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 .
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