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5.71. Below is given a proof of a result. What result is proved? Proof Let a, b, c E Z such that a² + b² = c². Assume, to the contrary, that a, b and c are all odd. By (the contrapositive of) Theorem 3.12 O, a? and b? are both odd. Therefore, a? = 2r +1 and b = 2s+ 1 for integers rand s. Thus, a? + b² = 2r + 2s + 2 = 2 (r + s +1). Since r +s+1 is an integer, a² + b² is even. However, a? + b² = c² and c2 is odd. This is a contradiction.

5.71. Below is given a proof of a result. What result is proved? Proof Let a, b, c E Z such that a² + b² = c². Assume, to the contrary, that a, b and c are all odd. By (the contrapositive of) Theorem 3.12 O, a? and b? are both odd. Therefore, a? = 2r +1 and b = 2s+ 1 for integers rand s. Thus, a? + b² = 2r + 2s + 2 = 2 (r + s +1). Since r +s+1 is an integer, a² + b² is even. However, a? + b² = c² and c2 is odd. This is a contradiction.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5.71. Below is given a proof of a result. What result is proved? Proof Let a, b, CE Z such that a² + b² = c². Assume, to the contrary, that a, b and c are all odd. By (the contrapositive of) Theorem 3.12 O, a? and b? are both odd. Therefore, a? = 2r +1 and b2 = 2s +1 for integers r and s. Thus, a² + b² = 2r + 2s +2 = 2 (r + s+1). Since r+s+1 is an integer, a? + b? is even. However, a? +b² = c² and c2 is odd. This is a contradiction.
Transcribed Image Text:5.71. Below is given a proof of a result. What result is proved? Proof Let a, b, CE Z such that a² + b² = c². Assume, to the contrary, that a, b and c are all odd. By (the contrapositive of) Theorem 3.12 O, a? and b? are both odd. Therefore, a? = 2r +1 and b2 = 2s +1 for integers r and s. Thus, a² + b² = 2r + 2s +2 = 2 (r + s+1). Since r+s+1 is an integer, a? + b? is even. However, a? +b² = c² and c2 is odd. This is a contradiction.
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