Suppose $a, b \in \mathbb{Z}$, $a &gt; b$, and $\gcd (a,b) = 1$.  What can be said about $\gcd (a+b,a-b)$?  Is it true in general that $\gcd (a+b,a-b) \leq 2$?
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
I'd like to be able to easily transfer large files wirelessly, fast, between Android devices - preferably without requiring root.  As a secondary objetive, it wouldn't hurt if the same solution could be used, to transfer files between the Android devices and a Windows PC.   Specifically, I'm trying to exchange files between my Nexus 7 (Android 4.2.2) and Samsung Galaxy S2 (Android 4.0.4).   Bluetooth is simply too slow, so that's out of the question Transfer via the Internet is also not an option, hence so is cloud-based solutions Only Nexus 7 supports NFC, so that's out too  looked promising and very simple, but does not work on "Devices that don't have WiFi Tethering setting, such as the Nexus 7"   Wi-Fi direct ( and up) seems to be a decent lowest common denominator between my two devices, however...  Galaxys S2's native Send via Wi-Fi Direct sees the Nexus, but seems to get stuck in the "Connecting..." most times. When there finally seems to be breakthrough, I get "Waiting..." and a spinning circle. Even if it worked, it would only allow me to send from the Galaxy S2 to the Nexus 7, as a similar option os not present on the Nexus 7.  J4velin's  is no longer available in the Play Store, with the following explanation:     WiFi Direct drivers are broken on many devices, which led to users leaving negative reviews for this app. As I don't see any sense in keeping a free app published when only getting 1-star-ratings (for an issue which only your device manufacturer can fix), the app is no longer available at the Play Store.   Nick Adams'  seems flaky at best. Invations from Nexus 7 to Galaxy S2 are never received. Invitations from Galaxy S2 are mostly received, but even when accepting promptly, the peers doesn't always show as connected. When they finally do, usually after quite a long wait, I can select a file to send but it's never transferred -  "Transfer failed" it says almost immediately in the notification bar.  budius inc's  is working slightly better, but still very unreliable. I actually managed to (occasionally) send files both ways, but never when attempting connection from the Nexus 7. To send from the Nexus 7 to the Galaxy S2, I had to go through this akward manoeuvre:   Share the file on the Nexus 7 via the WiFi Shoot intent Start the Receive WiFi Shoot app on the Galaxy S2 Invite the Nexus 7 from the Galaxy S2 (never the other way around!) Accept the invitation on the Nexus 7 Press the Shoot button on the Galaxy S2   Take the whole switching back and forth between devices, and garnish with random delays or failures in any step of the process...  Is there really no easy, reliable, fast way to transfer files directly between different Android devices?  Generally, my experience with Wi-Fi Direct is all but positive...   A device often won't even see a device lying next to it Sometimes, it will see the other device, only to have it disappear from the list, before I have the chance to connect When I do have a chance, invations are not realiably received (from Galaxy S2 to Nexus 7 mostly reliable, but never worked the other way around) When invitations are received (and accepted), a partnership rarely manages to be setup ("invited" or "connecting") And even when that happens, some apps can't seem to transfer files through it...   *Sigh*  Maybe J4velin's explanation pretty much sums up the sad state of Wi-Fi Direct on Android, which otherwise seemed as the best solution to my quest? Are all limits solvable without L'Hôpital Rule or Series Expansion If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$ Suppose that a function $f:[a,b]\to\mathbb{R}$ is continuous and nonnegative. Prove that if $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$.   I've been trying to prove it using the extreme value theorem, continuity and the upper and lower sums but can't come up with something tight enough. More specifically, I've been trying to relate $|f(x)-f(t)|&lt;\epsilon$ to the sum of $M_k-m_k(x_k-x_{k-1})$ WoeUSB Error Code 256 with NTFS formatted USB I tried to create a Windows 10 USB boot medium with WoeUSB. I formatted a USB drive with an NTFS partition, but WoeUSB complains with:  Installation failed! Exit code: 256 Log: WoeUSB v@@WOEUSB_VERSION@@ ============================== Mounting source filesystem... Error: File "/media/woeusb_source_1543626298_6098/sources/install.wim" in source image has exceed the FAT32 Filesystem 4GiB Single File Size Limitation and cannot be installed.  You must specify a different --target-filesystem. Refer: https://github.com/slacka/WoeUSB/wiki/Limitations#fat32-filesystem-4gib-single-file-size-limitation for more info. 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For example:   public class Genotype {   private boolean bits[];   private int rating;   private int length;   private Random random;    public Genotype() {              //  &lt;= THIS is the bandit, this one right here     random = new Random();   }    /* creates a Random genetoype */   public Genotype(int length, Random r) {     random = r;     this.length = length;     bits = new boolean[length];      for(int i=0;i&lt;length;i++) {         bits[i] =random.nextBoolean();     }   }    /* copy constructor */   public Genotype(Genotype g,Random r) {     random = r;     bits = new boolean[g.length];     rating = g.rating;     length = g.length;      for(int i=0;i&lt;length;i++) {         bits[i] = g.bits[i];     }    } }   The first constructor doesn't seem to be a "real" constructor, it seems as though in every case one of the other constructors will be used. So why is that constructor defined at all? 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Please look at this code:  create table #t1(   id int identity (1,1),   val varchar(10) );   insert into #t1 values ('a'); insert into #t1 values ('b'); insert into #t1 values ('c'); insert into #t1 values ('d');   Now, whenever you execute this  select *,      ( select top 1 val from #t1 order by NEWID()) rnd  from #t1 order by 1;   you will get a result with where all rows have the same random value. e.g.  id          val        rnd ----------- ---------- ---------- 1           a          b 2           b          b 3           c          b 4           d          b   I know a way using a cursor to loop throw the rows and get different random values, but that is not performant.   A clever solution to this is   select t1.id, t1.val, t2.val from #t1 t1     join (select *, ROW_NUMBER() over( order by NEWID()) lfd from #t1) as t2 on  t1.id = t2.lfd    But I simplified the query. The real query looks more like  select *,      ( select top 1 val from t2 where t2.x &lt;&gt; t1.y order by NEWID()) rnd  from t1 order by 1;   and the simple solution doesn't fit. I'm looking for a way to force repeated evaluation of  ( select top 1 val from #t1 order by NEWID()) rnd    without the use of cursors.  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