advertisement

Math 345/645 - Weekly homework 3 The problems on this assignment should be written up impeccably and turned in on Tuesday, February 9. This assignment needs to be typed in LATEX. A problem with a ∗ is extra-credit for undergraduates and required for graduate students. 1. Let {Fn } be the Fibonacci sequence, defined by F0 = 0, F1 = 1, F2 = 1, and Fn = Fn−1 +Fn−2 for n ≥ 2. Prove that for all n ≥ 1, gcd(Fn , Fn−1 ) = 1. 2. Prove the following extension of Theorem 1.45. Suppose that n is a positive integer and n a, b, c ∈ Z with ca ≡ cb (mod n). Prove that a ≡ b (mod (c,n) ). 3. ∗ Suppose that a and b are positive integers and gcd(a, b) = 1. Prove that there are no non-negative integers x and y so that ax + by = ab − a − b. Prove that if N is an integer with N > ab − a − b, then there are non-negative integers x and y so that ax + by = N . 1