I found this problem in Discrete Mathematics and Its Applications while studying for an exam and am having trouble solving it. Prove that for $a,b\in N^*, a \ge b, 2\gcd(a,b) = \gcd(a+b, a-b)$ if $a$ and $b$ are odd. My current idea was to use the principles in the Euclidean Algorithm to somehow prove it, but I'm running into a wall. Any help would be great, thanks!
I want to show that for $\gcd(a,b) = 1$ $a,b \in Z$   $\gcd(a+b, a-b) = 1$ or $\gcd(a+b, a-b) = 2$ holds.      I think the first step should look something like this:      $d = \gcd(a+b, a-b) = \gcd(2a, a-b)$   From here I tried to proceed with two cases. 1: $a-b$ is even, which leads to $\gcd(a+b, a-b) = 2$ 2: $a-b$ is odd, which leads to $\gcd(a+b, a-b) = 1$  My main problem I think is, that I do not know how I should include $\gcd(a,b) = 1$ in the proof.  Any help is appreciated. Thx in advance. Cherio Woltan
Ephemeral error message: "cryptsetup: unknown fstype, bad password or options?" I activated disk encryption for my Ubuntu 15.10 64bit installation and get every time after entering the (correct) password the error message:  cryptsetup: unknown fstype, bad password or options?  But after few seconds (without any further interaction) this error message is replaced by the expected message:  cryptsetup: sda5_crypt set up successfully  That's it and the system is started.  Any suggestions what could be the problem? Is it merely cosmetic or serious? + or @ mark after running 'ls -al' In Mac OS X, run 'ls -al' gives me something like this.   drwxrwxrwx+  4 smcho  staff     136 May  5 09:18 Public drwxr-xr-x+  6 smcho  staff     204 Feb  1  2010 Sites drwxrwxrwx   9 smcho  staff     306 Feb  2  2010 backup drwxr-xr-x@ 36 smcho  staff    1224 Sep  4 22:51 bin    What's the + or @ at the end of the first column means? Is this unique to Mac, or common in UNIX?   ADDED  After Michael Mrozek's answer, I ran 'ls -ale' to get the following.   drwx------+ 66 smcho  staff    2244 Aug 30 13:40 Library  0: group:com.apple.sharepoint.group.3 allow search  1: group:everyone deny delete drwxr-xr-x   3 smcho  staff     102 Sep  4 15:01 Mail drwx------+ 13 smcho  staff     442 Aug 28 17:55 Movies  0: group:everyone deny delete drwx------+  6 smcho  staff     204 Jul  9 09:37 Music  0: group:everyone deny delete drwx------+ 11 smcho  staff     374 Aug 28 16:55 Pictures  0: group:everyone deny delete drwxr-xr-x   3 smcho  staff     102 Mar 18 15:43 Projects drwxrwxrwx+  4 smcho  staff     136 May  5 09:18 Public  0: group:everyone deny delete drwxr-xr-x+  6 smcho  staff     204 Feb  1  2010 Sites  0: group:everyone deny delete   What those appended messages mean? Why do I have them for some of the files? I don't remember doing anything particular for them. How to reinstall Cortana? With reference to:    I removed the Cortana packages from my running Windows 10 using the method posted by magicandre1981 and using  win6x_registry_tweak.   My question is how can I re-install Cortana (or for that matter any other package removed similarly)?  I have opened up the install.wim image (converted from a Win 10 install.esd file) and found Cortana in the system apps folder, but don't know how / what to use to re-install it.  Any help please? What possessive forms are used for mutual 1st person ownership? I want to talk to someone about the house that my wife and I own.  Saying, for example, "My wife's and my house is awesome," sounds a bit funny to me.  What's the best way to express this?  Clarification  I'm asking specifically about the grammar of multiple nouns in possessive form.  I'm particularly curious if it's possible to do this with a first-person pronoun (me).  I am capable of rephrasing this in other ways - my question is not how to express the idea, but about this particular grammatical construction, if it is even legal. complex number question involving modulus Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\frac{|z − w|}{|1 − z^*w|} = 1$$ [Hint: Note that $|a|^2 = aa^*$.]  Hey guys, couldn't get my thinking cap on for this question. Any helpful input? Will appreciate it! Is there an official rule that the minimum damage can't be negative? A slightly humorous thought occurred to me (a recovering rules lawyer) after reading .     "None of the answers actually specify that 0 is the minimum damage."   Logically it wouldn't make sense for an attack to apply negative damage or healing, except in specific cases like hitting a flame elemental with a fire attack. Ruling that you could do negative damage would lead to an exploit to avoid costs of healing. As such I highly doubt any DM would make such a ruling. I definitely wouldn't except for the most absurd and humorous of games were failure is the expected result not success.  Another possibility (as suggested by Secespitus) is that negative damage would reflect back on the attacker. This could be seen as punching a strongman so hard (or in just the wrong way) as to injure your own hand.  This would also play well in a failure is the only option game, and if there is a rule suggesting that it would be an appropriate answer to this question as well.  Take the case of a person with less than 8 Strength, no levels in monk, and does not have a tavern brawler.  Would they do -1(1 - 2) damage on unarmed strikes? If so what would the result of the negative damage be?   "Healing" the target because such a puny hit actually raises their fighting spirits and makes them more capable of fighting Reflective Damage, the attacker actually takes 1 point of damage, representing injury to his hand or other damage   I am curious to know if there is an official rule stating that the minimum damage an attack can do is 0 or is there something in the damage calculation that I am missing that would otherwise prevent negative damage? I understand that multivariate regression models contain more than 1 dependent variable, but what is the difference between running a multivariate regression with dependent a and b to a set of independent variables, rather than running two regression models, one for a and one for b against the same set of independent variables? I've found plenty of posts about changing the visibility of a block programatically.  My problem is the opposite. Given a block's ID/delta, how do I find out if it should be displayed at the current path?  I have a field collection that allows you to append a block using the Block Reference module (I preprocess the field collection and use my own template, so the logic has been taken away from the Block Reference module). It works, but I only want it to kick in if the "Show block on specific pages" rule says that block should be displayed. htl5 voice recording with isualization Just noticed this:  Someone @replied to me in a question which appropriately showed up in my feed.  I go to click on that notice in the feed, and it presents me with "Loading the question failed."  I'd show a picture, but I don't think that it's entirely relevant - I believe it's reproducible if an answer is posted, someone makes a reply, the answer is removed, then that someone else goes back in the feed to try and revisit the post.  Alternatively, why not synchronize the interface between what a 10K user sees on the site, and what they see in the app?  As a 10K user, I would expect to see deleted questions and answers. I have a group $G$ with $p^2$ elements, where $p$ is a prime number. Some (potentially) useful preliminary information I have is that there are exactly $p+1$ subgroups with $p$ elements, and with that I was able to show $G$ has a normal subgroup $N$ with $p$ elements.   My problem is showing that $G$ is abelian, and I would be glad if someone could show me how.   I had two potential approaches in mind and I would prefer if one of these were used (especially the second one).   First: The center $Z(G)$ is a normal subgroup of $G$ so by Langrange's theorem, if $Z(G)$ has anything other than the identity, it's size is either $p$ or $p^2$. If $p^2$ then $Z(G)=G$ and we are done. If $Z(G)=p$ then the quotient group of $G$ factored out by $Z(G)$ has $p$ elements, so it is cylic and I can prove from there that this implies $G$ is abelian. So can we show theres something other than the identity in the center of $G$?  Second: I list out the elements of some other subgroup $H$ with $p$ elements such that the intersection of $H$ and $N$ is only the identity (if any more, due to prime order the intersected elements would generate the entire subgroups). Let $N$ be generated by $a$ and $H$ be generated by $b$. We can show $NK= G$, i.e every element in G can be wrriten like $a^k b^l $. So for this method, we just need to show $ab=ba$ (remember, these are not general elements in the set, but the generators of $N$ and $H$).   Do any of these methods seem viable? I understand one can give very strong theorems using Sylow theorems and related facts, but I am looking for an elementary solution (no Sylow theorems, facts about p-groups, centrailzers) but definitions of centres and normalizers is fine. Let $r$ be a relation on $A \times A$ such that $(a,b) r (c,d) \iff ad = bc.$ How can I show that this relation is transitive, ie. $(a, b)r(c,d)$ and $(c,d)r(e, f) \implies (a,b)r(e,f)$?  I tried to say that $(a,b) r (c,d)$ means that $c=ka$ and $d=kb,$ $k$ a coefficient, such that $ad = bc \iff akb = bka,$ and going on from there, but I'm not sure this is valid for all values of $a, b, c,$ and $d$. How can I show the transitivity of the relation? I am trying to teleport a player to certain coordinates once they enter an area.  The idea is that they walk into a really small building, but once they turn a corner, they get teleported into a larger building.   Please refer to the adventure map named '5 Cubed', if you need help understanding what I am asking. Trying to identify pre-1970s short story involving multiple humans absorbed into an alien blob How to enter the Default Keyring password via the command line? Is there a way to enter the default keyring password using the command line?   For instance:  You have a remote setup of Ubuntu 10.10 thats set to auto login. You don't want to remove  the keyring password.  All right the system boots up and logs in automatically, then asks for the keyring password now at this point you can create ssh connections but you can't remote desktop.   What can you do to enter the keyring password at this point?  Also, to better clarify, this is from a remote connection using the command line. What is the simplest organism that was observed to learn: change its behavior permanently in response to some event/stimuli in a way that this change persists even if the event does not happen again? Test if a website is alive from a C# application I am looking for the best way to test if a website is alive from a C# application. Background My application consists of a Winforms UI, a backend WCF service and a website to publish content to the UI and other consumers. To prevent the situation where the UI starts up and fails to work properly because of a missing WCF service or website being down I have added an app startup check to ensure that all everything is alive. The application is being written in C#, .NET 3.5, Visual Studio 2008 Current Solution Currently I am making a web request to a test page on the website that will inturn test the web site and then display a result. WebRequest request = WebRequest.Create(&quot;http://localhost/myContentSite/test.aspx&quot;); WebResponse response = request.GetResponse();  I am assuming that if there are no exceptions thown during this call then all is well and the UI can start. Question Is this the simplest, right way or is there some other sneaky call that I don't know about in C# or a better way to do it. I'm asked to used induction to prove Bernoulli's Inequality: If $1+x&gt;0$, then $(1+x)^n\geq 1+nx$ for all $n\in\mathbb{N}$. This what I have so far:  Let $n=1$. Then $1+x\geq 1+x$. This is true. Now assume that the proposed inequality holds for some arbitrary $k$, namely that $$1+x&gt;0\implies (1+x)^k\geq 1+kx,~\forall~k\in\mathbb{N}\setminus\{1\}$$ is true. We want to show that the proposed inequality holds for $k+1$. Thus multiplication of $(1+x)$ on each side of the above inequality produces the following result: $$(1+x)(1+x)^k\geq (1+kx)(1+x)\implies (1+x)^{k+1}\geq 1+x+kx+kx^2\cdots\cdots\cdots$$  I'm not sure where to go from here. This question is the flip-side of the commonly asked food temperature safety concerns. I'd like to know, if a meat-based soup stock (chicken/pork/beef/etc) were to be held constantly at a safe hot temperature of 165F/73C-180F/82C, how long could it stay at this temperature and still be considered safe to eat?  Obviously, it would reduce over time, but suppose it's reconstituted with water regularly so that it could continue being held... Is there ANY sort of time limit (even completely unreasonable spans like, months, years, etc)? Can a regular salaried (w2) employee opt to pay quarterly taxes? As a consultant, I got used to paying quarterly taxes. Now that I'm a regular w2 salaried employee, I'm subject to normal withholding on each pay statement. I would like to keep making exact quarterly payments instead of having a (generally incorrect) amount held back each paycheck.   I haven't been able to find any information on the 1040-es that would make me think I'm ineligible to do so, but I suspect that it may be the case. I'd have to stop paycheck withholdings by updating my w4 to either have a high number of exemptions or mark myself as exempt from taxes, which I assume is where the rub lies.   So, as a regular salaryman, is it possible for me to pay taxes exclusively through quarterly payments in lieu of traditional withholdings?