Below is given a proof of a result. What result is provedProof Let a, b, c Z such that
Chapter 5, Problem 5.63(choose chapter or problem)
Below is given a proof of a result. What result is proved?Proof Let a, b, c Z such that a2 + b2 = c2. Assume, to the contrary, that a, b and c are all odd. Thena = 2r + 1, b = 2s + 1 and c = 2t + 1, where r,s, t Z. Thus,a2 + b2 = (4r 2 + 4r + 1) + (4s2 + 4s + 1)= 2(2r 2 + 2r + 2s2 + 2s + 1).Since 2r 2 + 2r + 2s2 + 2s + 1 is an integer, it follows that a2 + b2 is even. On the other hand,c2 = (2t + 1)2 = 4t2 + 4t + 1 = 2(2t2 + 2t) + 1.Since 2t 2 + 2t is an integer, it follows that c2 is odd. Therefore, a2 + b2 is even and c2 is odd, contradictingthe fact that a2 + b2 = c2.
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