Show that if $(x,y) =1$ then $(x-y, x+y)$ is either $1$ or $2$.
If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $
Batch export Photoshop layers to individual PNG files Torsion elements of a group aren't necessarily a subgroup In a question from an exam in an undergraduate group theory course, we were asked to prove or disprove that the set of all Torsion elements of a group is necessarily a subgroup.  I knew that the set of Torsion elements is closed under the inverse operation, but was later told that it is not closed under multiplication, therefore disproving the claim. However, I couldn't find any examples of a group $G$ and two elements $a,b$ such that both $a$ and $b$ have a finite order, but $ab$ doesn't. I know that for this to happen $G$ must be an infinite and non-Abelian group, but still couldn't find a valid example.   What are some examples of groups/elements fulfilling the aforementioned property? Which is more appropriate in the following sentence?     "I asked him to not judge her according to (or: based on) her beauty"   vs     "I asked him not to judge her according to (or:based on) her beauty"   or maybe you wouldn't choose either of the above, but something else? CUDA Driver API vs. CUDA runtime I would like my sections to start on a new page if they do not fit on the current page. In other words:  if a section fits on the current page print it, if it does not fit start printing it on the next page.  I would like this to be done automatically.  \usepackage{titlesec} seems to be the solution, but I can not get it to do the job.   Should it be done with titlesec? how ? Is there a better approach? 'pagesize' attribute ignored in hot network list Steps to reproduce:   Open  in Guest/Incognito mode or with cookies disabled. This will have 30 questions per page by default. Click the  link and as expected, you'll see questions 1-50 with page size of 50. Go the the  and see the bug: you're in the second page of 30 items per page, i.e. you'll see questions 31-60 instead 51-100 as expected.     It is very annoying. Please fix it.  My user-agent is: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.15; rv:70.0) Gecko/20100101 Firefox/70.0  There are no plugins installed affecting the site. It happens both when logged in or not.  The mechanism used to work before, but some weeks/months ago it stopped, enough to bother me to post here :)  Here are all modified "privacy" settings from about:support page:  privacy.clearOnShutdown.offlineApps true privacy.clearOnShutdown.openWindows true privacy.clearOnShutdown.siteSettings    true privacy.cpd.offlineApps true privacy.cpd.openWindows true privacy.cpd.passwords   true privacy.cpd.siteSettings    true privacy.donottrackheader.enabled    true privacy.history.custom  true privacy.item.cookies    true privacy.sanitize.migrateClearSavedPwdsOnExit    true privacy.sanitize.migrateFx3Prefs    true privacy.sanitize.pending    [{"id":"shutdown","itemsToClear":["cache","cookies","offlineApps","history","formdata","downloads","sessions","siteSetti privacy.sanitize.sanitizeInProgress ["cache","cookies","offlineApps","history","formdata","downloads","sessions","siteSettings","openWindows"] privacy.sanitize.sanitizeOnShutdown true privacy.sanitize.timeSpan   0 privacy.trackingprotection.enabled  true privacy.trackingprotection.introCount   20 What are the correct settings for min-memory and max-memory in this use case?  The server has 8GB ram, dual Intel Xeon processors, running Windows Server 2008 R2 / Sql Server 2008 Standard Edition. It's running several databases ranging from 30GB - 5GB in size.  Originally the memory usage was set to the default settings (min=0 max=2,147,483,647). On these settings most of the memory usage was taken up by sqlservr.exe and the server would eventually need to be restarted every day or two. It would run normally at first but within a day start to timeout on simple operations like looking up a record using the primary key.   I have changed min=4,096 and max=6,144. This results in only 1.4GB memory usage. However now all four cpus are running at 50-60% cpu usage constantly. Tasks are taking roughly 1/3rd longer to execute, although the server is much more stable. Fitting an infinite collection of balls in an infinite dimensional unit ball Given an infinite dimensional normed linear space, how would one show that it is possible to fit an infinite collection of non-overlapping balls of radius $\frac{1}{4}$ in the unit ball?  I guess one can immediately reduce the problem to a normed linear space of countably infinite dimensions. The solution seems clear if the concept of orthogonality exists, but not every normed linear space has an inner product so it's not possible to apply something like Gram-Schmidt to produce an orthogonal basis. Is there any way around this, or there another approach that can be used? I'm using ubuntu 14.04 LTS. When I start my pc and click on any bookmark from the file manager it shows this error:   Unable to find the requested file. Please check the spelling and try again. Unhandled error message: Error when getting information for file '/media/angel/DOC/edu': No such file or directory   But when I open DOC drive and again clink on the same bookmark it works properly. Can anybody tell me why is this happening and how to solve this issue?   Edit: Output of lsblk is following: (ubuntu is installed in sda5)  NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT sda      8:0    0 119.2G  0 disk  ├─sda1   8:1    0  91.3G  0 part  ├─sda2   8:2    0     1K  0 part  ├─sda5   8:5    0  24.1G  0 part / └─sda6   8:6    0   3.8G  0 part [SWAP] sdb      8:16   0 465.8G  0 disk  ├─sdb1   8:17   0  93.2G  0 part  ├─sdb2   8:18   0     1K  0 part  ├─sdb5   8:21   0  93.2G  0 part  ├─sdb6   8:22   0  89.3G  0 part  ├─sdb7   8:23   0  93.2G  0 part  ├─sdb8   8:24   0  93.2G  0 part  └─sdb9   8:25   0   3.8G  0 part  sr0     11:0    1  1024M  0 rom     After opening the disk drive the output of lsblk is bellow:   NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT sda      8:0    0 119.2G  0 disk  ├─sda1   8:1    0  91.3G  0 part  ├─sda2   8:2    0     1K  0 part  ├─sda5   8:5    0  24.1G  0 part / └─sda6   8:6    0   3.8G  0 part [SWAP] sdb      8:16   0 465.8G  0 disk  ├─sdb1   8:17   0  93.2G  0 part  ├─sdb2   8:18   0     1K  0 part  ├─sdb5   8:21   0  93.2G  0 part  ├─sdb6   8:22   0  89.3G  0 part /media/angel/DOC ├─sdb7   8:23   0  93.2G  0 part  ├─sdb8   8:24   0  93.2G  0 part  └─sdb9   8:25   0   3.8G  0 part  sr0     11:0    1  1024M  0 rom   The output of sudo blkid gives the following information about sdb6:  /dev/sdb6: LABEL="DOC" UUID="04280A22280A12FA" TYPE="ntfs" UFW Allows 22 for IPv4 and IPv6 but SSH Disconnects When Enabling sudo ufw disable followed by sudo ufw enable kicks me out of SSH  DMESG reports  [UFW BLOCK] IN=eth0 OUT= MAC=30:........ SRC=192.168.1.me DST=192.168.1.server LEN=52 TOS=0x00 PREC=0x00 TTL=128 ID=15776 DF PROTO=TCP SPT=55640 DPT=22 WINDOW=253 RES=0x00 ACK URGP=0  I can log back in without having to change rules via the console (UFW still enabled).  This started after upgrading Xenial (16.04) from kernel 4.4 to 4.15 (HWE). Upgrading to 18.04.1 did not solve the issue.  Versions:   iptables v1.6.1 ufw 0.35 4.15.0-29-generic #31-Ubuntu Ubuntu 18.04.1 LTS   UFW status verbose is (some rules were omitted, but they are all ALLOW)  Logging: on (low) Default: deny (incoming), allow (outgoing), disabled (routed) New profiles: skip  To                         Action      From --                         ------      ---- 22                         ALLOW IN    Anywhere 22 (v6)                    ALLOW IN    Anywhere (v6)   Why is this happening, or at least, how to revert to the expected behavior?  I looked at , and I am not sure it applies, but here's /etc/ufw/before.rules  # # rules.before # # Rules that should be run before the ufw command line added rules. Custom # rules should be added to one of these chains: #   ufw-before-input #   ufw-before-output #   ufw-before-forward #  # Don't delete these required lines, otherwise there will be errors *filter :ufw-before-input - [0:0] :ufw-before-output - [0:0] :ufw-before-forward - [0:0] :ufw-not-local - [0:0] # End required lines   # allow all on loopback -A ufw-before-input -i lo -j ACCEPT -A ufw-before-output -o lo -j ACCEPT  # quickly process packets for which we already have a connection -A ufw-before-input -m conntrack --ctstate RELATED,ESTABLISHED -j ACCEPT -A ufw-before-output -m conntrack --ctstate RELATED,ESTABLISHED -j ACCEPT -A ufw-before-forward -m conntrack --ctstate RELATED,ESTABLISHED -j ACCEPT  # drop INVALID packets (logs these in loglevel medium and higher) -A ufw-before-input -m conntrack --ctstate INVALID -j ufw-logging-deny -A ufw-before-input -m conntrack --ctstate INVALID -j DROP  # ok icmp codes for INPUT -A ufw-before-input -p icmp --icmp-type destination-unreachable -j ACCEPT -A ufw-before-input -p icmp --icmp-type source-quench -j ACCEPT -A ufw-before-input -p icmp --icmp-type time-exceeded -j ACCEPT -A ufw-before-input -p icmp --icmp-type parameter-problem -j ACCEPT -A ufw-before-input -p icmp --icmp-type echo-request -j ACCEPT  # ok icmp code for FORWARD -A ufw-before-forward -p icmp --icmp-type destination-unreachable -j ACCEPT -A ufw-before-forward -p icmp --icmp-type source-quench -j ACCEPT -A ufw-before-forward -p icmp --icmp-type time-exceeded -j ACCEPT -A ufw-before-forward -p icmp --icmp-type parameter-problem -j ACCEPT -A ufw-before-forward -p icmp --icmp-type echo-request -j ACCEPT  # allow dhcp client to work -A ufw-before-input -p udp --sport 67 --dport 68 -j ACCEPT  # # ufw-not-local # -A ufw-before-input -j ufw-not-local  # if LOCAL, RETURN -A ufw-not-local -m addrtype --dst-type LOCAL -j RETURN  # if MULTICAST, RETURN -A ufw-not-local -m addrtype --dst-type MULTICAST -j RETURN  # if BROADCAST, RETURN -A ufw-not-local -m addrtype --dst-type BROADCAST -j RETURN  # all other non-local packets are dropped -A ufw-not-local -m limit --limit 3/min --limit-burst 10 -j ufw-logging-deny -A ufw-not-local -j DROP  # allow MULTICAST mDNS for service discovery (be sure the MULTICAST line above # is uncommented) -A ufw-before-input -p udp -d 224.0.0.251 --dport 5353 -j ACCEPT  # allow MULTICAST UPnP for service discovery (be sure the MULTICAST line above # is uncommented) -A ufw-before-input -p udp -d 239.255.255.250 --dport 1900 -j ACCEPT  # don't delete the 'COMMIT' line or these rules won't be processed COMMIT   PS: I didn't expect this to "fix" the issue but just for reference I changed the port SSHD listens on (and the corresponding rule) and the problem persists. How do I convert a light circuit with a single pole switch to use two 3-way switches? I have a box in the hall which includes a 3-way switch (with another switch at the other end of the hall), a single pole switch to the garage lights and a single pole switch to the exterior lights on the garage.  The single pole switch to the garage is wired with a 3 wire plus ground. The white and black wires feed the garage lights from the switch and the red wire feeds the garage ceiling outlets and is hot all the time.  In the garage I would like to add a 3-way switch for the garage lights and another 3-way switch for the exterior garage lights.  In other words I would like to be able to turn on both of those light locations from both the inside hall and the garage as well. I know that I will of course have to replace the existing single pole switches in the hall with 3-way switches and add a junction box in the garage with 2 3-way switches in it.  The thing that I don't know is what wiring connections I need to make.  Can you help? Find the radius of the inscribed circle of a right triangle. Find the radius of the inscribed circle of a right triangle. The triangle's height is $\sqrt{6} + \sqrt{2}$ while the bisector of the right angle is 4.  This seemed like a generic similar triangles problem but it isn't that simple when I tried to solve it. Any help?  EDIT: Here is a picture of the problem Is there a max length for an array in C++?  Is it a C++ limit or does it depend on my machine? Is it tweakable? Does it depend on the type the array is made of?  Can I break that limit somehow or do I have to search for a better way of storing information? And what should be the simplest way?  What I have to do is storing long long int on an array, I'm working in a Linux environment. My question is: what do I have to do if I need to store an array of N long long integers with N > 10 digits?  I need this because I'm writing some cryptographic algorithm (as for example the p-Pollard) for school, and hit this wall of integers and length of arrays representation. Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$. I want to prove that:  $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$. Is there a way to force Google Photos to recognize new faces? At 12 o'clock, the hour hand and minute hand of the clock can be swapped, and the clock still gives the same time, but at 6 o'clock, it can not be swapped. So in what cases when we swap the hour and the minute hand position does a clock still give a valid time? What's the difference between a projection and a datum? How to send parameter from jsp page to jasper file Twisted serialport dataReceived() provides fragmented data If I'm playing the Hilltop farm in co-op - what farmer's mining level is used for the hilltop spawns?