There are $a,b \in \mathbb Z, a \neq 0$ and $d$ which is the smallest number in $\mathbb N$ for that  $d = l \cdot a + k \cdot b$ is true.  ($k,l \in \mathbb Z$)  I think that $d|b$ and $d|a$ is true, but I can't find proof for it, can you?
I'm trying to prove that if there are to numbers $n,m$ (natural numbers), and their smallest common multiple is $k$, so that $k = n·i$ and $k = m·j$ for some $i,j$ natural numbers, any common multiple $q$ of $n,m$ is a multiple of $k$, so that there exists a $g$ so that $q=k·g$.  Logically it seems correct, but I can't seem to prove it... Any help?
Theme Activate Hook It's not difficult to show that  $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$  On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$.  Squaring the first power series and comparing terms gives us  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}2^{-2n}=1$$  that is,  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$$  My question: is there a more direct, combinatorial proof of this identity?  I've been racking my brains trying to come up with one but I'm not having much success. I have installed Ubuntu Server on an old desktop, and I'm new to this.  I had set it up and was trying to connect through Putty but that was not working.  When I tried ifconfig there was no eth0 IP address, and logging onto the router showed the only device connected by ethernet is the printer.  However, the ethernet cable is plugged into the server and is showing the green light by it.  I tried opening /etc/networking/interfaces but this shows the following:  source /etc/network/interfaces.d/*  # The loopback network interface auto lo iface lo inet loopback   I tried adding 'auto eth0' after this and saving but made no difference, and have tried sudo ifconfig eth0 up which didn't do anything either.  sudo lshw -C network says that the network is DISABLED.  Any ideas to help get this up and running?  EDIT: Running command lspci -knn | grep Eth -A3 results in the following:  02:05.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL-8110SC/8169SC Gigabit Ethernet [10ec:8167] (rev 10)          Subsytem: Gigabyte Technology Co., Ltd GA-MA69G-S3H Motherboard [1458:e000]          Kernel driver in use: r8169          Kernel modules: r8169 Insert result of a parallellized select query into a table, in Postgresql According to ,      "Even when it is in general possible for parallel query plans to be   generated, the planner will not generate them for a given query if any   of the following are true:       The query writes any data or locks any database rows. If a query contains a data-modifying operation either at the top level or within   a CTE, no parallel plans for that query will be generated. This is a   limitation of the current implementation which could be lifted in a   future release."   Indeed, when I try to insert result of a parallel SELECT query into a table (either by SELECT.. INTO either by WITH..SELECT..INTO, the query is not executed as parallel query.  My question is: Is there any way to trick the Postgresql so that a SELECT query is executed as parallel query and then its result inserted into a table? Height and minimal number of generators of an ideal How can I determine the height and the least number of generators of the ideal $ I=(xz-y^2,x^3-yz,z^2-x^2y) \subset K[x,y,z] $?      I tried to calculate the dimension of the vector space $I/I\mathfrak m$ with $\mathfrak m=(x,y,z)$, but I'm not able to find it.  I think the height is $2$, because the affine variety should be a curve, and also I know that the ideal is prime, so I need a chain $0\subset \mathfrak p \subset I$. Maybe $(xz-y^2)?$ Is it prime? I've been messing around with Unity and broke something, how do I "start over"? If $(Tx \mathbin{|} x) = 0$ for all $x$ then $T = 0$ Let $T$ be a linear operator on an inner product vector space $V$. I'd like to prove that if $$(Tx|x)=0 \quad \forall x \in V$$ then $T$ is the null operator. I can't figure out this proof using contradiction, do I have to choose an appropriate vector? Why does my computer get less battery life with Ubuntu compared to Windows? Last year I purchased a Dell Inspiron 5315 laptop shipping with Windows 7 preinstalled. I installed Ubuntu 10.10 alongside it with dual-booting. On upgrading to Ubuntu 11.10, I was upset to find I now get less than one hour of battery life! Windows 7 gives me 2.5 hours of running time.  Is there any possible explanation why  my battery is running out faster with Ubuntu? Can anybody tell me better power options? Spinor dotted and undotted indices Answering your own question? Accept it. What are path and contour integrals? I am very confused on what path and contour integrals actually represent. I tried looking for an answer on Google but the only examples were related to Quantum Mechanics. For instance, real Riemann integrals can be thought as &quot;the area under the curve&quot;. What about path/contour integrals? Also, is there a difference between path and contour integrals, or are they the same thing? I've been thinking of contour integrals as path integrals where the path is a &quot;closed loop&quot;, but I'm not sure if that's the case. It's all so confusing, and not being able to visualise these concepts makes it feel like I'm just memorising formulas and theorems without any reason. I have been thinking about path integrals as integrating over a curve in three dimensions. For example, in an $xyz$ coordinate system (with $y$ pointing “up”), we have a path on the $x$-&quot;$z$&quot; plane and integrate over that path. That is, instead of integrating over and along &quot;$y=0$&quot;, we integrate over a certain path. But I don't know if this is a correct way to think of path integrals. This issue has been frustrating me a lot because I'm now studying theorems that involve path/contour integrals and I just can't understand what they really do or mean. By path integral I mean $$\int_{\gamma\vert_{[a,b]}}f(z)\hspace{0.2em}dz = \int_{a}^{b}f(\gamma(t))\cdot\gamma'(t)\hspace{0.2em}dt$$ where $f:U\subseteq\mathbb{C}\rightarrow\mathbb{C}$ is a function that satisfies all conditions for complex integration and $\gamma:[a,b]\rightarrow U\subseteq\mathbb{C}$ is at least a piece-wise regular path. Any help would be appreciated. Releasing bound ports on process exit How do I make sure that a socket bound to a port is properly released on process exit such that the port can be reused without bind() failing with EADDRINUSE? I've written a tiny program which just creates a socket, binds it to a fixed port, waits for a connection and then immediately terminates. When I rerun the program, the bind() call fails with EADDRINUSE, but if I wait a few minutes, it succeeds.  Is there a way I can explicitly "unbind" the socket, thereby freeing the port number? Down Arrow Key is not working What package should I use to typeset units? Copying protected files between servers in one line? I'd like to copy squid.conf from one server to another.   The servers don't talk to each other. I'd like to go through my workstation. Both servers have the file, so it will be overwritten on the target. The files have 600 permission and are owned by root. root login via ssh is disabled (PermitRootLogin no). I'd like to do it in one line, if possible, since it will be a part of a setup guide.   I know to do  ssh source 'tar czpf - -C /etc/squid/ squid.conf' | \     ssh target 'tar xzpf - -C /etc/squid/'   to copy files between servers and preserve permissions. However, in this case I will get "Permission denied".  I also know I can do this:  ssh -t source 'sudo cat /etc/squid/squid.conf'   This way the -t allows sudo to ask for the admin password before outputing the content of the file.  The problem is, I don't know how to combine those techniques into something that will ask for the sudo password on each server, and transfer the file to its destination. Is this possible?  UPDATE: Here's the best I could come up with:  ssh -t source 'sudo tar czf /tmp/squid.tgz -C /etc/squid squid.conf' &amp;&amp; \ ssh source 'cat /tmp/squid.tgz' | \     ssh target 'cat &gt;/tmp/squid.tgz' &amp;&amp; \ ssh -t source 'sudo rm /tmp/squid.tgz' &amp;&amp; \ ssh -t target \     'sudo tar xzf /tmp/squid.tgz -C /etc/squid &amp;&amp; sudo rm /tmp/squid.tgz'   Calling this a one-liner seems like a stretch. I think I'll just break it down to separate steps in the setup guide. How to Draw 2 Functions In Tex How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$$  I tried to prove that by proving that the following summation in in absolute value converges so the original one converges too, but that's incorrect.  Any other ideas? I was reading the Math.random() javadoc and saw that random is only psuedorandom.    Is there a library (specifically java) that generates random numbers according to random variables like environmental temperature, CPU temperature/voltage, or anything like that? I have seen the following methods of putting JavaScript code in an &lt;a&gt; tag:  function DoSomething() { ... return false; }    &lt;a href="javascript:;" onClick="return DoSomething();"&gt;link&lt;/a&gt; &lt;a href="javascript:DoSomething();"&gt;link&lt;/a&gt; &lt;a href="javascript:void(0);" onClick="return DoSomething();"&gt;link&lt;/a&gt; &lt;a href="#" onClick="return DoSomething();"&gt;link&lt;/a&gt;   I understand the idea of trying to put a valid URL instead of just JavaScript code, just in case the user doesn't have JavaScript enabled. But for the purpose of this discussion, I need to assume JavaScript is enabled (they can't login without it).  I personally like option 2 as it allows you to see what's going to be run&ndash;especially useful when debuging where there are parameters being passed to the function. I have used it quite a bit and haven't found browser issues.  I have read that people recommend 4, because it gives the user a real link to follow, but really, # isn't "real". It will go absolutely no where.  Is there one that isn't support or is really bad, when you know the user has JavaScript enabled?  Related question: . Sum of Cosine/Exponential