91Ó°ÊÓ

(a) Is the set of natural numbers closed under division? (b) Is the set of rational numbers closed under division? (c) Is the set of nonzero rational numbers closed under division? (d) Is the set of positive rational numbers closed under division? (e) Is the set of positive real numbers closed under subtraction? (f) Is the set of negative rational numbers closed under division? (g) Is the set of negative integers closed under addition?

An integer \(a\) is said to be a type 0 integer if there exists an integer \(n\) such that \(a=3 n\). An integer \(a\) is said to be a type 1 integer if there exists an integer \(n\) such that \(a=3 n+1\). An integer \(a\) is said to be a type 2 integer if there exists an integer \(m\) such that \(a=3 m+2\). (a) Give examples of at least four different integers that are type 1 integers. (b) Give examples of at least four different integers that are type 2 integers. (c) By multiplying pairs of integers from the list in Exercise (9a), does it appear that the following statement is true or false? If \(a\) and \(b\) are both type 1 integers, then \(a \cdot b\) is a type 1 integer.

Pythagorean Triples. Three natural numbers \(a, b,\) and \(c\) with \(a

More Work with Pythagorean Triples. In Exercise (13), we verified that each of the following triples of natural numbers are Pythagorean triples: $$ \begin{array}{lll} \cdot 3,4, \text { and } 5 & \bullet 8,15, \text { and } 17 & \bullet 12,35, \text { and } 37 \\ \cdot 6,8, \text { and } 10 & \bullet 10,24, \text { and } 26 & \bullet 14,48, \text { and } 50 \end{array} $$ (a) Focus on the least even natural number in each of these Pythagorean triples. Let \(n\) be this even number and find \(m\) so that \(n=2 m .\) Now try to write formulas for the other two numbers in the Pythagorean triple in terms of \(m\). For example, for \(3,4,\) and \(5, n=4\) and \(m=2,\) and for 8\. \(15,\) and \(17, n=8\) and \(m=4 .\) Once you think you have formulas. test your results with \(m=10\). That is, check to see that you have a Pythagorean triple whose smallest even number is 20 . (b) Write a proposition and then write a proof of the proposition. The proposition should be in the form: If \(m\) is a natural number and \(m \geq 2\), then \(\ldots \ldots\)