a means that either b
= a or a + (-b) is positive. 2. Recall from the reading in Section 3.0 that by definition, b a means that either b
= a or a + (-b) is positive.
Restating the question in an if--then form we obtain: If a and b are
negative integers and a | b, then b a.
To prove an if--then statement (p 
q) is true, you can assume that p is true and show that q
must also be true.
Proof Suppose that a and b are negative integers and a | b.
By the definition of divisibility, there exists an integer k such that b = a · k. Also, since a and b are both negative, we know that k > 0.
| If k = 1, then | b = a. |
||
| If k > 1, then | a + (-b) | = | a + [-(a · k)] |
| = | a (1 - k) |
We know that a is a negative integer,
and since k > 1, we know that (1 - k) must also be a negative integer. Therefore
the product a(1 - k) must be positive. Hence either b = a or
a + (-b) is positive, that is, b a.