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key1fa02 - COT3100C-01 Fall 2002 S Lang Solution keys to...

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COT3100C-01, Fall 2002 S. Lang Solution keys to Assignment #1 (40 pts.) 9/6/2002 1. (16 pts., 4 pts. each part) Recall the following definitions and theorems about integers: Definition . An integer a is even if a = 2 b for some integer b . (That is, there exists an integer b such that a = 2 b .) Definition . An integer a is odd if a = 2 b + 1 for some integer b . (That is, there exists an integer b such that a = 2 b + 1.) Definition . An integer a is a divisor of integer b , denoted a | b , if a 0 and there exists integer c such that b = ac . Theorem. Each integer is either even or odd (but not both). Theorem. The sum of two odd integers is even. Theorem. If the product of two integers is even, then at least one of them is even. (Equivalently, the product of two odd integers is odd.) Theorem. If a | b and b | c , then a | c . Use these definitions and theorems (and other appropriate algebra laws) to prove each of the following questions, where all variables refer to integers: (a) If a is odd, then 3 a + 4 b is odd. Proof: Since a is odd (by assumption), and 3 is odd, so the product 3 a is odd (by a theorem given above). Thus, 3 a = 2 c + 1 --- (1), for some integer c , according to the definition of odd. Therefore, 3 a + 4 b = (2 c + 1) + 4 b , according to (1) = 2( c + 2 b ) + 1 --- (2) which implies 3 a + 4 b is odd by the definition of odd, since ( c + 2 b ) in (2) is an integer. (b) If a b is odd, then exactly one of the two integers a , b is odd; that is, either a is even and b is odd, or b is even and a is odd.
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