If $gcd(a,b)=d$ and $a^2+b^2=c^2$, show that $gcd(a,c)=gcd(b,c)=d$. This is what I have so far. We know that $d|a$ and $d|b$ which implies that $d|c$, but that is not enough to show that $gcd(a,c)=gcd(b,c)=d$, only that they are at least $d$. Also $a^2=c^2-b^2$ and $b^2=c^2-a^2$. Any hints or solutions are greatly appreciated.
Let $a, b, c \in \mathbb{N}$. Show that if $a^2+b^2=c^2$ then $gcd(a,b)=gcd(a,c)=gcd(b,c)$.  Using basic divisibility criteria I couldn't advance anything.
Do's and Don't's of Undergraduate Research I was fortunate enough to get a position as a researcher for the Mayo Clinic's SURF Program this year. My PI's lab focus is on the the immune system's role in CNS axonal and neuronal injury, specifically through the lens of how innate and adaptive immune effectors interact w/ infected neurons.  Although I do research under a professor at my college and I volunteered for a state university lab during the previous summer, this is my first REU/SURF opportunity, and I REALLY want to make a good impression. Here are my questions:  What are the do's and don't's in terms of being a skilled and efficient researcher?  Since I am still an undergrad, I know that I will be a less useful asset to the lab than a grad student or post-doc, but what can I do as an undergrad to not burden my colleagues and PI?  Thank you all for your help! Wishing you all the very best! I am trying to run grep against a list of a few hundred files:  $ head -n 3 &lt;(cat files.txt) admin.php ajax/accept.php ajax/add_note.php   However, even though I am grepping for a string that I know is found in the files, the following does not search the files:  $ grep -i 'foo' &lt;(cat files.txt)  $ grep -i 'foo' admin.php The foo was found   I am familiar with the -f flag which will read the patterns from a file. But how to read the input files?  I had considered the horrible workaround of copying the files to a temporary directory as cp seems to support the &lt;(cat files.txt) format, and from there grepping the files. Shirley there is a better way. I've made a roughness 0 glass pane, when I'm in lookdev mode it's reflecting a background that I didn't put there. I know I could just ignore it, but I'd rather be able to see what it'll mostly look like when I'm in the later stages. I also just want to know how on Earth I managed to set this background without realising it  Reflection completely ignores any objects I put in front of it. If I placed this glass inside my house, it would still just reflect trees (This is the point where I felt I couldn't ignore it - reflections are going to play a big part in my project and I don't want to be in render mode all the time)  The nodes are the default glass BSDF, except I set roughness to 0  The glass is set to 20mm thick, if that makes a difference  The background moves with my perspective and moves when I relocate the glass  I have a few HDRI's on my PC, but none look remotely like this   Also, I'm using cycles as I find Evee buggy as hell SELECT DISTINCT on multiple columns Supposing we have a table with four columns (a,b,c,d) of the same data type.  Is it possible to select all distinct values within the data in the columns and return them as a single column or do I have to create a function to achieve this? My ListView (of my ViewFilipper) fills the parent and buttons on the bottom get hidden Why does my Mac OS X 10.6 kernel run in 32-bit mode? I have a MacBook Pro (5,1) with 4GB of memory running 10A432 - but it's running the 32-bit kernel.  Any ideas why it wouldn't be running in 64-bit mode, and how I can get it to do so? If $z_n \to z$ then $(1+z_n/n)^n \to e^z$ We are dealing with $z \in \mathbb{C}$. I know that $$ \left(1+ \frac{z}{n} \right)^n \to e^{z} $$ as $n \to \infty$. So intuitively if $z_n \to z$ then we should have $$ \left(1+ \frac{z_n}{n} \right)^n \to e^{z}. $$ If $z_n \in \mathbb{R}$ I would be happy writing $$ \exp \left(n \log\left(1+ \frac{z_n}{n} \right) \right) = \exp \left(z_n \frac{\log\left(1+ \frac{z_n}{n} \right)-\log(1+0)}{\frac{z_n}{n}-0} \right) \to \exp(z \cdot 1) $$ where here we are using the definition of derivative. But if $z \in \mathbb{C}$ the logarithm is multivalued and I'm not sure that the same calculation is allowed. Is there a different way to show this for complex $z$? I'm currently building a town in Minecraft Pocket Edition and I'm wondering how many blocks long and how many blocks wide the world is. Create file and its parent directory I know that touch command creates a file:  touch test1.txt   but how I can create a file and its full path?  for example my desktop contains nothing:  ~/Desktop/$ ls ~/Desktop/$   and I want to create 1.txt in ~/Desktop/a/b/c/d/e/f/g/h/1.txt. Can I do this with a simple command like:  $ touch ~/Desktop/a/b/c/d/e/f/g/h/1.txt   instead of create full path manually and then create the file? In bash, how can I convert a Unicode Codepoint [0-9A-F] into a printable character? I have a list of Unicode codepoints, but I don't know of a "simple" way to convert these hex values into the actual characters they represent...    I've heard that zsh has echo -e '\u0965', but I use bash 4.1.  Is there something as simple as the zsh method, for bash? Two players, $A$ and $B$, alternately and independently flip a coin and the first player to get a head wins. Assume player $A$ flips first.  If the coin is fair, what is the probability that $A$ wins?   So $A$ only flips on odd tosses. So the probability of winning would be $$ P =\frac{1}{2}+\left(\frac{1}{2} \right)^{2} \frac{1}{2} + \cdots+ \left(\frac{1}{2} \right)^{2n} \frac{1}{2}$$  Is that right? It seems that if $A$ only flips on odd tosses, this shouldn't matter. Either $A$ can win on his first toss, his second toss, ...., or his $n^{th}$ toss. So the third flip of the coin is actually $A$'s second toss. So shouldn't it be $$P = \frac{1}{2} + \left(\frac{1}{2} \right)^{2} + \left(\frac{1}{2} \right)^{3} + \cdots$$ How to show that every Boolean ring is commutative? Why is there no consistency in the plural forms of words ending on an "f" (e.g. safe, roof, dwarf, etc.)? The plural form of leaf is leaves, although according to  leafs is also correct. Dwarf can be pluralized as either dwarfs or dwarves. Conversely, the words roof and safe are pluralized as roofs and safes.  Is there any logic underpinning these variations? Is there a historical trend where words that were previously pluralized as -ves are now being increasingly pluralized as -fs? In a certain case, the user will be asked to confirm if he/she wishes to leave the page, due to editing a post, even when he/she has stopped editing.  To reproduce:   Start editing a post (I've only experienced this with a question) Another user finishes editing the post before you and the message "This post has been edited" appears Clicking on the message bar will take you out of editing mode and will show you the updated post from the other user's edits Trying to navigate away from the page will then ask for your confirmation How to fix "The system is running in low-graphics mode" error? Note: This is an attempt to create a canonical question that covers all instances of &quot;low-graphics mode&quot; error that occurs to a user, including but not limited to installation of wrong drivers, incorrect or invalid lightdm greeters, low disk space, incorrect installation of graphics card like ATI and Nvidia, incorrect configuration of xorg.conf file while setting up multiple monitors among others. If you are experiencing the &quot;low-graphics mode&quot; error when trying to login but none of the following answers work for you, please do ask a new question and then update the answers of this canonical question as and when your new question gets answered.   When I try to boot into my computer, I am getting this error:  The system is running in low-graphics mode Your screen, graphics cards, and input device settings could not be detected correctly. You will need to configure these yourself.   How do I fix the failsafe X mode and login into my computer?  Answer index: Question mark and colon in JavaScript Trying to install Atom. Unable to start Clangd language server Is it possible to keep publishing under my professional (maiden) name, different from my married legal name? Book wherein humans were most dangerous/aggressive of numerous species Move the Add Favorite / Ignored Tags boxes to the top It seems that most people who use the ignored tags feature end up with a fairly long list of those tags.  Although , it is sometimes necessary to add new ignored tags manually (typically when adding some-tag*).  It would be a lot easier if the add box was above the long list of ignored tags rather than below. I suppose the same would apply to the favorites tags for consistency's sake.