Let $R$ be a Noetherian ring, $I$ an ideal and $M$ a finite $R$-module. Let $0=N_1\cap \dots \cap N_s$ in $M$ be a primary decomposition with $N_i$ $p_i$-primary submodules of $M$. Show that $\Gamma_I(M)=\bigcap\limits_{I\not\subset p_i} N_i$.
Proof: Let $0=\bigcap\limits_{i=1}^s N_i$. By HW 7.1a, $$ \bigcup_{n\geq 1} \left(0\underset{R}{:}I^n\right) = 0^{(I^n)} = \bigcap\limits_{I^n\not\in p_i} N_i $$
And now I am stuck.
I know that I need to use this fact:
Fact 1: Show that $\Gamma_I(M)=0$ if and only if $I$ contains a non zerodivisor on $M$.
but I don't know how. Also for reference purposes: HW 7.1a is
Fact 2: Let $R$ be Noetherian ring and $I, J$ $R$-ideals. Write $I^{\langle J\rangle}=\bigcup\limits_{n\geq 1} (I:J^n)$, which is called the saturation of $I$ with respect to $J$. Show
a. If $I=\bigcap\limits_{i=1}^m q_i$ with $p_i$-primary, then $I^{\langle J \rangle}=\bigcap\limits_{J \not\subset p_i} q_i$.
b. $I^{\langle J \rangle}$ is the unique largest $R$-ideal that coincides with $I$ locally on the open set $\mathrm{Spec}(R)\setminus V(J)$.