all damage that would be dealt by Searing Blood this turn and 2) put a +1/+1 counter on Elvish Mystic.  Searing Blood resolves, deals no damage, and goes to the graveyard.  Anthony casts and resolves  to kill Elvish Mystic.  Searing Blood triggers from the graveyard, because Elvish Mystic died. Does the prevention effect created by Dromoka's command still apply to it after it changes zones? In other words, will Searing Blood deal 3 damage to Nathan? Is there a word with phobia suffix for fear of waking others up? I usually come home after basketball at like 12:30 a.m. and at this time my mom and sister usually sleep (my Dad though is usually up). When I come home, I don't go to sleep right away and will sometimes eat or do something else. When I do so, I do it in stealth, so that I don't wake up anyone. I do out of two things, to be polite as I do not want to wake anybody up and also the fact that if I make a loud noise and wake my mom up, my mom will get mad and lecture me. Does this count as a phobia? In addition, if it does what would the name of it be. Table of Contents for set of PDF files How to make Table of Contents for a set of PDF files?  Here's my code!  \documentclass{book} \usepackage{pdfpages} \usepackage[pdfauthor={&lt;somename&gt;},pdftitle={&lt;sometitle&gt;},pdftex]{hyperref} \usepackage{fancyhdr} \usepackage{tocloft} \usepackage[margin=0.5in,bottom=0.75in,top=0.75in]{geometry}   \pagestyle{fancy} \renewcommand{\headrulewidth}{0.4pt} \fancyhead[L]{X Y Z} \fancyhead[R]{} \begin{document}  \tableofcontents \newpage  \clearpage \phantomsection \includepdfset{pages=-,pagecommand=\thispagestyle{fancy}} \addcontentsline{toc}{chapter}{SAMPLE PDF 1\\ \textit{description}} \includepdf[pages=-]{template1.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 2\\ \textit{description}} \includepdf[pages=-]{template2.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 3\\ \textit{description}} \includepdf[pages=-]{template3.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 4\\ \textit{description}} \includepdf[pages=-]{template4.pdf}  \clearpage \phantomsection \addcontentsline{toc}{chapter}{SAMPLE PDF 5\\ \textit{description}} \includepdf[pages=-]{template5.pdf}  \end{document}   Here's the output.    My problem is that the page number is not aligned with the title, it is aligned with the description. Also I'd like to put dots.   My desired output looks like this: Plurality of verb depending on plurality of list elements This question is also open on EL&amp;U:   Considering the amount of controversy it aroused, I believe it's past "Learners'" level.    An edit was suggested to my sentence.     There was were an orange, some grapes, two apples and a small pile of cherries on the plate.   In my native language plurality of the verb always follows plurality of the first element on the list. There were an orange,... sounds awkward to me, no matter what follows up. My simple solution was reordering:     There were some grapes, an orange, two apples and a small pile of cherries on the plate.   But that's not the first time I faced this situation and I'd like to know what the rules of grammar say about that — was my editor overzealous or am I trying to copy rules of my language that don't apply in English? Let $l \in\mathbb{R} $ and $ f:(0,\infty) \to \Bbb R $ be a function such that $ \lim_{x\to \infty} xf(x)=l $ .  Prove that  $ \lim_{x\to \infty} f(x)=0 $.  Any help would be appreciated - I found this difficult to prove Is there any way to specify a suggested filename when using data: URI? If for example you follow the link:  data:application/octet-stream;base64,SGVsbG8=  The browser will prompt you to download a file consisting of the data held as base64 in the hyperlink itself.  Is there any way of suggesting a default name in the markup?  If not, is there a JavaScript solution? DML is not allowed in a constructor. Is there a known reason behind this limit?  I have been searching for the answer &amp; haven't found anything which addresses the question. May be it's not documented anywhere(Or google doesn't know where it is). Objective-C: Use singleton vs. use class as an object? I want to know if my server is running Subversion 1.5.  How can I find that out?  Also would be nice to know my SVN client version number. svn help hasn't been helpful.  Note: I don't want my project's revision number, etc. This question is about the Subversion software itself. Proving the given quadratic form is chi-squared $k$ Suppose $\underline{X}$ is an $m$-dimensional vector following multivariate Normal distribution i.e. $\underline{X}$~$N_m(\underline{\mu},\Sigma)$ where $\Sigma$ is positive definite. Let $B$ be a symmetric $m$ by $m$ matrix, having the following properties:  $B\Sigma B=B$  $\underline{\mu}'B\underline{\mu}=0$  $trace(B\Sigma)=k$, where $k&lt;m$   Prove that $\underline{X}'B\underline{X}$ follows chi-squared ($k$).  I proceeded a lot but got stuck in one last step where I think linear algebra is troubling me. Kindly have  a look at my &quot;solution&quot;.  The second condition can be written as $\underline{\mu}'B\Sigma B\underline{\mu}=0\implies B\underline{\mu}=\underline{0}$ as $\Sigma$ is Positive definite. Then after a little algebra and considering condition 2, we can write $\underline{X}'B\underline{X}=(\underline{X}-\underline{\mu})'B(\underline{X}-\underline{\mu})=(\underline{X}-\underline{\mu})'B\Sigma B(\underline{X}-\underline{\mu})$ But $\Sigma$ is positive definite hence we can have a symmetric matrix $\sqrt{\Sigma}$ defined such that $\sqrt{\Sigma}\sqrt{\Sigma}=\Sigma$. Using this, we can write $(\underline{X}-\underline{\mu})'B\Sigma B(\underline{X}-\underline{\mu})=(\sqrt{\Sigma}B(\underline{X}-\underline{\mu}))'(\sqrt{\Sigma}B(\underline{X}-\underline{\mu}))$ We want to show that this follows chi-squared ($k$). Now, it is obvious that as $(\underline{X}-\underline{\mu})$~$N(\underline{0},\Sigma)$, $\sqrt{\Sigma}B(\underline{X}-\underline{\mu})$~$N(\underline{0},\sqrt{\Sigma}B\Sigma B\sqrt{\Sigma})=N(\underline{0},\sqrt{\Sigma}B\sqrt{\Sigma})$ Now, $trace(\sqrt{\Sigma}B\sqrt{\Sigma})=trace(B\Sigma)=k$ and $\sqrt{\Sigma}B\sqrt{\Sigma}$ is idempotent (verified). So its eigenvalues are only $0$ or $1$. Since its trace is $k$, there are $k$ eigenvalues with value $1$ and other $m-k$ eigenvalues are $0$. Now here is the thing I want to prove: I want to show that$\sqrt{\Sigma}B\sqrt{\Sigma}$, which is the covariance matrix, is diagonal and has only $k$ $1$'s and remaining $m-k$ diagonal entries $0$. The reason is, then I will be able to say that $k$ components of $\sqrt{\Sigma}B(\underline{X}-\underline{\mu})$ are independent standard normals whose sum of squares is our quadratic form and thus it follows chi-squared ($k$). But I really could not show that $\sqrt{\Sigma}B\sqrt{\Sigma}$ has the desired property i.e. it is diagonal with rank $k$. But this is crucial, it is the finale. If this does not happen, then some components will be correlated with each other and we cannot say that our quadratic form has chi-squared ($k$). I would really appreciate if someone helped me to prove this final part. As I understand it, the hosts file is one of several system facilities that assists in addressing network nodes in a computer network. But what should be inside it? When I install Ubuntu by default   127.0.0.1 localhost    will be there. Why?   How does /etc/hosts work in case of JVM systems like Cassandra? When is DNS alternative, I guess not on a single computer? Integral vanishes on all intervals implies the function is a.e. zero I am having trouble with the following problem:      $f:\mathbb{R}\to \mathbb{R}$ is a measurable function such that for all $a$:   $$\int_{[0,a]}f\,dm=0.$$   Prove that $f=0$ for $m$ almost every $x$ (here $m$ is the Lebesgue measure).    I have no problem proving this for $f$ non-negative, or under the assumption that $f$ is integrable. But the question only assumes that $f$ is measurable and no more.  My idea was the usual thing; we look at the set of points where $f$ is positive and negative and assume one of these has measure greater than zero. Then I wanted to estimate one of these by an open set, look at the integral on the open set and show that it had to be greater than zero, a contradiction. But a key part of this attack is the assumption of the absolute continuity of the integral, which only holds in the case where $f$ is integrable.  Alternatively, if it were integrable one could simply estimate $f$ by a continuous function, where the result is quite obvious.  Ultimately we are going to show that $f$ is integrable, but it is not clear to me how to show this before showing it is zero a.e. So there must be a simpler way. Does anyone have suggestions? How does accepting an answer work? What does it mean to accept an answer? How do I accept an answer, and what are the rules? Can I accept my own self-answer? What are the reputation benefits? Which answer should I accept? Should I upvote the accepted answer? What should I do if none of the answers are suitable?      For more information, see "" in the . I didn't find the difference between them by  googling  Edit: Thx litterally, authenticate, means prove its identity, which is a just a subprocess of global access authorization btw I forgot to mention "Sign In", that's another confusing term for me ;)  Edit 2 other useful answers: How to completely remove Garage Band from macOS? Proof of Steinitz Theorem I want a source containing the proof of Steinitz Isomorphism Theorem stating:     For any Dedekind domain $R$ and any two nonzero ideals $I$ and $J$ of $R$ we have $I⊕J≅R⊕IJ$.   Thanks! Suppose that $f:M \to N$ is a smooth map of two manifolds and $E$ is a vector bundle over $N$. Given a connecton $\nabla$ on $E$ one can pull it back to $f^*N$ (which is a bundle over $M$) and obtain a connection $f^* \nabla$.   If $E=TN$ is a tangent bundle one can consider the Levi-Civita connection $\nabla$, with respect to the chosen metric $g$ on $N$. It is characterised uniquely as torsion free metric connection. Given a map $f:M \to N$ one can pullback the metric $g$ to $f^*g$. If it happens that $f^*g$ is again a metric (which is the case for example for immersions) one can consider the Levi Civita connection on $M$ with respect to $f^*g$. I wonder how these two constructions are related, in particular      Is there a way to pullback affine connections to get again affine connection (i.e. connections on tangent bundle)?     Additionally I would like to know:     What is the relation between Christoffels symbols for the connection and for its pullback? Gduated symbology not taking attribute into account using QGIS I m working on a project on QGIS. I have a point shapefile vector that represents my store. In the attribute table, i have bunch of data such as size, unit per transaction and most importantly store turnover. When I want to graduate the symbology, the turnover isn't present in the list I can choose from. Does someone know why, and how to fix it? \mathbb{Z} yields undefined control sequence error I have a USB flash drive which is no longer recognized by my computer. Windows Disk Management and DiskPart report No Media with no storage space (0 bytes) on the drive and I cannot partition or format the drive:     If the drive appears in Windows Explorer, trying to access it returns an error message indicating that there is no disk inserted, such as the following:     Please insert a disk into drive X:.   Various disk partitioning and data recovery utilities don't recognize the drive or only give a generic name for the drive and cannot access the contents of the drive.  What can I do? How do I recover data from the drive?     This question comes up often and the answers are usually the same. This post is meant to provide a definitive, canonical answer for this problem. Feel free to edit the answer to add additional details.