HAPPY NEW YEAR $2020$ Remainder Problem I framed a new question just now. What is the Remainder when the number $20^{20}$ is divided by $2020$  My try:  $$\frac{20^{20}}{2020}=\frac{20^{19}}{101}$$  Now Consider: $$20^{18}=(400)^9=(404-4)^9=101k-2^{18}$$  Now i was trying to find Remainder without calculator or by manual division.
Modular exponentiation by hand ($a^b\bmod c$) How do I efficiently compute $a^b\bmod c$:  When $b$ is huge, for instance $5^{844325}\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$? When $(a,c)\ne1$, for instance $6^{103}\bmod 14$?  Are there any other tricks for evaluating exponents in modular arithmetic?  This is being asked in an effort to cut down on duplicates, see  and .
LAN not working on 17.04 Actually yesterday I've upgraded from Ubuntu 16.04 to 17.04 and I got a problem that, if connect LAN cable it does not showing anything and  remains disconnected. But i can connect with WiFi. I don't know what's the problem. Actually its showing "device not managed" under the "Ethernet Network" and both were disabled. I've searched for the solution, but did not found yet. Currently my bibliography says "Literatur" (german). But I want something completely different there. How can I achieve that? How to pass the output of one command as the command-line argument to another? So I have a script that, when I give it two addresses, will search two HTML links:  echo "http://maps.google.be/maps?saddr\=$1\&amp;daddr\=$2" | sed 's/ /%/g'   I want to send this to wget and then save the output in a file called temp.html. I tried this, but it doesn't work. Can someone explain why and/or give me a solution please?  #!/bin/bash url = echo "http://maps.google.be/maps?saddr\=$1\&amp;daddr\=$2" |  sed 's/ /%/g' wget $url Salutation for two doctors (not married) I currently have a setup where I send a char using a Tx of 434MHz and an Uno to a Mega with a Rx. The Mega counts how many times it receives the char and then if it falls below a certain number it triggers an alarm. Is this a viable way to measure the distance between two microcontrollers while indoors or is there a better way.  Transmitter (Mega)  #include &lt;SoftwareSerial.h&gt;  int rxPin=2; //Goes to the Receiver Pin int txPin=5; //Make sure it is set to pin 5 going to input of receiver  SoftwareSerial txSerial = SoftwareSerial(rxPin, txPin); SoftwareSerial rxSerial = SoftwareSerial(txPin, rxPin); char sendChar ='H';  void setup() {   pinMode(rxPin, INPUT);   pinMode(txPin,OUTPUT);   txSerial.begin(2400);   rxSerial.begin(2400); }  void loop() {   txSerial.println(sendChar);   Serial.print(sendChar); }   Receiver  #include &lt;SoftwareSerial.h&gt;  //Make sure it is set to pin 5 going to the data input of the transmitter int rxPin=5;  int txPin=3; //Don't need to make connections int LED=13; int BUZZ=9; int t=0;  char incomingChar = 0; int counter = 0;  SoftwareSerial rxSerial = SoftwareSerial(rxPin, txPin);  void setup() {   pinMode(rxPin, INPUT); //initilize rxpin as input   pinMode(BUZZ, OUTPUT); //initilize buzz for output   pinMode(LED, OUTPUT); //initilize led for output   rxSerial.begin(2400); //set baud rate for transmission   Serial.begin(2400); //see above }  void loop() {   for(int i=0; i&lt;200; i++) {     incomingChar = rxSerial.read(); //read incoming msg from tx      if (incomingChar =='H') {       counter++; //if we get bit "h" count it       }     delay(5); //delay of 10 secs   }   Serial.println(incomingChar);   Serial.println(counter); //prints the the bits we recieved    if(counter&lt;55) {     //if we receive less than 100 bits than print out of range triggers alarm     Serial.println("out of range");      tone(BUZZ,5000,500);digitalWrite(LED,HIGH);   }   else {     noTone(BUZZ);digitalWrite(LED, LOW);     //if we get more than 100 bits than we are within range turn off alarm     Serial.println("in range");    }    counter = 0;   incomingChar=0; } I am interested in finding out how confident my model (say Logistic Regression) is in predicting the label of a new data point. For example, if it is not confident, I better abstain from making a prediction.   Logistic Regression outputs probabilities, which gives you a notion that there is confidence in prediction. But in fact, it is not. $P(y|x)$ being 0.3, doesn't tell me that the model is confident in its prediction or not. All, we can say is that it believes 100% that class A is 0.3 and other class is 0.7.  Can we use confidence intervals of LR as some sort of confidence in prediction? E.g., larger the difference between upper and lower bound for a particular data point, less confident it is, and vice-versa?  If not, how can we build confidence in our prediction? Can anybody guide me to some paper or field of study? Would a wormhole be able to move? Windows 7 Automatic Log-on with a Default User Account? I have two User Accounts for my Windows 7 with no password requirement. I frequently use one of those accounts; so I would like Windows to log-on automatically with that account without the need for user input.  P.S. I don't prefer installing/using a third-party software for that purpose; especially those that place at low levels before Windows log-on. When and where do I have the highest chance of seeing auroras? I'm wondering what time of year is best, and what locations might be best, for catching the aurora.  I'm currently thinking of February up at Sweden's ice hotel, but since the whole point of the trip is so that my wife can see the Aurora, I want to pick a location that affords the best chances to see one.  Edit: Just adding a link so you can see realtime aurora activity in the northern hemisphere    I know this is anecdotal, but I ended up going to Iceland in late february during the new moon and saw the aurora 3 out of 6 nights I was there.  There was activity every single night but the biggest problem was weather.  It was cloudy almost every single night. Why do I get "Permission denied (publickey)" when trying to SSH from local Ubuntu to a Amazon EC2 server? I have an instance of an application running in the cloud on Amazon EC2 instance, and I need to connect it from my local Ubuntu. It works fine on one of local ubuntu and also laptop. I got message "Permission denied (publickey)" when trying to access SSH to EC2 on another local Ubuntu. It's so strange to me.  I'm thinking some sort of problems with security settings on the Amazon EC2 which has limited IPs access to one instance or certificate may need to regenerate.  Does anyone know a solution? I'm looking for an example of $2$ non isomorphic groups $G_1,G_2$  that are finitely generated and presented that have the same finite quotients (up to isomorphism) .  Thanks Functions Which are non differentiable on a Given Set. Does assigning objects to null in Java impact garbage collection? If I get rid of the last Pokemon I have from a family do I still keep the candies from that family when I catch another one of the same family? I have the following content in in /etc/udev/rules.d/81-external-disk.rules:  ENV{ID_FS_UUID}=="6826692e-79f4-4423-8467-cef4d5e840c5", RUN{program}+="/bin/mount -o nofail,x-systemd.device-timeout=1 -t ext4 -U 6826692e-79f4-4423-8467-cef4d5e840c5 /backup/external"   After running:  udevadm control --reload ; udevadm trigger /dev/sdb1   It does nothing at all. However if II change the mount command for something such as /bin/touch /tmp/xyz it works.  Versions:  [root@helsinki rules.d]# rpm -qa | grep udev libgudev1-219-19.el7_2.12.x86_64 python-pyudev-0.15-7.el7_2.1.noarch [root@helsinki rules.d]# rpm -qa | grep systemd systemd-libs-219-19.el7_2.12.x86_64 systemd-219-19.el7_2.12.x86_64 systemd-sysv-219-19.el7_2.12.x86_64 [root@helsinki rules.d]# cat /etc/redhat-release  CentOS Linux release 7.2.1511 (Core) Is a sentence always grammatically incorrect if it has no verb? Is the following grammatically correct? My friend says the second sentence is grammatically incorrect, but couldn't explain why.     I have always been fascinated by statistics. The different ways in   which you could look at data and infer knowledge from it. Dual edge detector I am designing a hobby project that will run on a battery so I am trying to reduce power consumption. The SoC I am using has the ability to go into a deep sleep mode where it consumes very little power, and then can be interrupted by a falling edge on a pin.  I want to generate a high-low-high pulse when the state of a reed switch changes in either direction. I have been reading a lot and it seems that a one-shot monostable circuit can be built from a couple of transistors and an RC circuit to generate an output pulse when an edge is detected. I understand how that works. However, I can't figure out the best way to also pulse on the opposite edge.  One thought, was to just combine two one-shots that are configured to trigger on rising and falling edges respectively, and use the transistors at the output to pull the common pull-up low (kind of like a NOR gate).  Since I am a hobbyist, I am mostly using passive components. What is the "real" way of doing this in a low-power configuration using logic devices?  Edit: I figure I should mention something about the timing requirements. Normally the reed switch will be either open or closed for many seconds at minimum before changing states.  Edit: Here is an example of the circuit I was using for detecting only the falling edge:    And here is the ~20ms pulse it was producing for ~200ms button press simulating the reed switch (Channel 1 is Vbe for T1 and Channel 2 is measured at RST): Cardinality of cartesian square Given an infinite set $A$ -  does the cardinality of $A$ equal to the cardinality of $A^2$? How do I test a private function or a class that has private methods, fields or inner classes? How do I unit test (using xUnit) a class that has internal private methods, fields or nested classes? Or a function that is made private by having  (static in C/C++) or is in a private () namespace?  It seems bad to change the access modifier for a method or function just to be able to run a test. I need to (permanent) redirect all of the following:   www.example.com www.example.com/folder sub.example.com sub.example.com/folder   Basically, all URLs that are reachable without being intended to: www.new-example.com, I want to do this within Apache and using the mod_rewrite module via the .htaccess file.