MATH 211 Homework, Spring 2008

Math 211 - Spring 2006-2007 Maple Assignment 2 Due date: May 10, 2007 Important Instructions: Important Instructions: Important Instructions: • Students are allowed to work in groups consisting of at most 3 students. • Students will be interviewed based on which the assignment grades will be finalized • Assignment submissions should be on Moodle • Penalty of 20 points per day after May, 10 2007 • If you have any question pass by bliss 206 Tuesday and Thursday anytime between 11:00 and 2:00 Exercise 1. Exercise 1. Exercise 1. Write a procedure CompositeNumbers(n) which takes as input an integer n and prints all the composite numbers less than n. Test the procedure and save the results. Exercise 2. Exercise 2. Exercise 2. 1. Write a procedure PrimeFactors1(n) which takes as input an integer n and returns a set containing all the prime numbers which divide n. For example: • PrimeFactors1(10) should return 2, 5 • PrimeFactors1(396) should return 2, 3, 11 • PrimeFactors1(5940) should return 2, 3, 5, 11 2. Write a procedure PrimeFactors2(n,A) which takes as input an integer n and a set A containing the result returned by PrimeFactors1(n) and returns a new set containing the multiplicity of each of the elements of A. For example: 1 • PrimeFactors2(10,PrimeFactors1(10)) should return 1, 1 since 10 = 21 ∗ 51 • PrimeFactors1(396,PrimeFactors1(396)) should return 2, 2, 1 since 396 = 22 ∗ 32 ∗ 111 • PrimeFactors1(5940,PrimeFactors1(5940)) should return 2, 3, 1, 1 since 5940 = 22 ∗ 32 ∗ 51 ∗ 111 3. Write a procedure myLCM(a,b) which takes as input two integers a and b and returns their least common multiple by using PrimeFactors2. 4. Write a procedure myGCD(a,b) which takes as input two integers a and b and returns their greatest common divisor by using PrimeFactors2. 5. Test each procedure and compare your results with the built-in gcd and lcm functions of Maple. Exercise 3. Exercise 3. Exercise 3. Let P(n) = P(n − 1) + n2 with P(1) = 1 1. Write an iterative procedure myIterativeEval(n) which takes as input an integer n and returns the value of P(n). 2. Write a recursive procedure myRecursiveEval(n) which takes as input an integer n and returns the value of P(n). 3. Test each procedure twice Note. Note. Note. For the first two problems use the following functions of Maple: 1. isprime(n) - takes as input an integer n and returns true if n is prime and false if not. 2. nextprime(n) - takes as input an integer n and returns the next largest prime. 3. mod(a,b) - takes as input two integers and returns the remainder computed upon dividing a by b. GOOD LUCK ⌣ 2