strayer_p22 - 22 43 45 46 47 48 49 50 51 52 53 Chapter 1...

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Unformatted text preview: 22 43. 45. 46. 47. 48. 49. 50. 51. 52. 53. Chapter 1 Divisibz'lity and Factorization (b) Let a,b1, b2, . . . , bneZ with (a, b) = (a, b2) = - - - = (a, b,,) = 1. Prove that (a,b1b2‘-'bn) : 1 (a) Let a, b, c 52 with (a, b) = 1. Prove that if a lc and b [c, then ab [c. (b) Provide a counterexample to show why the statement of part (a) does not hold if (a, b) 5.5 1. (c) Let a, a2, . . . , a,,, c 52 with al, a2, . . . , an pairwise relatively prime. Prove that if a,- l c for each i, then alaz - - -a,, [C. (a) Let a, b, c eZ with (a, b) : 1 and a I be. Prove that a I c. (b) Provide a counterexample to show why the statement of part (a) does not hold if (a, b) 7a 1. Let a, b, C, and d be positive integers. If b 7e d and (a, b) : (C, d) = 1, prove that z + 57' e: Z. Let a, b, c, and d be integers with b and d positive and (a, b) 2 (c, d) = 1. A mistake often made when first encountering fractions is to assume that I“; + 57' : ,‘j 17‘}. Find all solutions of this equation. (a) Let a, b EZ and let m be a nonnegative integer. Prove that (a, b) = 1 if and only if (a'", b) = 1. (b) Let a, b 52 and let m and n be nonnegative integers. Prove that (a, b) = 1 if and only if (a'", b") : 1. Let a, b eZ. Prove that (a, b) = 1 if and only if (a + b, ab) = 1. Prove that in any eight composite positive integers not exceeding 360, at least two are not relatively prime. Prove that every integer greater than 6 can be expressed as the sum of two relatively prime integers greater than 1. Let a1, a2, . . . , an 62 with a, 7e 0. Prove that (01! a2) a3) - ' ' i an) 2 ((al! a2)! a3) ' ' ' ’ an) (This method can be used generally to compute the greatest common divisor of more than two integers.) Use this method to compute the greatest common divisor of each set of integers in Exercise 34. Let a1, (12, . . . , an 52 with a, 7A 0 and let 6 be a nonzero integer. Prove that (ca1,ca2, c113, . . . ,ca,,) : lcl (a1, a2, a3, . . . , an). Let n 62. Prove that the integers 6n — 1, 6n + 1, 6n + 2, 6n + 3, and 6n + 5 are pairwise relatively prime. ___-___ The Euclidean Algorithm Our current method for finding the greatest common divisor of two integers is to list all divisors of each integer, find those divisors common to the two lists, and then choose the greatest such common divisor. Surely this method becomes unwieldy for large integers! (For example, what is the greatest ...
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