elementary number theory - 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?