By Hodge, Jonathan K.; Schlicker, Steven; Sundstrom, Ted
""This ebook arose from the authors' method of instructing summary algebra. They position an emphasis on energetic studying and on constructing scholars' instinct via their research of examples. ... The textual content is equipped in the sort of approach that it's attainable firstly both earrings or groups.""
-Florentina Chirteş, Zentralblatt MATH 1295
""This booklet arose from the authors' method of educating summary algebra. They position an emphasis on energetic studying and on constructing scholars' instinct via their research of examples. ... The textual content is prepared in this kind of manner that it's attainable firstly both jewelry or groups.""
-Florentina Chirteş, Zentralblatt MATH 1295
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Extra info for Abstract Algebra : An Inquiry Based Approach
Furthermore, the Division Algorithm implies that there exist integers q and r such that a = dq + r and 0 ≤ r < d. Thus, r = a − dq = a − (am + bn)q = a(1 − mq) + b(nq), which implies that either r ∈ L or r = 0. Since r < d, it cannot be the case that r ∈ L. Thus, it must be that r = 0, which implies that d | a. A similar argument establishes that d | b as well. Now suppose that for some integer k, k | a and k | b. Then there exist integers h and j such that d = am + bn = (kh)m + (kj)n ? = k(hm + jn), and so k | d.
Gcd(a, b) divides every linear combination of a and b. • gcd(a, b) is positive. These three facts together imply that gcd(a, b) is not only a positive linear combination of a and b, but in fact the smallest positive linear combination of a and b. 11 states a useful consequence. 10. Let a and b be integers, not both zero. Then gcd(a, b) is equal to the smallest positive linear combination of a and b. 11. Let a and b be integers, not both zero. Then gcd(a, b) = 1 if and only if there exist integers x and y such that ax + by = 1.
Then d = gcd(a, b) if and only if all of the following conditions hold: (i) d | a and d | b. (ii) If k is an integer such that k | a and k | b, then k | d also. (iii) d is positive. 16. Let a, b, and d be integers, with a and b not both zero. (a) Suppose d = gcd(a, b). 15 are automatically satisfied. Then use Bezout’s Identity to prove condition (ii). 15. Explain why there cannot exist an integer k > d such that k | a and k | b. 17. Throughout this investigation, the definitions and theorems involving gcd(a, b) have all assumed that a and b are not both zero.
Abstract Algebra : An Inquiry Based Approach by Hodge, Jonathan K.; Schlicker, Steven; Sundstrom, Ted