Donald's "Theorem" states that if x and y are even integers, then ry must be a perfect square. (Note that z is a perfect square if z = m² for some integer m. Some perfect squares you know are 0, 1, 4, 9, 16, 25, ..) This "theorem" is in fact false. (a) Find a pair of even integers x and y that shows this "theorem" is false. (b) Donald's proof of his "theorem" follows. What is the flaw in his proof? Please describe the flaw clearly and concisely. "Proof: Suppose x and y are even integers. Then by definition x = 2k and y = some integer k. Now, xy = (2k)(2k) = 4k² = (2k)². Let us defined j = 2k, so that j is an integer. Thus, xy = j² where j is an integer, so xy is a perfect square." 2k for
Donald's "Theorem" states that if x and y are even integers, then ry must be a perfect square. (Note that z is a perfect square if z = m² for some integer m. Some perfect squares you know are 0, 1, 4, 9, 16, 25, ..) This "theorem" is in fact false. (a) Find a pair of even integers x and y that shows this "theorem" is false. (b) Donald's proof of his "theorem" follows. What is the flaw in his proof? Please describe the flaw clearly and concisely. "Proof: Suppose x and y are even integers. Then by definition x = 2k and y = some integer k. Now, xy = (2k)(2k) = 4k² = (2k)². Let us defined j = 2k, so that j is an integer. Thus, xy = j² where j is an integer, so xy is a perfect square." 2k for
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.4: Mathematical Induction
Problem 46E
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Transcribed Image Text:Donald's "Theorem" states that if x and y are even integers, then ry must be a perfect
square. (Note that z is a perfect square if z = m² for some integer m. Some perfect
squares you know are 0, 1, 4, 9, 16, 25, ..) This "theorem" is in fact false.
(a) Find a pair of even integers x and y that shows this "theorem" is false.
(b) Donald's proof of his "theorem" follows. What is the flaw in his proof? Please
describe the flaw clearly and concisely.
"Proof: Suppose x and y are even integers. Then by definition x = 2k and y =
some integer k. Now, xy = (2k)(2k) = 4k² = (2k)². Let us defined j = 2k, so that j
is an integer. Thus, xy = j² where j is an integer, so xy is a perfect square."
2k for
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