Download PDF by Hodge, Jonathan K.; Schlicker, Steven; Sundstrom, Ted: Abstract Algebra : An Inquiry Based Approach

By Hodge, Jonathan K.; Schlicker, Steven; Sundstrom, Ted

ISBN-10: 1466567082

ISBN-13: 9781466567085

""This ebook arose from the authors' method of educating summary algebra. They position an emphasis on lively studying and on constructing scholars' instinct via their research of examples. ... The textual content is geared up in this kind of means that it really is attainable first of all both earrings or groups.""
-Florentina Chirteş, Zentralblatt MATH 1295

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""This e-book arose from the authors' method of instructing summary algebra. They position an emphasis on lively studying and on constructing scholars' instinct via their research of examples. ... The textual content is geared up in this sort of manner that it really is attainable firstly both jewelry or groups.""
-Florentina Chirteş, Zentralblatt MATH 1295

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Is the converse true? Verify your answer. (7) Let a be any integer. What is gcd(a, a + 2)? Prove your answer. (8) Let a and x be integers, with x > 0. Prove that gcd(a, a + x) = gcd(a, x). (9) Let a and b be integers, not both zero. Prove that if gcd(a, b) = 1, then gcd(a + b, ab) = 1. Is the converse true? Verify your answer. 9 unique? Explain. Investigation 4 Prime Factorization Focus Questions By the end of this investigation, you should be able to give precise and thorough answers to the questions listed below.

6. Suppose, to the contrary, that r ≥ a. (a) Beginning with the fact that r = b − aq, show that r − a ∈ S. ) (b) Why would your answer to part (a) be a contradiction? ) (c) Try to explain the reasoning from parts (a) and (b) in the context of an elementary school division problem. What does the set S represent? How is r chosen from S, and what would happen intuitively if r was not less than a? Now we must show that the q and r we have found are unique. In particular, we want to show that if there are integers q ′ and r′ for which b = aq ′ + r′ and 0 ≤ r′ < a, then it must be the case that q ′ = q and r′ = r.

Before we do so, however, it is useful to note two important observations that suggest an even stronger version of the theorem. The first is that gcd(a, b) not only can be written as a linear combination of a and b, but also divides every linear combination of a and b. 14. ) 29 GCDs and Linear Combinations The second observation is that the greatest common divisor of two integers is always positive. This is true because 1 divides every integer, and so the greatest common divisor of any two integers must be greater than or equal to 1.

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Abstract Algebra : An Inquiry Based Approach by Hodge, Jonathan K.; Schlicker, Steven; Sundstrom, Ted


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