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.
Advanced Engineering Mathematics
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Transcribed Image Text:**IV. Parity of Integers**
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 \( 4 \mid (x^2 - y^2) \), then \( x \) and \( y \) have the same parity.
b. State and prove the converse of "part a" of this problem.
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Can you re-explain how you did the cases like you did and why they prove they'll always be the same parity for part a as I do not understand it...
Original Question:
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.
Solution
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