Why are most divisibility exercises only for positive integers? I've been doing some exercises and most of them are about positive integers. Here are a few examples:
(1) Show that if $a,b$ are positive integers then $ab = \gcd(a,b) \text{lcm}(a,b)$.
(2) Let $a,b$ positive integers and $d = \gcd(a,b)$ and $m = \text{lcm}(a,b)$. If $t$ divides both $a$ and $b$ then prove that $t$ divides $d$. If $s$ is a multiple of both $a$ and $b$ show that $s$ is a multiple of $m$.
(3) Let $n$ and $a$ be positve integers and $d=\gcd(a,n)$. Show that the equation $ax \pmod n=  1 $ has a solution if and only if $d=1$.
And so on. 
But it seems to me that these statements should also be true for negative integers. For example if $a=-2$ and $b=-3$ then $ab = 6$ and $\gcd(-2, -3) = 1$ and $\text{lcm}(-2,-3) = 6$ so that claim (1) seems to apply also to negative integers.

What will go wrong in general for negative integers? Do these really
  only hold for positive integers?

 A: Generally, "partially ordered sets" make the world a lot simpler.
If you have a relation, like $\mid$, with the following properties, it is a partial order:


*

*$a\mid a$

*$a\mid b$ and $b\mid c$ implies $a\mid c$

*$a\mid b$ and $b\mid a$ implies $a=b$.


If you have (1) and (2) without (3), you have what is called a "pre-order." A pre-order is just a little messier than an partial order.
The integers under divisibility is a pre-order. The positive integers under divisibility is a partial order.
Every pre-order can be made into a partial order by essentially treating $a,b$ with $a\mid b$ and $b\mid a$ as "the same." But this requires the notion of equivalence classes, which is probably too far for this answer. In this case, you can define divisibility questions where $n$ and $-n$ are considered the same for all purposes. But the properties of this set of pairs under divisibility is exactly the same as the properties of the positive integers under divisibility, and thus it is much easier to just restrict to those.
When you get to other types of rings, you'll have more complicated sets of elements in each divisibility class.
A: It's possible to work with negative integers, but more care has to be taken. For example, define $\gcd(8,-12)$ as an integer $d$ which is a common divisor of $8$ and $-12$, but which is a multiple of any common divisor of $8$ and $-12$. Then there are two possibilities for  $\gcd(8,-12)$, namely $4$ and $-4$. As you see, gcd's and lcm's will generally only be defined up to sign. It's sort of as if you had to say $\sqrt{16} = \pm 4$ all the time.
