MAT 125a, HW1 Solutions 5 (a) (i) First assume x B cfw_iI Xi . Since x B cfw_iI Xi we have that x B and j I such that x Xj . Since x B and x Xj we have x B Xj iI (B Xi ). So x B cfw_iI Xi implies x iI (B Xi ) hence B cfw_iI Xi iI (B Xi ). (ii) Next ass
MAT 115A
University of California
Fall 2010
Homework 1
due October 6, 2010 1. Rosen 1.1 #5, pg. 13 Use the well-ordering principle to show that 3 is irrational. 2. Rosen 1.3 #8, pg. 27 Use induction to prove that
n k=1 n(n+1) 2 2
k3 =
.
3. Rosen 1.4 #7, p
MAT 115A
University of California
Fall 2010
Homework 2
due October 13, 2010 1. Rosen 3.1 #4, pg. 74 Use the sieve of Eratosthenes or Sage to nd all primes less than 200. 2. Rosen 3.1 #5, pg. 74 Find all primes that are the dierence of the fourth powers of
MAT 115A
University of California
Fall 2010
Homework 3
due October 20, 2010 1. Rosen 3.3 #20, pg. 95 Let a1 , a2 , . . . , an be integers not all equal to zero. Is it true that the greatest common divisor of these integers (a1 , . . . , an ) is the least
MAT 115A
University of California
Fall 2010
Homework 4
due October 27, 2010 1. Rosen 3.6 #4 (a)-(c), pg. 131 Using the Fermat factorization method, factor each of the following positive integers: 8051, 73, 46009. 2. Rosen 3.6 #20, pg. 132 n Find all prime
MAT 115A
University of California
Fall 2010
Solutions Homework 1
1. Rosen 1.1 #5, pg. 13 Use the well-ordering principle to show that 3 is irrational. Solution: Suppose that 3 were rational. Then there would exist positive integers a and b with 3 = a . He
MAT 115A
University of California
Fall 2010
Solutions Homework 2
1. Rosen 3.1 #4, pg. 74 Use the sieve of Eratosthenes or Sage to nd all primes less than 200. Solution: The primes less than 200 are 2, 3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,
MAT 115A
University of California
Fall 2010
Solutions Homework 3
1. Rosen 3.3 #20, pg. 95 Let a1 , a2 , . . . , an be integers not all equal to zero. Is it true that the greatest common divisor of these integers (a1 , . . . , an ) is the least positive in