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# WA 5 - Nicola Hodges David Walnut MATH 290-002 Writing...

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Nicola Hodges David Walnut MATH 290-002 October 28, 2010 Writing Assignment 5 1. Use a form of induction to prove that for all natural numbers n≥2, eithe prime or n has a prime factor p with p ≤ √2. Proof by Complete Induction. Base Case. If n=2, then clearly n is prime and the result holds. Induction Step. Assume it holds for all numbers in the set {2,3,…, n-1}, we will show that it also holds for n. If n is prime, then clearly we are done. Suppose n is not prime, then it has factors other than 1 and itself. Let n=ab, where a and b are natural numbers and 1 < a,b < n. By assumption, the result holds for a and b. It must be true that a ≤ √n or b ≤ √n because if both a >√n and b >√n, then n= ab > (√n) 2 = n which is illogical. So, if both a and b are prime, then n has a prime factor p with p ≤ √n and the results holds. Suppose n has at least one of a or b that is not prime, for clarity assume that it is a. By the induction hypothesis, a has a prime factor p such that p

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WA 5 - Nicola Hodges David Walnut MATH 290-002 Writing...

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