I need help proving the following:

1. Let $\displaystyle R$ be a Noetherian ring, $\displaystyle I$ an ideal, and $\displaystyle N \subseteq M$ be $\displaystyle R$-modules. Let $\displaystyle \frac{M}{N}$ be a finitely generated $\displaystyle R$-module. Prove that $\displaystyle \frac{IM}{IN}$ is a finitely generated $\displaystyle R$-module.

2. Let $\displaystyle R$ be a Noetherian ring, $\displaystyle I$ an ideal of R. Prove that $\displaystyle M$ is a finitely generated $\displaystyle R$-module iff $\displaystyle \frac{M}{\text{Nil}(R)M}$ is a finitely generated $\displaystyle \frac{R}{\text{Nil}(R)}$-module.

The first problem seems to be straightforward. Can I just take the generators and multiply them by $\displaystyle I$ to show this is finitely generated? Thanks for any suggestions with this one.

As for the second one, I am not seeing how to use the first part in this problem. I am guessing that I just let $\displaystyle I=\text{Nil}(R)$ for one direction; but the other direction is confusing me.