$N$ and $K$ are submodules of $M$ with $I=Ann(N)$ and $J=Ann(K)$, then show that annihilator of intersection of $N$ and $K$ contains $I+J$. Give example to show that the inclusion may be strict.
Hint: How do you usually prove inclusion? you should start by picking $i\in I$ and $j\in J$. For any $m\in N\cap K$, $im=0$ and $jm=0$ (why?), so what can you conclude about $i+j$?