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IV. Two integers x and y are said to be of the same parity if x and y are both even or are both odd. The integers x and y are of opposite parity if one of x and y is even and the other is odd. For example, 5 and 13 have the same parity while 8 and 11 are of opposite parity. a. Prove that for all integers x and y, if 41(x² - y²) then x and y have the same parity. b. State and prove the converse of "part a" of this problem.

IV. Two integers x and y are said to be of the same parity if x and y are both even or are both odd. The integers x and y are of opposite parity if one of x and y is even and the other is odd. For example, 5 and 13 have the same parity while 8 and 11 are of opposite parity. a. Prove that for all integers x and y, if 41(x² - y²) then x and y have the same parity. b. State and prove the converse of "part a" of this problem.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.1: Postulates For The Integers (optional)
Problem 25E: 25. Prove that if and are integers and, then either or. (Hint: If, then either or, and similarly...
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IV. Two integers x and y are said to be of the same parity if x and y are both even or are both odd. The integers x and y are of opposite parity if one of x and y is even and the other is odd. For example, 5 and 13 have the same parity while 8 and 11 are of opposite parity. a. Prove that for all integers x and y, if 41(x² − y²) then x and y have the same parity. b. State and prove the converse of "part a" of this problem.
Transcribed Image Text:IV. Two integers x and y are said to be of the same parity if x and y are both even or are both odd. The integers x and y are of opposite parity if one of x and y is even and the other is odd. For example, 5 and 13 have the same parity while 8 and 11 are of opposite parity. a. Prove that for all integers x and y, if 41(x² − y²) then x and y have the same parity. b. State and prove the converse of "part a" of this problem.
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