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.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter2: Functions And Graphs
Section2.2: The Rectangular Coordinate System And Graphing Lines
Problem 103E
Related questions
Question
![=
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc06c0cac-2e9c-4d45-9a45-e5decd8c209b%2F951a22d7-93fb-4fc8-9668-65547f9ee908%2Fnrc891p_processed.jpeg&w=3840&q=75)
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc06c0cac-2e9c-4d45-9a45-e5decd8c209b%2F951a22d7-93fb-4fc8-9668-65547f9ee908%2F9anrsmr_processed.jpeg&w=3840&q=75)
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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