Oblemns for Solution: 1. Recall that Euler's phi function (or called Euler's totient function) φ(...
blems for Solution: Recall that Euler's phi function (or called Euler's totient function) o(n) is defined as the number of integers m in the range 1 S m S n such that m and n are relatively prime, ie, gcd(mn) = 1. Find a formula for (n), n 2. (Hint: Factor n as the product of prime powers, ie., n llis] pr., where p's are distinct primes and c, 1, blems for Solution: Recall that Euler's phi function (or called...
For any positive integer n, Euler’s totient or phi function, Φ(n), is the number of positive integers less than n that are relatively prime to n.? What is Φ(55) ?
Exercise 2. Let φ denote the Euler totient function. (i) Prove that for all positive integers m and n, if m,n are relatively prime (coprime), then φ(mn-o(m)o(n) (ii) Is the converse true? Prove or provide a counter-example.
2. (a) In lecture, we saw Euler’s Totient Function Φ(n), which is defined as the number of positive integers less than or equal to n that are relatively prime to n. Suppose I want to compute Φ(84), as per lecture. I know that 84 = 14 × 6, so I compute Φ(14) and multiply it by Φ(6). Do I get the right result? Briefly, why does this work or not work? Your answer should be brief, but not as simple...
(3.5) Summing the Euler S-function (n): The Euler 6-function 6(n) counts the number of positive integers less than or equal to n, which are relatively prime with n. Evaluate 4(d), and prove that your answer is correct. (3.4) Relatively Prime Numbers and the Chinese Re- mainder Theorem: Give an example of three positive integers m, n, and r, and three integers a, b, and c such that the GCD of m, n, and r is 1, but there is no...
Write code for RSA encryption package rsa; import java.util.ArrayList; import java.util.Random; import java.util.Scanner; public class RSA { private BigInteger phi; private BigInteger e; private BigInteger d; private BigInteger num; public static void main(String[] args) { Scanner keyboard = new Scanner(System.in); System.out.println("Enter the message you would like to encode, using any ASCII characters: "); String input = keyboard.nextLine(); int[] ASCIIvalues = new int[input.length()]; for (int i = 0; i < input.length(); i++) { ASCIIvalues[i] = input.charAt(i); } String ASCIInumbers...
8. Define (n) to be the number of positive integers less than n and n. That is, (n) = {x e Z; 1 < x< n and gcd(x, n) = 1}|. Notice that U (n) |= ¢(n). For example U( 10) = {1, 3,7, 9} and therefore (10)= 4. It is well known that (n) is multiplicative. That is, if m, n are (mn) (m)¢(n). In general, (p") p" -p Also it's well known that there are relatively prime, then...
We begin by formally defining the arithmetic function v(n) first introduced irn Example 1(b). Definition 3: Let neZ with n > 0. The number of positive divisors function, denoted v(n), is the function defined by In other words, v(n) is the number of positive divisors of n. [The notation here is chosen by this author for ease of remembrance: v (lowercase Greek letter nu) represents the "number" of positive divisors. However, the number of positive divisors function is denoted variously...
You may import the following library functions in your module: from fractions import gcd from math import floor from random import randint You may also use: • the built-in pow() function to compute modular exponents efficiently (i.e., ak mod n can be written in Python as pow(a,k,n)), • the sum() function returns the sum of a list of integers (sum(1,2,3,4) returns 10). problem 1 a. Implement a function invPrime(a, p) that takes two integers a and p > 1 where...
use MatLab to answer these questions 1. (10 points) Create an m-file called addup.m Use a for loop with k = 1 to 8 to sum the terms in this sequence: x(k) = 1/3 Before the loop set sumx = 0 Then add each term to sumx inside the loop. (You do not need to store the individual values of the sequence; it is sufficient to add each term to the sum.) After the loop, display sumx with either disp()...