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280hw3 - On ={1,3 2n-1 Note that it follows that O is the...

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Computer Science 280: Homework 3 Homework 3: 2/13/08 (due 2/20/08) The following problems are all taken from the handout on Number Theory from Rosen's book: Section Number Points Comments 2.4 6 4 Use the definition of divisibility! 12(a),(b) 2 14 5 Don't just give the answer. Prove it! Hint: Think in terms of prime factors. (There's a reason that the problem is in this chapter ...) 16 3 20 5 28(a),(b) 2 30(a),(b) 2 2.5 22(c),(d) 6 Extra problems: 1. [5 points] Do problem 0.2, 33 in DAM3 (DAM2: 0.4, 24), then prove (by induction) that your formula for f ( n ) in part (b) is correct. 2. [8 points] Let S be the smallest set such that has the following two properties: S1. 1 is in S , and S2. if x is in S then x +2 is in S . Define On as inductively as follows: { O1 = {1} { O(n+1) = On {2n+1}. Let O = On (note that is a union symbol, in case it doesn't come our right on your browser). (a) Prove that by induction that
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Unformatted text preview: On = {1,3, . .., 2n-1}. Note that it follows that O is the set of odd numbers. (b) Prove that O = S . (Recall that this means you have to prove that S is a subset of; O and O is a subset of S . To show that S is a subset of O , use the fact that S is the smallest set satisfying S1 and S2. To show that O is a subset of S , prove by induction that On is a subset of S .) 3. [5 points] Define a function h inductively as follows. { h(1) = h(2) = 1 { h(n) = h(n-1)^2 + h(n-2) if n > 2. The function h grows quickly after the first few values. (a) Compute h(5). (b) Prove that for all n > 1 that the greatest common divisor of h(n) and h(n-1) is 1 (i.e., h(n) and h(n-1) are relatively prime). (Hint: induction is a good approach here). Page 1 of 1 Computer Science 280: Homework 3 10/18/2008 http://www.cs.cornell.edu/courses/cs280/2008sp/280hw3.html...
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