The question:   Let $a,b \in  \mathbb{Z}$ with $a$ and $b$ not both zero, and let $d = (a,b)$. Prove that $d$ is the least positive integer linear combination of $a$ and $b$; ie ,for all $c \in  \mathbb{Z}^{+}$, if $c$ is an integer linear combination of a $a$ and $b$, then $d \leq c$  I tried finding similar examples of this proof on stack exchange and google but found nothing.
Given $a_1, a_2, a_3, \ldots , a_n$ not all zero, show $\gcd(a_1, a_2, a_3, \ldots , a_n)$ is the least positive integer that can be expressed in the form $a_1x_1+a_2x_2+ \ldots +a_nx_n$. Also deduce $a_1x_1+a_2x_2+ \ldots +a_nx_n = b$ has solutions $\iff \gcd(a_1, a_2, a_3, \ldots , a_n)|b$.   I know that $ax+by=n$ has integer solutions $\iff \gcd(a,b)|n$. I'll need to use induction to show $a_1x_1+a_2x_2+ \ldots +a_nx_n = \gcd(a_1, a_2, a_3, \ldots , a_n)$ has solutions, and then I will be able to say that as $\gcd(a_1, a_2, a_3, \ldots , a_n) | a_1x_1+a_2x_2+ \ldots +a_nx_n $ for any $x_1,x_2, x_3 \ldots x_n$ then $c \leq \gcd(a_1, a_2, a_3, \ldots , a_n)$ and $c \geq \gcd(a_1, a_2, a_3, \ldots , a_n)$, and thus $c=\gcd(a_1, a_2, a_3, \ldots , a_n)$.
I have to resit a calculus exam and for some reason set proofs were never my best friend...  Anyway, on a practice exam I encountered the following proof:  $$A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$$  When I draw a Venn-diagram it seems quite obvious but I couldn't manage to write the proof down properly.  If someone could help me, that'd be great! Since I updated the system to 10.11.5, the Facebook message notification is not showing message text.   Is this a bug in this software upgrade of OSX? I would like to understand a couple of fact on maximum likelihood estimators (MLEs) for logistic regressions.   Is it true that, in general, the MLE for logistic regression is biased? I would say "yes". I know, for example, that sample dimension is related to the asymptotic bias of MLEs.   Do you know any elementary examples of this phenomenon? If the MLE is biased, is it true that the covariance matrix of the MLEs is the inverse of the Hessian of the maximum likelihood function?   edit: I have met this formula quite often and without any proof; it seems a quite arbitrary choice to me. Do you recommend using semicolons after every statement in JavaScript? In many situations, JavaScript parsers will insert semicolons for you if you leave them out. My question is, do you leave them out?  If you're unfamiliar with the rules, there's a description of semicolon insertion on the . Here's the key point:     If the first through the nth tokens of a JavaScript program form are grammatically valid but the first through the n+1st tokens are not and there is a line break between the nth tokens and the n+1st tokens, then the parser tries to parse the program again after inserting a virtual semicolon token between the nth and the n+1st tokens.   That description may be incomplete, because it doesn't explain @Dreas's example. Anybody have a link to the complete rules, or see why the example gets a semicolon? (I tried it in JScript.NET.)   is related, but only talks about a specific scenario. In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading! Does someone knows if it exists some ppa packages source for the newest Ubuntu version 19.10 i had problem with that when i upgraded from Ubuntu 19.04 to 19.10 i didn't work? The script works fine if I run it with sh filename, but it ignores all the --exclude parameters if its run by cron:  #!/bin/sh TIMESTAMP=$(date +%u)  WEEK=`date +"%V"` if [ $(($WEEK%2)) -eq 0 ]; then     echo "even";     BACKUP_DIR="/mnt/backup/2_$TIMESTAMP"     mkdir -p "$BACKUP_DIR/www"  else     echo "odd";     BACKUP_DIR="/mnt/backup/1_$TIMESTAMP"     mkdir -p "$BACKUP_DIR/www"   fi  if [ $(date +%u) -gt 6 ]; then     zip -r -y $BACKUP_DIR/www/web.zip /var/www/ --exclude='var/www/\.opcache/*' --exclude='var/www/utils/phpThumb/cache/*' else     zip -r -y $BACKUP_DIR/www/web.zip /var/www/ --exclude='var/www/mycompany/files/*' --exclude='var/www/\.opcache/*' --exclude='var/www/mycompany/szamlazz/xml_szamlak/*' --exclude='var/www/mycompany/files_admin/szamlazz/*' --exclude='var/www/mycompany/files_admin/chatFiles/*' --exclude='var/www/mycompany/utils/phpThumb/cache/*' --exclude='*/backup-guard/*' --exclude='var/www/mycompany/mailer/*' --exclude='var/www/mycompany/attachments/*' fi  exit 0   The above script should create a larger backup on sunday, and a partial on weekdays, but it just zips everything. Can't delete folder in Windows 7 Python/Matplotlib - Is there a way to make a discontinuous axis? Fibonacci addition law $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$ Showing a tensor is not a simple tensor dist$(x,A)=0$ if and only if $x\in \overline{A}$ (closure of $A$) What are the differences between Factor Analysis and Principal Component Analysis? It seems that a number of the statistical packages that I use wrap these two concepts together. However, I'm wondering if there are different assumptions or data 'formalities' that must be true to use one over the other. A real example would be incredibly useful. What formats supporting animation are suitable for the web? What formats supporting animation are suitable for the web and what are the pros and cons of each. Just curious whether the process takes a while, and whether it scrapes through history to determine if it was once valid or not. Currently, I have 6 0-vote accepted answers and 20 1+-vote answers (), and that makes it > 5 and > 20% but not for long if voting goes up or down from here. ;-P  Update:  Special thanks to Grace Note for pointing out the  to follow. I upgraded this to a bug because it's been 48 hours during which the criteria supposedly were met. 6 0-votes, 20-23 1+ votes during the whole period. The criteria only last so long, so it may go out of scope soon. If it only checks every 3 days, I'm not sure if that's intentional or a bug. I have 4GB RAM. When I open Firefox, IntelliJ IDEA or VS Code and some other application my memory is about used up thus my machine hangs and I can't do anything. I can't even close any applications.    Date and time are shown in the top bar so that I can view it any time without any thing typing.   If I would view memory status in this way without typing anything then I can make a decision whether to open an application or whether this application may put my machine in hanging state or not.    Is it possible in Ubuntu 18.04 LTS? What's a (big) word or phrase to describe a book that is so good that I cannot put down when I start to read it? Find the sum to n terms as well as the sum to infinity of the series: Find the sum to $n$ terms as well as the sum to infinity of the series:  $$\frac{1}{3} + \frac{1}{4\cdot 2!} + \frac{1}{5\cdot 3!} +\cdots.$$  I was trying this question many times but I didn't get the answer. I was using geometric series formula, but it was not applicable here...I don't know where I have to start, and I don't know any hint about this series....  If anybody helps me, I would be very thankful to him. There is this addon that was built called Window Generator 2. It can be . However, when I go to the download link it shows up in a new tab as a bunch of code.  How can I download and use this addon in Blender? I was wondering how far in imaging physics had gotten. Do we hold the technology to actually take a picture of, say, an alpha particle, or even a single atom?  I realise we aren't talking about camera pictures, so what kind of imaging techniques have taken images/is most likely to be the technique to take an image of a single atom?