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Then d = "THEOREM": Suppose a, b E N, and d GCD(d, b²). “Proof”": By hypothesis, we have that dla and db, so there are integers s and dt. Then d² = d²s and so d²|a². Similarly, d²|b². Thus d² is a a = ds and b = mon divisor of a and b², as desired.

Then d = "THEOREM": Suppose a, b E N, and d GCD(d, b²). “Proof”": By hypothesis, we have that dla and db, so there are integers s and dt. Then d² = d²s and so d²|a². Similarly, d²|b². Thus d² is a a = ds and b = mon divisor of a and b², as desired.

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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= GCD(a, b). Then d² = GCD(d, b²). e) "THEOREM": Suppose a, b E N, and d "Proof": By hypothesis, we have that dla and db, so there are integers s and I with dt. Then d² = a = ds and b = d'3² and so d²|a². Similarly, d²|b². Thus d² is a com mon divisor of d² and b², as desired.
Transcribed Image Text:= GCD(a, b). Then d² = GCD(d, b²). e) "THEOREM": Suppose a, b E N, and d "Proof": By hypothesis, we have that dla and db, so there are integers s and I with dt. Then d² = a = ds and b = d'3² and so d²|a². Similarly, d²|b². Thus d² is a com mon divisor of d² and b², as desired.
PROOF EVALUATION (This type of exercise will appear occasionally): Each of the follow- ing 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: G 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.
Transcribed Image Text:PROOF EVALUATION (This type of exercise will appear occasionally): Each of the follow- ing 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: G 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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