$A$ and $B$ are positive integers such that $\gcd(A,B)=9, \text{lcm}(A,B)=180, A<B.$ How many such $A$'s are there? $A$ and $B$ are positive integers such that $$\gcd(A,B)=9, \text{lcm}(A,B)=180, A&lt;B.$$ How many such $A$'s are there?  So from $\text{lcm}(A,B)=\frac{AB}{\gcd(A,B)} \Rightarrow 180 = \frac{AB}{9} \Rightarrow 1620=AB.$ Prime factoring $1620= 2^2\cdot3^4 \cdot5 = 4\cdot81\cdot5$, from here it seems that $81$ is the largest prime factor and the two other possbile prime factors are $4$ and $5$. Just following my intuition this got me to the right answer which was $2$, however I cannot find any justification for this. Is my reasoning correct or could the be a setting where this would fail? If so what would be the alternatives?
Find $x,y$ given $\gcd(x,y)$ and ${\rm lcm}(x,y)$ 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 $GCD(x,y), \ LCM(x,y)$
How could I define function with function such as Coefficient? Checking if a worksheet-based checkbox is checked Simple extension fields If I am correct simple extension fields are extensions generated by one element. I have learned that this means that elements of a simple extension can be written as powers of that element as long as the power is greater than the degree of that element. Also this means that all elements of the simple ext. Can be written in terms of decending powers of the generating element where the highest power is the degree of the generator minus 1. This concept seems to work for generating elements whose irreductable in a certain field is a descending degree polynomial. Looking at the construction of the complex numbers by starting with $\Bbb R$ (and doing $\Bbb R [x]/\langle x^2+1\rangle$), it is said that all elements are of the form $a+b\cdot \text{generator}$. But when the generator is evaluated for the irreductable, $x^2+1$, we see... $(\text{generator})^2=-1$. Thus there is no first power of gen. Can someone explain where all elements of the extension just explained are of the form $a+b\cdot \text{generator}$ comes from or explain this situation for anytime where the irreduc ( gen, F)  is not a descending degree polynomial. I hope my question is clear.  Thank you When I restart httpd, I get the following error.  What am I missing?  [root@localhost ~]# service httpd restart Stopping httpd:                                            [  OK  ] Starting httpd: Syntax error on line 22 of /etc/httpd/conf.d/sites.conf: Invalid command 'SSLEngine', perhaps misspelled or defined by a module not included in the server configuration   I have installed mod_ssl using yum install mod_ssl openssh  Package 1:mod_ssl-2.2.15-15.el6.centos.x86_64 already installed and latest version Package openssh-5.3p1-70.el6_2.2.x86_64 already installed and latest version   My sites.conf looks like this  &lt;VirtualHost *:80&gt; #    ServerName shop.itmanx.com     ServerAdmin webmaster@itmanx.com      DocumentRoot /var/www/html/magento     &lt;Directory /var/www/html&gt;         Options -Indexes         AllowOverride All     &lt;/Directory&gt;      ErrorLog logs/shop-error.log     CustomLog logs/shop-access.log &lt;/VirtualHost&gt;  &lt;VirtualHost *:443&gt;     ServerName secure.itmanx.com     ServerAdmin webmaster@itmanx.com      SSLEngine on     SSLCertificateFile /etc/httpd/ssl/secure.itmanx.com/server.crt     SSLCertificateKeyFile /etc/httpd/ssl/secure.itmanx.com/server.key     SSLCertificateChainFile /etc/httpd/ssl/secure.itmanx.com/chain.crt      DocumentRoot /var/www/html/magento     &lt;Directory /var/www/html&gt;         Options -Indexes         AllowOverride All     &lt;/Directory&gt;      ErrorLog logs/shop-ssl-error.log     CustomLog logs/shop-ssl-access.log     &lt;/VirtualHost&gt; My Custom Skin won't Change How to remove duplicate values from a multi-dimensional array in PHP Seeking open raw LiDAR or laser scanner data? I have a PC with an OEM install of Win 8 updated to 8.1.  It has 3 physical hard drives in it.  The drive where Windows is installed has a LOT of partitions on it already.    Can I install Ubuntu on a different physical hard drive than Windows, and be able to dual-boot?  I am able to boot to a live USB of Ubuntu 14.04, so I think I'm about ready to proceed.  But if I can make my installation easier by using a different physical drive, I'd like to do that. I recently ran across elliptic curve crypto-systems:    (Brown University)  (Wikipedia)  (IEEE)  (RSA.com)   It seemed to me to be great alternative to RSA as the de-facto cryptosystems to be used in banking and financial systems and in the public key infrastructure for certificates, but is not used! If someone can explain why this is not done, it would be very helpful. A comparison between traditional RSA and an elliptic curve cryptology would be helpful.  To begin with:  Advantage of RSA:   Well established.   Advantages of elliptic curve:   Shorter keys are as strong as long key for RSA (see the IEEE paper) Low on CPU consumption. Low on memory usage. How to start a new Linux distro? sudo credential caching on by default Is it possible to split micro USB to USB cable so two cables (data + and data -) go to the PC with USB and two other power cables go to power brick? Get text from pressed button Title restated: Show that $A\in\mathbb{C}_n$ is normal $\iff$ $tr(A^*A) = \sum_{i = 1}^n|\lambda_i|^2$, where $\lambda_1,...,\lambda_n$ are the eigenvalues of $A$.  This question comes from "Matrices and Linear Transformations" by Charles Cullen. I'm studying for an exam and am trying to do some problems but I'm having trouble with this one. Any help would be appreciated.  Thank you. How to run long time process on Udev event? I want to run a ppp connection when my USB modem is connected, so I use this udev rule:  ACTION=="add", SUBSYSTEM=="tty", ATTRS{idVendor}=="16d8",\     RUN+="/usr/local/bin/newPPP.sh $env{DEVNAME}"  (My modem appears in /dev as ttyACM0)  newPPP.sh:  #!/bin/bash /usr/bin/pon prov $1 &gt;/dev/null 2&gt;&amp;1 &amp;   Problem:  The udev event fires, and newPPP.sh is running, but newPPP.sh process is killed after ~4-5s. ppp does not have time to connect (its timeout is 10s for dial up).  How can I run a long time process, that will not be killed?  I tried using nohup, but it didn't work either.  System: Arch Linux  Update  I found a solution , thanks to .  I use at now to run my job detached from udev process.  But the one question remains unanswered: Why do nohup and &amp; not work? Finding $\int_0^{\infty}xe^{-\lambda x} \, dx$ How to integrate:  $$\int_0^\infty x \, \lambda e^{-\lambda x} \, dx$$    I tried using integration by parts:  Let $u = x$ and $dv = \lambda e^{-\lambda x} \, dx$  Then $du = dx$  And $v = - \lambda e ^{-\lambda x}$  Correct so far?   Then   $$\begin{aligned} uv - \int v \, du &amp;= -\lambda x e^{-\lambda x} - \int (- \lambda e^{- \lambda x}) dx \\ &amp;= -\lambda x e^{-\lambda x} - \lambda e^{-\lambda x} \end{aligned}$$    The correct answer from lecture notes      UPDATE 2  Let $u=\lambda x$ and $dv = e^{-\lambda x} dx$  Then $du = \lambda \, dx$  $$dv = e^{-\lambda x} \, dx$$  Let $y = -\lambda x$  Then $dy = -\lambda \, dx$  So $dx = -\frac{1}{\lambda} dy$  $$\begin{aligned} v &amp;= \int e^u \cdot - \frac{1}{\lambda} du \\ &amp;= - \frac{1}{\lambda} e^{-\lambda x} \end{aligned}$$  But here I seem to have an extra $- \frac{1}{\lambda}$ in $v$?  If I continue using integration by parts, I get:   $$-x e^{-\lambda x} + \color{red}{\frac{1}{\lambda}} \int e^{-\lambda x} \, dx$$ Ctrl -C, Ctrl -V not working in different keyboard layout When I am in English, Ctrl+C, Ctrl+V works fine when I need to copy paste. But when I turn to Greek, Ctrl+C and Ctrl+V and basically no shortcut almost works in Greek.  So each time I have to alt shift and go back to English in order to user Ctrl+C to copy something. How can I apply this to the Greek keyboard profile?  EDIT  I also realized that when using NetBeans IDE 7.4,  I can copy and paste in Greek but I can't use the tonic system of the language over the words. In Aptana studio I can't use Copy and paste Ctrl+C, Ctrl+V When using gedit everything works fine. Different software use different keyboard layout settings? Factor analysis and regression Power Off button is gone on Ubuntu 20.04 LTS I recently installed Ubuntu 20.04 LTS on my laptop and I noticed that I no longer have the Power Off button but rather a menu that I need to click on and then I need to click on Power Off... and then Power Off. Is there a way to get the button back? How can I search within a manpage?