Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
**Theorem**: Suppose \( a, b \in \mathbb{N} \), and \( d = \text{GCD}(a, b) \). Then \( d^2 = \text{GCD}(a^2, b^2) \). **Proof**: By hypothesis, we have that \( d \mid a \) and \( d \mid b \), so there are integers \( s \) and \( t \) with \( a = ds \) and \( b = dt \). Then \( a^2 = d^2s^2 \) and so \( d^2 \mid a^2 \). Similarly, \( d^2 \mid b^2 \). Thus \( d^2 \) is a common divisor of \( a^2 \) and \( b^2 \), as desired.
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Transcribed Image Text:**Theorem**: Suppose \( a, b \in \mathbb{N} \), and \( d = \text{GCD}(a, b) \). Then \( d^2 = \text{GCD}(a^2, b^2) \). **Proof**: By hypothesis, we have that \( d \mid a \) and \( d \mid b \), so there are integers \( s \) and \( t \) with \( a = ds \) and \( b = dt \). Then \( a^2 = d^2s^2 \) and so \( d^2 \mid a^2 \). Similarly, \( d^2 \mid b^2 \). Thus \( d^2 \) is a common divisor of \( a^2 \) and \( b^2 \), as desired.
**Proof Evaluation** (This type of exercise will appear occasionally): Each of the following is a proposed "proof" of a "theorem". However, the "theorem" may not be a true statement, and even if it is, the "proof" may not really be a proof. You should read each "theorem" and "proof" carefully and decide and state whether or not the "theorem" is true. Then: - If the "theorem" is false, find where the "proof" fails. (There has to be some error.) - If the "theorem" is true, decide and state whether or not the "proof" is correct. If it is not correct, find where the "proof" fails.
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Transcribed Image Text:**Proof Evaluation** (This type of exercise will appear occasionally): Each of the following is a proposed "proof" of a "theorem". However, the "theorem" may not be a true statement, and even if it is, the "proof" may not really be a proof. You should read each "theorem" and "proof" carefully and decide and state whether or not the "theorem" is true. Then: - If the "theorem" is false, find where the "proof" fails. (There has to be some error.) - If the "theorem" is true, decide and state whether or not the "proof" is correct. If it is not correct, find where the "proof" fails.
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We modify the missing arguments in the proof and explain in detail below.

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