First I will post the question that I am working on.

I am not good at proofs (even elementry proofs such as these ones). I was wondering if someone could take a look at my work and perhaps confirm whether my proofs are adequate and/or make some suggestions.

First I will start off with a basic definition of a divisor:

An integer, a, not equal to zero, is called a divisor of an integer b if there exists an integer c such the b = a c.

i) If a|b then -a|b.

Assume a|b. Then b = a c for some integer c by def.

Let c = -k where -1, k are integers. Then

b = a (-k) = - a k.

Since k is an integer, then by def., if a|b then b = -a k.

Similarly,

ii) If a|b then a|-b.

Assume a|b. Then b = a c for some integer c by def.

Let c = -k where -1, k are integers. Then

b = a c = a -k = -a k = -(a k)
-b = --(a k)
-b = a k

Once again since k is an integer, then by def., if a|b then -b = a k.

Also

iii) If a|b then -a|-b.

I am sort of stuck on this one. I am not yet sure how to show

If a|b then -a|-b.

I thought

b = a c,
-b = -a c

By definition, -a|-b if -b = -a c for some integer c. Since c is an integer, then by def. if a|b then -a|-b.

Part iii) seems pretty weak to me. In fact all look pretty weak now.