Solved: Fill in the blanks in the following proof that for all integers a and b, if a

Chapter 4, Problem 14

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Fill in the blanks in the following proof that for all integers a and b, if a | b then a |(b). Proof: Suppose a and b are any integers such that (a) . By definition of divisibility, there exists an integer r such that (b) . By substitution. b = ar = a(r). Let t = (c) . Then t is an integer because t = (1)r, and both 1 and r are integers. Thus, by substitution, b = at, where r is an integer, and so by definition of divisibility, (d) , as was to be shown.

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