I've just started reading Ribenboim's "13 lectures on FLT" and in it he makes sure to first explain the statement of the theorem and point out that, effectively, it has to be proven only for the exponents $4$ and $p$, where $p$ is an odd prime. This was clear to me, but after this he goes on to point out 2 further "cases" of the theorem:
The statement of Fermat's last theorem is often subdivided further into two cases:
The first case holds for the exponent $p$ when there do not exist integers $x,y,z$ such that $p\nmid xyz$ and $x^p+y^p=z^p$. The second case holds for the exponent $p$ when there do not exist integers $x,y,z$, all different from $0$, such that $p\mid xyz$, $gcd(x,y,z) = 1$ and $x^p+y^p=z^p$.
If I understood these statements correctly, the goal is to make the distinction for the case when $p$ divides none of the three terms (the first case) and the case when it divides only one (the second case). Why is it necessary to do this? What effect on the problem does the amount of terms divisible by the exponent have?