Problem 14E

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 t is an integer, and so by definition of divisibility, (d), as was to be shown.

SOLUTION

Step 1

Given result is, for all integers