The greatest common divisor (GCD) of two integers a and b is defined as the largest integer that can divide both a and b without a remainder. For example, the GCD of 30 and 54 is 6, whereas the GCD of 7 and 5 is 1. The following procedure was developed by Euclid to compute the greatest common divisor of two positive integers a and b. In this exercise, we will prove the correctness of this algorithm. procedure Euclidean(a, b) 1 x ← a 2 y ← b 3 while x 6= y do 4 if x > y then 5 x ← x − y 6 else 7 y ← y − x 8 return x (a) State the loop invariant for the while loop in this procedure.
(b) Prove the loop invariant.
(c) Prove that procedure Euclidean always terminates provided that a and b are positive integers.
(d) Using the termination property of your loop invariant, prove that procedure Euclidean computes and returns the greatest common divisor of a and b.

Design a class tictactoe that: holds the following information about the game: two-dimensional array (3 by 3), and winner. add additional variables as needed. includes the functions to perform the various operations on objects. for example, function to print the board, getting the move, checking if move is valid, determining if there is a winner after each move. add additional operations as needed. includes constructor(s). write the functions of the class, and write a program that uses the class. the program should declare an object of type tictactoe. the program will create the board and store it in the array. the program will allow two players to play the tic-tac-toe game. after every valid move update the array, check if there is a winner. if there is no winner and no tie, then print the board again to continue.

Create a string list. 2. use console. readline() to collect values of firstname, lastname, street, city, state, zip, save them to list. 3. write a simple linq statement, call method uppercasewords() to change first letter to uppercase. 4. create a foreach statment to display the information. public static string uppercasewords(string value) { char[] array = value. tochararray(); if (array. length > = 1) { if (char. islower(array[0])) { array[0] = char. toupper(array[0]); } } for (int i = 1; i < array. length; i++) { if (array[i - 1] == ' ') { if (char. islower(array[i])) { array[i] = char. toupper(array[i]); } } } return new string(array);

Create a class hand in its own module. one object of class hand represents a hand of cards and so one object stores a number of card objects. for this assignment you will submit three separate modules, one with the definition of class card, one with the definition of class hand and one with the main application that thoroughly tests class hand. class hand must contain the following four methods: 1) , numcardsinhand) takes an integer as parameter and initializes a hand object with numcardsinhand card objects inside it. these card objects are generated randomly. for simplicity, assume an infinite number of decks of cards.2) bjvalue(self) returns the blackjack value for the whole hand of cards3) ) returns a string containing all the cards in the hand4) hitme(self) adds one randomly generated card to the handcreate a main program in its own module that thoroughly tests class hand. you will have three modules/files to upload to your etudes assignment submission: card. py, hand. py and the module that contains your main program. two alternatives for extra credit - you cannot get credit for both! (+1 point): after you have thoroughly tested the class hand and all of its methods, add code to your main program that stores one hand object as a pickle file and reads it back into a new hand object. you are only eligible for this extra credit if class hand has all four of the methods above working. or(+2 points): after you have thoroughly tested the class hand and all of its methods, add code to your main program that stores one hand object as a text file on the disk and reads it back into a new hand object. you are only eligible for this extra credit if class hand has all four of the methods above working. notes: -start by making any and all modifications suggested by my comments to your previous submission of class card from assignment #6 "a robust card object"-once your class card is tested and working well, you will not make any further modifications to it for the purposes of class hand.-you can keep the test code for class card intact. if it is indented inside an if __name__ == "__main__", then it will not be executed when your main program's module imports it.-to save time, write and test one method for class hand at a time.-under no circumstances are you to attempt the extra credit until you are completely finished with writing and testing all the methods in class hand.