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m25-hw1-new - S i that contains at least n 1 of the numbers...

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Math 25: Advanced Calculus Fall 2010 Homework # 1 Due Date: Friday, October 1 Reading: Read Appendix A of the textbook. Complete the following problems from the textbook 1 : A.5.2: Show that there are infinitely many prime numbers. A.6.1: Prove the following assertion by contraposition: If x is irrational, then x + r is irrational for all rational numbers r . A.7.2: Every prime number greater than 2 is odd. Is the converse true? In addition, please complete the following problems: 1.1: Fix natural numbers n and k , and let m = n · k + 1. Suppose you have m natural numbers a 1 ,a 2 ,...,a m N that are partitioned into k distinct sets S 1 ,S 2 ,...,S k (so each number a i belongs to exactly one of the sets S j ). Prove that there exists some set
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Unformatted text preview: S i that contains at least n + 1 of the numbers a i . 1.2: An integer p is prime if the following conditions are satisfied: 1. p 6 = ± 1, and 2. whenever a · b is divisible by p , either a is divisible by p or b is divisible by p . If p is a prime number, show that √ p is irrational. 1 For the problems from the textbook, you may use, without proof, the following facts about prime numbers: • a natural number p is prime if and only if its only proper divisors are 1 and p (and p 6 = ± 1), and • an integer n ≥ 2 can be written as a product of prime numbers. 1...
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