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We will simply hint at the richness of this set of numbers in the next exercises. E4. Verify that the Fibonacci sequence gives the size of the rabbit population each month. E5. The Fibonacci numbers and the golden ratio : (i) The golden ratio is the ratio b: a so that b: a = (a + b):b. The ratio of the golden ratio was considered by the ancient Greeks to be the most perfect ratio for the lengths of the sides of rectangles, such as portraits. Show that if b:a is the golden ratio, then b]a = (1 + j"S)/2.
That would prove the theorem by Induction (2). We prove that if Pl' . . Pn = ql . . qm' then Pi means of the following important lemma: = qj for some j by A. The Fundamental Theorem of Arithmetic 49 Lemma 2. If P is a prime and P divides ab, then P divides a or P divides b. Proof of Lemma 2. This follows immediately from Corollary 3 of Section 3C, an application of Bezout's lemma, namely : if albe and (a, b) = 1, then ale. For suppose P divides be. If P is prime then either P divides b, or (p, b) = 1.
Show that (a, b, c) = ax and s are obtained by Euclid's algorithm. Show that -bI2:::;; r :::;; bl2 and -a12 :::;; s :::;; a12. E20. For a, b natural numbers, consider the set of numbers ar + bs for all integers r, s so that ar + bs ~ 1. Since this set is nonempty, by well-ordering it has a least element. Show that the least element of this set is the greatest common divisor of a and b. E21. Chicken nuggets used to be sold at a hamburger chain in packages of 6, 9, and 20 pieces. What is the largest number of pieces you could not order exactly?