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Question - Math 324 Assignment 2 (due September 24)

1. (1.5, #44)(5) Show that the nth Fibonacci number fn, is even if, and only if, 3 divides n.

2. (3.3, #6)(5) If (a, b) =1 and (a, c) = 1, show that (a, bc) = 1.

3. (2 + 3 + 2) With a = 2639 and b = 2145, find
i) (a, b), ii) x, y with ax + by = (a, b), and iii) [a, b].

4. (3.5, #8)(5) Show that every positive integer is the product of a square integer and a
squarefree integer. (An integer is squarefree if it is not divisible by any square > 1).

5. (6)(Continuing problem 1.5) Is the factorization of every L-number into L-primes
unique? Why?

6. (6) Show that | âˆ‘ 1 â‰¤ n â‰¤ N 1/n âˆ’ log(N) | is a bounded function o ...Read More

Solution Preview - f of the form 3k, 3k + 1, or 3k +2 it suffices to show that f 3k, f 3k +1 âˆ’ 1, and f 3k + 2 âˆ’ 1 are all even for every integer k â‰¥ 0. Proof by induction on k with base case true since f0, f1 âˆ’ 1, f2 âˆ’ 1 are all 0. Let k â‰¥ 1, and assume the induction hypothesis for k âˆ’ 1. Then (i) f3k = f3k âˆ’ 1 + f3k âˆ’ 2 = (f3(k âˆ’ 1) +2 âˆ’

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