Let $X\subset\mathbb CP^n$ be the complex hypersurface cut out as $F = 0$ for a homogeneous polynomial $F$ of degree $d$. $X$ as a divisor, defines a line bundle $\mathcal O_{\mathbb CP^n}(n)$ for some $n\in\mathbb Z$.
I wonder how to see $n=d$?
Also by the exact sequence $0\to \mathbb Z\to \mathcal O\to \mathcal O^*\to0$, we can get the injection $$\text {Picard}(\mathbb CP^n)\cong H^1(\mathbb CP^n, \mathcal O^*)\to H^2(\mathbb CP^n,\mathbb Z)\cong \mathbb Z$$
I wonder if $\mathcal O_{\mathbb CP^n}(n)\in H^1(\mathbb CP^n, \mathcal O^*)$ is mapped to $n\in H^2(\mathbb CP^n,\mathbb Z)\cong \mathbb Z$ under the injection?
You want compute the line bundle $[X]$ associated to the divisor $X$, where the cocycles $(g_{ij})_{i,j}$ of $[X]$ related to the open cover $U_j=\mathbb{C}P^N\setminus \{z_j=0\}$ are equal to
$g_{ij}=\frac{f_i}{f_j}$
Where $f_i$ is the local defining function of $X$ in the neighborhood $U_i$, so in your case it is
$f_i=\frac{F}{z_i^d}$
This means that the cocycles of the line bundle are
$g_{ij}=\frac{\frac{F}{z_i^d}}{\frac{F}{z_j^d}}=\frac{z_j^d}{z_i^d}$
But the cocycles of $\mathcal{O}(1)$ are exactly
$\frac{z_j}{z_i}$
So we get that
$[X]=\otimes_{j=1}^d\mathcal{O}(1)=\mathcal{O}(d)$