EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus,...
Recall that an integer >1 is called a prime when its only strictly positive factors are 1 and r. An integer > 1 is called composite when it's not a primec. (a) Show that a composite integer 2 < x < 150 must be a multiple of 2, 3, 5, 7, or 11 (b) Use the Sieve Method and a table with 15 rows and 10 columns to determine all primes between 2 and 150. (c) What's the largest prime...
Show that every positive integer n, there is a string of n consecutive integers where first integer is even, the second is divisible by a perfect square(other than 1), the third by a perfect cube(other than 1), etc..., and the nth is divisible by the nth power of an integer(other than 1). Then find an example for n = 3.
Part 15A and 15B (15) Let n E Z+,and let d be a positive divisor of n. Theorem 23.7 tells us that Zn contains exactly one subgroup of order d, but not how many elements Z has of order d. We will determine that number in this exercise. (a) Determine the number of elements in Z12 of each order d. Fill in the table below to compare your answers to the number of integers between 1 and d that are...
6.32 Theorem. If k and n are natural numbers with (k, d(n)) =I, then there exist positive integers u and v satisfving ku=(n)u The previous theorem not only asserts that an appropriate exponent is always availahle, but it also tells us how to find it. The numbers u and are solutions lo a lincar Diophantine cquation just like those we studied in Chapter 6.33 Exercisc. Use your observations so far to find solutions to the follow ing congruences. Be sure...
I have to use the following theorems to determine whether or not it is possible for the given orders to be simple. Theorem 1: |G|=1 or prime, then it is simple. Theorem 2: If |G| = (2 times an odd integer), the G is not simple. Theorem 3: n is an element of positive integers, n is not prime, p is prime, and p|n. If 1 is the only divisor of n that is congruent to 1 (mod p) then...
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...
Question 1 An array is NOT: A - Made up of different data types. B - Subscripted by integers. C - A consecutive group of memory chunks. D - None of the choices. Question 2 How many times is the body of the loop executed? int i=1; while(true) { cout << i; if(++i==5) break; } A - Forever B - 4 C - 5 D - 6 E - 0 Question 3 What is wrong with the following piece of...
1 L, as a dynamical system (Notes from Assignment #2) We take our definition of dynamical system to be an "object" along with a specific set of modifications that can be performed (dynamically) upon this object. In this case, the object is a bi-infinite straight road with a lamp post at every street corner and a marked lamp (the position of the lamplighter). There are two possible types of modifications: the lamplighter can walk any distance in either direction from...
This is the sequence 1,3,6,10,15 the pattern is addin 1 more than last time but what is the name for this patternThese are called the triangular numbers The sequence is 1 3=1+2 6=1+2+3 10=1+2+3+4 15=1+2+3+4+5 You can also observe this pattern x _________ x xx __________ x xx xxx __________ x xx xxx xxxx to see why they're called triangular numbers. I think the Pythagoreans (around 700 B.C.E.) were the ones who gave them this name. I do know the...
Chapter overview 1. Reasons for international trade Resources reasons Economic reasons Other reasons 2. Difference between international trade and domestic trade More complex context More difficult and risky Higher management skills required 3. Basic concept s relating to international trade Visible trade & invisible trade Favorable trade & unfavorable trade General trade system & special trade system Volume of international trade & quantum of international trade Commodity composition of international trade Geographical composition of international trade Degree / ratio of...