By Goodman F.M.

ISBN-10: 0979914205

ISBN-13: 9780979914201

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**Sample text**

ALGEBRAIC THEMES line, with the integer points marked, and wrapping it around the circumference of the clock, with 0 on the number line coinciding with 0 on the clock face. Then the numbers : : : ; 3n; 2n; n; 0; n; 2n; 3n; : : : on the number line all overlay 0 on the clock face. The numbers : : : ; 3n C 1; 2n C 1; n C 1; 1; n C 1; 2n C 1; 3n C 1; : : : on the number line all overlay 1 on the clock face. In general, for 0 Ä k Ä n 1, the numbers : : : ; 3n C k; 2n C k; n C k; k; n C k; 2n C k; 3n C k; : : : on the number line all overlay k on the clock face.

Let’s recall the process of long division of polynomials by working out an example. Let p D 7x 5 C 3x 2 C x C 2 and d D 5x 3 C 2x 2 C 3x C 4. d /. The first contribution to q is 75 x 2 , and p 7 2 x dD 5 The next contribution to q is p 7 . x2 5 7 . x2 5 14 25 x, 14 x/ d D 25 The next contribution to q is p 14 4 x 5 14 x 25 77 125 , 21 3 x 5 13 2 x C x C 2: 5 and 77 3 x 25 23 2 81 x C x C 2: 25 25 and 77 39 2 636 558 /d D x C xC : 125 125 125 125 Thus, qD 7 2 x 5 14 x 25 77 125 50 1. 12. d / 0. p/.

Proof. d; 0; : : : ; 0/Q 1 : It follows that d is the greatest common divisor of a1 ; : : : ; an . a1 ; : : : ; an /. 1. 7 using the well–ordering principle. (a) (b) If a ¤ 0, consider the set S of nonnegative integers that can be written in the form a sd , where s is an integer. Show that S is nonempty. By the well–ordering principle, S has a least element, which we write as r D a qd . Then we have a D qd C r. Show that that r < d. 36 1. 2. 9. m:n/. m; n/, then b divides y. 3. Suppose that a natural number p > 1 has the property that for all nonzero integers a and b, if p divides the product ab, then p divides a or p divides b.

### Algebra. Abstract and concrete by Goodman F.M.

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