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

ISBN-10: 1466567082

ISBN-13: 9781466567085

""This booklet 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 prepared in this type of method that it really is attainable firstly both jewelry or groups.""

-Florentina Chirteş, *Zentralblatt MATH* 1295

summary:

""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 one of these manner that it truly is attainable first of all both jewelry or groups.""

-Florentina Chirteş, *Zentralblatt MATH* 1295

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**Example text**

This implies that k ≤ d. Therefore, d = gcd(a, b). In the proof above, we showed that d was the greatest common divisor of a and b by first showing that d was a common divisor of a and b, and then arguing that any other common divisor k would also have to be a divisor of d. This allowed us to conclude that d was the largest of all of the common divisors of a and b. The next theorem formalizes this reasoning by stating an equivalent, and in some ways preferable, form of the definition of greatest common divisor.

Choosing m = b is particularly convenient, since b − ab = b(1 − a) ≥ 0. Thus, x = b − ab ∈ S. ) In either case, whether b ≥ 0 or b < 0, we have shown that S contains at least one element. 16 Investigation 2. Divisibility of Integers The set S is therefore a nonempty subset of the whole numbers, and so the Well-Ordering Principle allows us to conclude that S has a least element. Knowing that we want this least element to be our remainder, we will call it r. Furthermore, since r ∈ S, we can find an integer, say q, for which r = b − aq.

Let n be a natural number, and let a and b be integers. Then a ≡ b (mod n) if and only if a and b yield the same remainder when divided by n. 10. 11. Let n be a natural number, and let a and b be integers. (a) Use the Division Algorithm to write equations (together with the appropriate inequalities) that represent the result of dividing each of a and b by n. For convenience, use q1 , q2 , r1 , r2 to denote the resulting quotients and remainders. (b) If you haven’t already done so, write your equations from part (a) so that they are in the form a = .

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

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