By F. Goodman
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Additional info for Algebra. Abstract and Concrete
Assume inductively that the existence assertion holds for all nonnegative integers that are strictly smaller than a. a d / D q 0 d C r and 0 Ä r < d . q 0 C 1/d C r, and we are done. We deal with the case a < 0 by induction on jaj. If d < a < 0, take q D 1 and r D aCd . Suppose that a Ä d , and assume inductively that the existence assertion holds for all nonpositive integers whose absolute values are strictly smaller than jaj. a C d / D q 0 d C r and 0 Ä r < d . q 0 1/d C r. So far, we have shown the existence of q and r with the desired properties.
Hint: Use the existence of s; t such that sa C tb D 1. 11. Suppose that a and b are relatively prime integers and that x is an integer. Show that if a divides x and b divides x, then ab divides x. 12. Show that if a prime number p divides a product a1 a2 : : : ar of nonzero integers, then p divides one of the factors. 13. (a) (b) Write a program in your favorite programming language to compute the greatest common divisor of two nonzero integers, using the approach of repeated division with remainders.
Note that a D 0 if, and only if, jaj D 0. The integers, with addition and multiplication, have the following properties, which we take to be known. 1. (a) Addition on Z is commutative and associative. (b) 0 is an identity element for addition; that is, for all a 2 Z, 0 C a D a. (c) Every element a of Z has an additive inverse a, satisfying a C . a/ D 0. We write a b for a C . b/. (d) Multiplication on Z is commutative and associative. (e) 1 is an identity element for multiplication; that is, for all a 2 Z, 1a D a.
Algebra. Abstract and Concrete by F. Goodman