Chapter 5: Q35E (page 330)
Prove that divisible by 8 whenever n is an odd positive integer.
8 divides whenever n is a positive integer
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Prove that 21 divides whenever n is a positive integer.
In the proof of Lemma 1 we mentioned that many incorrect methods for finding a vertex such that the line segment is an interior diagonal of have been published. This exercise presents some of the incorrect ways has been chosen in these proofs. Show, by considering one of the polygons drawn here, that for each of these choices of , the line segment is not necessarily an interior diagonal of .
a) p is the vertex of P such that the angleis smallest.
b) p is the vertex of P with the least -coordinate (other than ).
c) p is the vertex of P that is closest to .
Suppose that is a simple polygon with vertices listed so that consecutive vertices are connected by an edge, and and are connected by an edge. A vertex is called an ear if the line segment connecting the two vertices adjacent tolocalid="1668577988053" is an interior diagonal of the simple polygon. Two ears and are called nonoverlapping if the interiors of the triangles with vertices and its two adjacent vertices and and its two adjacent vertices do not intersect. Prove that every simple polygon with at least four vertices has at least two nonoverlapping ears.
The well-ordering property can be used to show that there is a unique greatest common divisor of two positive integers. Let a and be positive integers, and let S be the set of positive integers of the form , where s and t are integers.
a) Show that s is nonempty.
b) Use the well-ordering property to show that s has a smallest element .
c) Show that if d is a common divisor of a and b, then d is a divisor of c.
d) Show that c I a and c I b. [Hint: First, assume that . Then , where . Show that , contradicting the choice of c.]
e) Conclude from (c) and (d) that the greatest common divisor of a and b exists. Finish the proof by showing that this greatest common divisor is unique.
Prove that if h > - 1 , then for all nonnegative integers n. This is called Bernoulli’s inequality.
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