Consider $\mathbb{F}_{p^{2}}$, the finite field with $p^{2}$ elements. What can one say about the number of elements $z$ of $\mathbb{F}_{p^{2}}$ with $\text{ord}(z)$ dividing $p+1$?
One know that for every $0\neq z\in\mathbb{F}_{p^{2}}$ we have $\text{ord}(z)|(p-1)(p+1)$, and that there are only $p-1$ elements with $\text{ord}(z)|p-1$. But the problem is that there could also be elements of which the order has a factor in $p-1$ and a factor in $p+1$.